Đạo hàm phân số là gì? Các nghiên cứu khoa học liên quan

Tóm tắt

Bài viết giới thiệu khái niệm đạo hàm phân số – mở rộng ý niệm đạo hàm bậc nguyên sang bậc thực và phức – làm nền tảng cho mô hình hóa các hệ thống có “nhớ” và cơ chế phân tán dị thường. Nội dung tập trung vào lịch sử phát triển, các định nghĩa Riemann–Liouville và Caputo, tính chất toán học, phương pháp tính toán, ứng dụng và thách thức nghiên cứu.

Nội dung chi tiết gồm bốn phần đầu: (1) lịch sử hình thành và hoàn thiện lý thuyết đạo hàm phân số từ thế kỷ XVII đến nay, (2) khái niệm cơ bản và quan hệ giữa tích phân phân số và đạo hàm phân số, (3) công thức định nghĩa Riemann–Liouville, (4) công thức định nghĩa Caputo. Phần hai (sẽ trình bày sau) làm rõ tính chất, phương pháp số, ứng dụng vào vật lý, sinh học, điều khiển và triển vọng tương lai.

Độc giả sau khi theo dõi phần này sẽ hiểu được nguồn gốc khái niệm, cách xây dựng đạo hàm phân số và sự khác biệt giữa hai định nghĩa kinh điển để áp dụng vào bài toán giá trị đầu và mô phỏng động lực học có độ nhớ dài.

Định nghĩa và lịch sử phát triển

Ý tưởng về đạo hàm bậc phân số bắt đầu khi Gottfried Wilhelm Leibniz đặt câu hỏi năm 1695: “Liệu có thể tính đạo hàm bậc ½ của hàm số hay không?” Dù trả lời chưa rõ ràng, câu hỏi này mở đường cho các nhà toán học thế kỷ XIX như Riemann, Liouville và Grünwald–Letnikov phát triển lý thuyết cơ bản.

Đến thập niên 1870, Bernhard Riemann đưa ra tích phân phân số đầu tiên, trong khi Joseph Liouville định nghĩa thuật toán tính đạo hàm phân số qua tích phân tường minh. Năm 1867, Grünwald và Letnikov độc lập đề xuất công thức giới hạn:

$\lim_{h\to 0} \frac{1}{h^\alpha}\sum_{k=0}^n (-1)^k \binom{\alpha}{k} f(x - kh) = D^\alpha f(x).$

Năm	Nhà toán học	Đóng góp
1695	Leibniz	Đề xuất ý tưởng đạo hàm bậc phân số
1867	Grünwald–Letnikov	Công thức giới hạn tính đạo hàm phân số
1872	Riemann	Định nghĩa tích phân phân số
1884	Liouville	Mở rộng tích phân phân số và công thức đạo hàm
1999	Podlubny	Xây dựng “Fractional Differential Equations” hệ thống hóa

Sự quan tâm gần đây tăng mạnh nhờ khả năng mô tả quá trình nhớ lâu dài (long-range memory) và lan truyền bất thường (anomalous diffusion) trong vật lý, sinh học và kinh tế.

Các khái niệm cơ bản về đạo hàm phân số

Đạo hàm phân số bậc α (α ∈ ℝ hoặc ℂ) tổng quát hóa phép vi phân bậc n (n ∈ ℕ) và phép tích phân bậc m (m ∈ ℕ), kết hợp qua quan hệ nghịch đảo:

$D^\alpha I^\alpha f(x) = f(x), \quad I^\alpha D^\alpha f(x) = f(x) - \sum_{k=0}^{n-1} \frac{f^{(k)}(a)}{k!}(x-a)^k.$

• Tuyến tính: \(D^\alpha[c_1 f + c_2 g] = c_1 D^\alpha f + c_2 D^\alpha g\).
• Không giao hoán: \(D^\alpha D^\beta \neq D^\beta D^\alpha\) nói chung.
• Quy tắc Leibniz tổng quát: $D^\alpha(uv)=\sum_{k=0}^\infty \binom{\alpha}{k}D^{\alpha-k}u\;D^k v.$

Việc chọn không gian hàm (Hölder, Sobolev) và điều kiện ban đầu/cận biên phù hợp quyết định tính hợp lệ của định nghĩa và tính tồn tại–duy nhất nghiệm bài toán vi phân phân số.

Định nghĩa Riemann–Liouville và Caputo

Trong hai định nghĩa kinh điển, đạo hàm phân số Riemann–Liouville và Caputo khác nhau ở việc đặt phép vi phân bậc nguyên trước hay sau tích phân:

Riemann–Liouville:

$D_{a}^{\alpha}f(x)=\frac{1}{\Gamma(n-\alpha)}\frac{d^n}{dx^n}\int_{a}^{x}(x-t)^{n-\alpha-1}f(t)\,dt,\quad n-1<\alpha<n.$

Caputo:

{}^{C}D_{a}^{\alpha}f(x)=\frac{1}{\Gamma(n-\alpha)}\int_{a}^{x}(x-t)^{n-\alpha-1}f^{(n)}(t)\,dt.\end{script>

Tiêu chí	Riemann–Liouville	Caputo
Vị trí vi phân	Sau tích phân	Trước tích phân
Điều kiện đầu	Cần điều kiện dạng phân số	Dễ áp dụng điều kiện cổ điển \(f^{(k)}(a)\)
Ứng dụng	Phương trình biến tích phân phức tạp	Bài toán giá trị đầu thực tế

Caputo được ưa chuộng hơn khi cần áp dụng điều kiện ban đầu truyền thống vì không yêu cầu tính toán đạo hàm phân số của hàm tại điểm a.

Tính chất toán học

Đạo hàm phân số kế thừa tính tuyến tính từ đạo hàm thông thường: với \(c_1,c_2\in\mathbb{R}\) và hàm đủ điều kiện, có $D^\alpha[c_1 f + c_2 g] = c_1 D^\alpha f + c_2 D^\alpha g.</span> Tuy nhiên, phép hợp (semigroup) chỉ giữ khi xét trong không gian hàm thích hợp: <span class="inline-equation"><span class="katex"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mi>D</mi><mi>α</mi></msup><msup><mi>D</mi><mi>β</mi></msup><mi>f</mi><mo>=</mo><msup><mi>D</mi><mrow><mi>α</mi><mo>+</mo><mi>β</mi></mrow></msup><mi>f</mi></mrow><annotation encoding="application/x-tex">D^\alpha D^\beta f = D^{\alpha+\beta} f</annotation></semantics></math></span></span> nếu và chỉ nếu các điều kiện về tính khả tích và khả vi đủ chặt chẽ.</p><p>Quy tắc nhân Leibniz mở rộng cho đạo hàm phân số được biểu diễn qua chuỗi vô hạn: <span class="inline-equation"><span class="katex"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mi>D</mi><mi>α</mi></msup><mo stretchy="false">(</mo><mi>u</mi><mtext> </mtext><mi>v</mi><mo stretchy="false">)</mo><mo>=</mo><msubsup><mo>∑</mo><mrow><mi>k</mi><mo>=</mo><mn>0</mn></mrow><mi mathvariant="normal">∞</mi></msubsup><mrow><mo fence="true">(</mo><mfrac linethickness="0px"><mi>α</mi><mi>k</mi></mfrac><mo fence="true">)</mo></mrow><mtext> </mtext><msup><mi>D</mi><mrow><mi>α</mi><mo>−</mo><mi>k</mi></mrow></msup><mi>u</mi><mtext>  </mtext><mtext> </mtext><msup><mi>D</mi><mi>k</mi></msup><mi>v</mi><mo separator="true">,</mo></mrow><annotation encoding="application/x-tex">D^\alpha(u\,v)=\sum_{k=0}^\infty \binom{\alpha}{k}\,D^{\alpha-k}u\;\,D^k v,</annotation></semantics></math></span></span> đem lại khả năng phân tích tương tác giữa hai hàm nhưng đòi hỏi kiểm soát sai số khi áp dụng thực tế.</p><ul><li><strong>Không giao hoán:</strong> nói chung \(D^\alpha D^\beta \neq D^\beta D^\alpha\), phụ thuộc vào điều kiện biên.</li><li><strong>Định lý truyền thống mở rộng:</strong> nếu \(f^{(n)}\) liên tục, có thể kết hợp tích phân và đạo hàm theo công thức nghịch đảo.</li></ul><h2>Các phương pháp tính toán</h2><p>Ba cách định nghĩa chính được sử dụng trong tính toán thực nghiệm và lý thuyết:</p><ol><li><strong>Định nghĩa Grünwald–Letnikov:</strong> <span class="inline-equation"><span class="katex"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mi>D</mi><mi>α</mi></msup><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><msub><mrow><mi>lim</mi><mo>⁡</mo></mrow><mrow><mi>h</mi><mo>→</mo><mn>0</mn></mrow></msub><mfrac><mn>1</mn><msup><mi>h</mi><mi>α</mi></msup></mfrac><msubsup><mo>∑</mo><mrow><mi>k</mi><mo>=</mo><mn>0</mn></mrow><mrow><mo stretchy="false">⌊</mo><mi>x</mi><mi mathvariant="normal">/</mi><mi>h</mi><mo stretchy="false">⌋</mo></mrow></msubsup><mo stretchy="false">(</mo><mo>−</mo><mn>1</mn><msup><mo stretchy="false">)</mo><mi>k</mi></msup><mrow><mo fence="true">(</mo><mfrac linethickness="0px"><mi>α</mi><mi>k</mi></mfrac><mo fence="true">)</mo></mrow><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo>−</mo><mi>k</mi><mi>h</mi><mo stretchy="false">)</mo><mi mathvariant="normal">.</mi></mrow><annotation encoding="application/x-tex">D^\alpha f(x)=\lim_{h\to 0}\frac{1}{h^\alpha}\sum_{k=0}^{\lfloor x/h\rfloor}(-1)^k\binom{\alpha}{k}f(x-kh).</annotation></semantics></math></span></span>Phương pháp này dễ cài đặt nhưng yêu cầu lưu giữ toàn bộ lịch sử giá trị hàm, gây tốn bộ nhớ.</li><li><strong>Định nghĩa Riemann–Liouville:</strong> sử dụng tích phân convolution với hạt nhân \((x-t)^{n-\alpha-1}\), thuận tiện khi áp dụng Laplace transform cho phương trình tuyến tính.</li><li><strong>Định nghĩa Caputo:</strong> phù hợp cho các bài toán giá trị đầu vì có thể dùng điều kiện ban đầu cổ điển \(f^{(k)}(a)\), thường được sử dụng trong mô hình sinh học và kỹ thuật.</li></ol><p>Khi triển khai số, thường kết hợp ba kỹ thuật chính để tính xấp xỉ đạo hàm phân số:</p><figure class="table"><table><thead><tr><th>Phương pháp</th><th>Ưu điểm</th><th>Nhược điểm</th></tr></thead><tbody><tr><td>Chuỗi Grünwald–Letnikov</td><td>Tri thức trực tiếp từ định nghĩa</td><td>Bộ nhớ và tính toán cao</td></tr><tr><td>Laplace/Fourier transform</td><td>Giảm bài toán vi phân phân số thành đa thức trong biến phụ</td><td>Chỉ áp dụng cho hệ tuyến tính và biên xác định</td></tr><tr><td>Predictor–Corrector (Adams)</td><td>Độ chính xác cao, ổn định với bước nhỏ</td><td>Phức tạp, tốn bộ nhớ lưu lịch sử</td></tr></tbody></table></figure><h2>Ứng dụng trong mô hình vật lý và sinh học</h2><p>Trong cơ học và truyền nhiệt, phương trình phân số mô tả quá trình khuếch tán dị thường (anomalous diffusion) với động lực bước Lévy thay vì Brownian motion. Ví dụ, phương trình phân số thời gian: <span class="inline-equation"><span class="katex"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mfrac><mrow><msup><mi mathvariant="normal">∂</mi><mi>α</mi></msup><mi>u</mi></mrow><mrow><mi mathvariant="normal">∂</mi><msup><mi>t</mi><mi>α</mi></msup></mrow></mfrac><mo>=</mo><mi>D</mi><mtext> </mtext><mfrac><mrow><msup><mi mathvariant="normal">∂</mi><mn>2</mn></msup><mi>u</mi></mrow><mrow><mi mathvariant="normal">∂</mi><msup><mi>x</mi><mn>2</mn></msup></mrow></mfrac></mrow><annotation encoding="application/x-tex">\frac{\partial^\alpha u}{\partial t^\alpha} = D\,\frac{\partial^2 u}{\partial x^2}</annotation></semantics></math></span></span> mô phỏng lan truyền nhiệt trong môi trường xốp hoặc sinh chất nhớ lâu (<a href="https://link.springer.com/article/10.1007/s00498-019-00212-9">Springer</a>).</p><p>Trong sinh học, mô hình phân số được áp dụng để miêu tả:</p><ul><li>Chuyển động vi khuẩn nhớ hướng (chemotaxis) có độ nhớ dài.</li><li>Lan truyền tín hiệu điện trong hệ thần kinh với hiệu ứng tích phân phân số thể hiện sự tích lũy và phân tán tín hiệu.</li></ul><h2>Ứng dụng trong xử lý tín hiệu và điều khiển</h2><p>Bộ lọc fractional-order filters ứng dụng đạo hàm phân số giúp giảm méo pha và điều chỉnh đáp ứng tần số linh hoạt hơn so với bộ lọc integer-order truyền thống. Ví dụ, bộ lọc PIαDβ mở rộng phương pháp PID để điều khiển hệ thống có động học phức tạp và nhớ lâu (<a href="https://ieeexplore.ieee.org/document/6413821">IEEE</a>).</p><p>Trong robot và tự động hóa, bộ điều khiển fractional-order cho phép tối ưu hóa thời gian đáp ứng và giới hạn quá độ, cải thiện ổn định theo phản hồi trong hệ thống cơ–điện tử.</p><h2>Phương pháp số và giả thiết tính toán</h2><p>Các phương pháp số thường yêu cầu giả thiết hàm đủ mịn và điều kiện biên thích hợp. Ví dụ:</p><ul><li><strong>Backward Euler phân số:</strong> Ổn định nhưng chỉ đạt thứ tự chính xác thấp, phù hợp bài toán thời gian dài.</li><li><strong>Predictor–Corrector Adams:</strong> Độ chính xác cao, cần lưu lịch sử và tính toán nhiều lần để sửa lỗi.</li><li><strong>Spectral methods:</strong> Hiệu quả với hàm mịn, đạt hội tụ nhanh nhưng kém với hàm không liên tục.</li></ul><figure class="table"><table><thead><tr><th>Phương pháp</th><th>Độ chính xác</th><th>Yêu cầu bộ nhớ</th></tr></thead><tbody><tr><td>Backward Euler</td><td>O(h^1)</td><td>Thấp</td></tr><tr><td>Predictor–Corrector Adams</td><td>O(h^{2-\alpha})</td><td>Cao</td></tr><tr><td>Spectral</td><td>Exponential</td><td>Trung bình</td></tr></tbody></table></figure><h2>Thách thức và vấn đề mở</h2><p>Vẫn còn thiếu những định lý toàn diện về tồn tại và duy nhất nghiệm cho phương trình vi phân phân số phi tuyến. Việc xây dựng không gian hàm phù hợp và điều kiện biên phức tạp (nonlocal boundary conditions) là hướng nghiên cứu cấp thiết.</p><p>Việc xác định chính xác tham số phân số từ dữ liệu thực nghiệm là bài toán nghịch đảo có tính không ổn định cao, đòi hỏi kết hợp regularization và phương pháp thống kê để ước lượng tham số tin cậy.</p><h2>Xu hướng nghiên cứu và triển vọng</h2><ul><li>Phát triển thuật toán AI và machine learning để ước lượng tham số α từ dữ liệu thời gian thực.</li><li>Tích hợp mô hình phân số với mạng lưới phức hợp và mô hình lan truyền dịch bệnh nhớ lâu (memory epidemic models).</li><li>Nghiên cứu tính ổn định và hội tụ của phương pháp số mới qua lý thuyết error analysis.</li><li>Mở rộng ứng dụng vào tài chính định lượng để mô tả chuyển động giá tài sản có tính phụ thuộc lịch sử dài hạn.</li></ul><h2>Tài liệu tham khảo</h2><ul><li>Podlubny I. <i>Fractional Differential Equations</i>. Academic Press; 1999.</li><li>Kilbas AA, Srivastava HM, Trujillo JJ. <i>Theory and Applications of Fractional Differential Equations</i>. Elsevier; 2006.</li><li>Machado JAT, Kiryakova V, Mainardi F. “Recent history of fractional calculus.” <i>Communications in Nonlinear Science and Numerical Simulation</i>. 2011;16(3):1140–1153. DOI:10.1016/j.cnsns.2010.06.019.</li><li>Diethelm K. <i>The Analysis of Fractional Differential Equations</i>. Springer; 2010.</li><li>Metzler R, Klafter J. “The Random Walk’s Guide to Anomalous Diffusion.” <i>Phys. Rep.</i> 2000;339(1):1–77. DOI:10.1016/S0370-1573(00)00070-3.</li></ul></div><div class="el-divider el-divider--horizontal" style="--el-border-style:solid;" role="separator"><!--v-if--></div><h2>Các bài báo, nghiên cứu, công bố khoa học về chủ đề đạo hàm phân số:</h2><div><div><!--[--><div class="entity-item-container" data-v-d071913d><a href="/publication/Self-Consistent-Molecular-Orbital-Methods-XII-Further-Extensions-of-Gaussian-Type-Basis-Sets-for-Use-in-Molecular-Orbital-Studies-of-Organic-Molecules/a5686c7f-c9e9-439f-b792-c51743636317" class="entity-title" data-v-d071913d><span data-v-d071913d> Phương pháp quỹ đạo phân tử tự nhất quán. XII. Phát triển bổ sung bộ cơ sở dạng Gaussian cho nghiên cứu quỹ đạo phân tử của các hợp chất hữu cơ</span><!----><span class="el-tag el-tag--warning el-tag--small el-tag--light" style="" data-v-d071913d><span class="el-tag__content"><!--[--><span class="icon enabled" style="color:inherit !important;cursor:unset;" data-v-d071913d data-v-f4e97fe2><i class="fas fa-robot" style="font-size:inherit;color:inherit !important;" data-v-f4e97fe2></i></span> Dịch bởi AI <!--]--></span><!--v-if--></span></a><div class="publish-info" data-v-d071913d><span data-v-d071913d><span data-v-d071913d>Journal of Chemical Physics</span> - </span> Tập 56 Số 5 <span data-v-d071913d> - Trang 2257-2261</span><span data-v-d071913d> - 1972</span></div><!----><div class="abstract-container" data-v-d071913d><div data-v-d071913d><div><span class=""><jats:p>Hai bộ cơ sở mở rộng (được gọi là 5–31G và 6–31G) bao gồm các hàm sóng nguyên tử được biểu diễn dưới dạng kết hợp tuyến tính cố định của các hàm Gaussian được trình bày cho các nguyên tố hàng đầu từ cacbon đến flo. Những hàm cơ sở này tương tự như bộ 4–31G [J. Chem. Phys. 54, 724 (1971)] ở chỗ mỗi lớp vỏ hóa trị được chia thành các phần bên trong và ngoài được mô tả tương ứng bằng ba và mộ...</span><span>... <span class="truncate-button"> hiện toàn bộ</span></span></div><!----></div></div><div class="keyword-container" data-v-d071913d><!--[--><span class="el-tag el-tag--primary el-tag--light" style="" data-v-d071913d><span class="el-tag__content"><!--[--> #quỹ đạo phân tử<!--]--></span><!--v-if--></span><span class="el-tag el-tag--primary el-tag--light" style="" data-v-d071913d><span class="el-tag__content"><!--[--> #hàm cơ sở Gaussian<!--]--></span><!--v-if--></span><span class="el-tag el-tag--primary el-tag--light" style="" data-v-d071913d><span class="el-tag__content"><!--[--> #cacbon<!--]--></span><!--v-if--></span><span class="el-tag el-tag--primary el-tag--light" style="" data-v-d071913d><span class="el-tag__content"><!--[--> #flo<!--]--></span><!--v-if--></span><span class="el-tag el-tag--primary el-tag--light" style="" data-v-d071913d><span class="el-tag__content"><!--[--> #năng lượng tổng<!--]--></span><!--v-if--></span><span class="el-tag el-tag--primary el-tag--light" style="" data-v-d071913d><span class="el-tag__content"><!--[--> #cân bằng hình học<!--]--></span><!--v-if--></span><span class="el-tag el-tag--primary el-tag--light" style="" data-v-d071913d><span class="el-tag__content"><!--[--> #phân tử đa nguyên tử<!--]--></span><!--v-if--></span><!--]--></div><div class="el-row" style="" data-v-d071913d data-v-5c117f8d><!--[--><div class="el-col el-col-24" style="" data-v-5c117f8d><!--[--><!--[--><span data-v-5c117f8d><!----></span><span data-v-5c117f8d><!----></span><span data-v-5c117f8d><!----></span><!--]--><!--]--></div><!--]--></div><div class="action-container" data-v-d071913d><button ariadisabled="false" type="button" class="el-button el-button--primary el-button--small is-round" style="" data-v-d071913d><!--v-if--><span class=""><!--[--><span class="icon enabled" style="color:inherit !important;cursor:unset;" data-v-d071913d data-v-f4e97fe2><i class="fas fa-quote-left" style="font-size:inherit;color:inherit !important;" data-v-f4e97fe2></i></span> 13956<!--]--></span></button><button ariadisabled="false" type="button" class="el-button el-button--primary el-button--small is-plain is-round" style="" data-v-d071913d><!--v-if--><span class=""><!--[--><a href="https://pubs.aip.org/jcp/article/56/5/2257/908679/Self-Consistent-Molecular-Orbital-Methods-XII" target="_blank" rel="nofollow noopener" data-v-d071913d><!--[--><span class="icon enabled" style="color:inherit !important;cursor:unset;" data-v-d071913d data-v-f4e97fe2><i class="fas fa-link" style="font-size:inherit;color:inherit !important;" data-v-f4e97fe2></i></span> Đi đến bài báo <!--]--></a><!--]--></span></button><button ariadisabled="false" type="button" class="el-button el-button--primary el-button--small is-text" style="" data-v-d071913d><!--v-if--><span class=""><!--[--><span class="icon enabled" style="color:inherit !important;cursor:unset;" data-v-d071913d data-v-f4e97fe2><i class="fas fa-quote-right" style="font-size:inherit;color:inherit !important;" data-v-f4e97fe2></i></span> Trích dẫn <!--]--></span></button><span class="save-button-container" data-v-d071913d data-v-c008b161><button ariadisabled="false" type="button" class="el-button el-button--primary el-button--small is-text" style="" data-v-c008b161><i class="el-icon" style=""><!--[--><!--[--><span class="icon enabled" style="color:inherit !important;cursor:unset;" data-v-c008b161 data-v-f4e97fe2><i class="fas fa-bookmark" style="font-size:inherit;color:inherit !important;" data-v-f4e97fe2></i></span><!--]--><!--]--></i><span class=""><!--[--> Lưu lại<!--]--></span></button></span><!--[--><!--]--></div></div><div class="entity-item-container" data-v-d071913d><a href="/publication/On-the-existence-of-solutions-for-a-multi-singular-pointwise-defined-fractional-q-integro-differential-equation/e54dd51e-e37d-4c97-9bb4-cc615d72e111" class="entity-title" data-v-d071913d><span data-v-d071913d>Về sự tồn tại của các nghiệm cho phương trình phân dị tổ hợp q-mẫu đa điểm được xác định điểm</span><!----><span class="el-tag el-tag--warning el-tag--small el-tag--light" style="" data-v-d071913d><span class="el-tag__content"><!--[--><span class="icon enabled" style="color:inherit !important;cursor:unset;" data-v-d071913d data-v-f4e97fe2><i class="fas fa-robot" style="font-size:inherit;color:inherit !important;" data-v-f4e97fe2></i></span> Dịch bởi AI <!--]--></span><!--v-if--></span></a><div class="publish-info" data-v-d071913d><span data-v-d071913d><span data-v-d071913d>Springer Science and Business Media LLC</span> - </span> <!----><span data-v-d071913d> - 2020</span></div><!----><div class="abstract-container" data-v-d071913d><div data-v-d071913d><div><span class=""><jats:title>Tóm tắt</jats:title><jats:p>Bằng cách sử dụng đạo hàm phân số loại Caputo và đạo hàm phân số loại Riemann–Liouville, chúng tôi nghiên cứu sự tồn tại của nghiệm cho một phương trình phân dị tổ hợp q-mẫu đa điểm được xác định tại các điểm với một số điều kiện biên. Trên thực tế, chúng tôi đưa ra một số kết quả bằng cách xem xét các điều kiện khác nhau và sử dụng một số kỹ thuật điểm cố đ...</span><span>... <span class="truncate-button"> hiện toàn bộ</span></span></div><!----></div></div><div class="keyword-container" data-v-d071913d><!--[--><span class="el-tag el-tag--primary el-tag--light" style="" data-v-d071913d><span class="el-tag__content"><!--[--> #phương trình phân số<!--]--></span><!--v-if--></span><span class="el-tag el-tag--primary el-tag--light" style="" data-v-d071913d><span class="el-tag__content"><!--[--> #đạo hàm q-phân số<!--]--></span><!--v-if--></span><span class="el-tag el-tag--primary el-tag--light" style="" data-v-d071913d><span class="el-tag__content"><!--[--> #điều kiện biên<!--]--></span><!--v-if--></span><span class="el-tag el-tag--primary el-tag--light" style="" data-v-d071913d><span class="el-tag__content"><!--[--> #tồn tại nghiệm<!--]--></span><!--v-if--></span><span class="el-tag el-tag--primary el-tag--light" style="" data-v-d071913d><span class="el-tag__content"><!--[--> #phương trình integro-diện tích<!--]--></span><!--v-if--></span><!--]--></div><div class="el-row" style="" data-v-d071913d data-v-5c117f8d><!--[--><div class="el-col el-col-24" style="" data-v-5c117f8d><!--[--><!--[--><span data-v-5c117f8d><!----></span><span data-v-5c117f8d><!----></span><span data-v-5c117f8d><!----></span><!--]--><!--]--></div><!--]--></div><div class="action-container" data-v-d071913d><button ariadisabled="false" type="button" class="el-button el-button--primary el-button--small is-round" style="" data-v-d071913d><!--v-if--><span class=""><!--[--><span class="icon enabled" style="color:inherit !important;cursor:unset;" data-v-d071913d data-v-f4e97fe2><i class="fas fa-quote-left" style="font-size:inherit;color:inherit !important;" data-v-f4e97fe2></i></span> 58<!--]--></span></button><button ariadisabled="false" type="button" class="el-button el-button--primary el-button--small is-plain is-round" style="" data-v-d071913d><!--v-if--><span class=""><!--[--><a href="https://boundaryvalueproblems.springeropen.com/articles/10.1186/s13661-020-01342-3" target="_blank" rel="nofollow noopener" data-v-d071913d><!--[--><span class="icon enabled" style="color:inherit !important;cursor:unset;" data-v-d071913d data-v-f4e97fe2><i class="fas fa-link" style="font-size:inherit;color:inherit !important;" data-v-f4e97fe2></i></span> Đi đến bài báo <!--]--></a><!--]--></span></button><button ariadisabled="false" type="button" class="el-button el-button--primary el-button--small is-text" style="" data-v-d071913d><!--v-if--><span class=""><!--[--><span class="icon enabled" style="color:inherit !important;cursor:unset;" data-v-d071913d data-v-f4e97fe2><i class="fas fa-quote-right" style="font-size:inherit;color:inherit !important;" data-v-f4e97fe2></i></span> Trích dẫn <!--]--></span></button><span class="save-button-container" data-v-d071913d data-v-c008b161><button ariadisabled="false" type="button" class="el-button el-button--primary el-button--small is-text" style="" data-v-c008b161><i class="el-icon" style=""><!--[--><!--[--><span class="icon enabled" style="color:inherit !important;cursor:unset;" data-v-c008b161 data-v-f4e97fe2><i class="fas fa-bookmark" style="font-size:inherit;color:inherit !important;" data-v-f4e97fe2></i></span><!--]--><!--]--></i><span class=""><!--[--> Lưu lại<!--]--></span></button></span><!--[--><!--]--></div></div><div class="entity-item-container" data-v-d071913d><a href="/publication/Theoretical-and-numerical-analysis-for-transmission-dynamics-of-COVID-19-mathematical-model-involving-Caputo-Fabrizio-derivative/2ba9e150-dd64-4a58-a871-e5fa632a24a4" class="entity-title" data-v-d071913d><span data-v-d071913d>Phân tích lý thuyết và số học cho động lực truyền bệnh COVID-19 dựa trên mô hình toán học liên quan đến đạo hàm Caputo–Fabrizio</span><!----><span class="el-tag el-tag--warning el-tag--small el-tag--light" style="" data-v-d071913d><span class="el-tag__content"><!--[--><span class="icon enabled" style="color:inherit !important;cursor:unset;" data-v-d071913d data-v-f4e97fe2><i class="fas fa-robot" style="font-size:inherit;color:inherit !important;" data-v-f4e97fe2></i></span> Dịch bởi AI <!--]--></span><!--v-if--></span></a><div class="publish-info" data-v-d071913d><span data-v-d071913d><span data-v-d071913d>Springer Science and Business Media LLC</span> - </span> <!----><span data-v-d071913d> - 2021</span></div><!----><div class="abstract-container" data-v-d071913d><div data-v-d071913d><div><span class=""><jats:title>Tóm tắt</jats:title><jats:p>Bài viết này tập trung vào nghiên cứu sự tồn tại và duy nhất của các nghiệm cho một mô hình toán học liên quan đến động lực truyền bệnh truyền nhiễm coronavirus-19 (COVID-19). Mô hình đã đề cập được xem xét với một đạo hàm dạng hạt nhân phi kỳ có chỉ số cấp thấp do Caputo–Fabrizio cung cấp. Để đạt được kết quả cần thiết về sự tồn tại và duy nhất của nghiệm c...</span><span>... <span class="truncate-button"> hiện toàn bộ</span></span></div><!----></div></div><div class="keyword-container" data-v-d071913d><!--[--><span class="el-tag el-tag--primary el-tag--light" style="" data-v-d071913d><span class="el-tag__content"><!--[--> #COVID-19<!--]--></span><!--v-if--></span><span class="el-tag el-tag--primary el-tag--light" style="" data-v-d071913d><span class="el-tag__content"><!--[--> #mô hình toán học<!--]--></span><!--v-if--></span><span class="el-tag el-tag--primary el-tag--light" style="" data-v-d071913d><span class="el-tag__content"><!--[--> #đạo hàm Caputo–Fabrizio<!--]--></span><!--v-if--></span><span class="el-tag el-tag--primary el-tag--light" style="" data-v-d071913d><span class="el-tag__content"><!--[--> #phương pháp lặp Picard<!--]--></span><!--v-if--></span><span class="el-tag el-tag--primary el-tag--light" style="" data-v-d071913d><span class="el-tag__content"><!--[--> #biến đổi Laplace<!--]--></span><!--v-if--></span><span class="el-tag el-tag--primary el-tag--light" style="" data-v-d071913d><span class="el-tag__content"><!--[--> #phân hoạch Adomian<!--]--></span><!--v-if--></span><!--]--></div><div class="el-row" style="" data-v-d071913d data-v-5c117f8d><!--[--><div class="el-col el-col-24" style="" data-v-5c117f8d><!--[--><!--[--><span data-v-5c117f8d><!----></span><span data-v-5c117f8d><!----></span><span data-v-5c117f8d><!----></span><!--]--><!--]--></div><!--]--></div><div class="action-container" data-v-d071913d><button ariadisabled="false" type="button" class="el-button el-button--primary el-button--small is-round" style="" data-v-d071913d><!--v-if--><span class=""><!--[--><span class="icon enabled" style="color:inherit !important;cursor:unset;" data-v-d071913d data-v-f4e97fe2><i class="fas fa-quote-left" style="font-size:inherit;color:inherit !important;" data-v-f4e97fe2></i></span> 24<!--]--></span></button><button ariadisabled="false" type="button" class="el-button el-button--primary el-button--small is-plain is-round" style="" data-v-d071913d><!--v-if--><span class=""><!--[--><a href="https://advancesindifferenceequations.springeropen.com/articles/10.1186/s13662-021-03316-w" target="_blank" rel="nofollow noopener" data-v-d071913d><!--[--><span class="icon enabled" style="color:inherit !important;cursor:unset;" data-v-d071913d data-v-f4e97fe2><i class="fas fa-link" style="font-size:inherit;color:inherit !important;" data-v-f4e97fe2></i></span> Đi đến bài báo <!--]--></a><!--]--></span></button><button ariadisabled="false" type="button" class="el-button el-button--primary el-button--small is-text" style="" data-v-d071913d><!--v-if--><span class=""><!--[--><span class="icon enabled" style="color:inherit !important;cursor:unset;" data-v-d071913d data-v-f4e97fe2><i class="fas fa-quote-right" style="font-size:inherit;color:inherit !important;" data-v-f4e97fe2></i></span> Trích dẫn <!--]--></span></button><span class="save-button-container" data-v-d071913d data-v-c008b161><button ariadisabled="false" type="button" class="el-button el-button--primary el-button--small is-text" style="" data-v-c008b161><i class="el-icon" style=""><!--[--><!--[--><span class="icon enabled" style="color:inherit !important;cursor:unset;" data-v-c008b161 data-v-f4e97fe2><i class="fas fa-bookmark" style="font-size:inherit;color:inherit !important;" data-v-f4e97fe2></i></span><!--]--><!--]--></i><span class=""><!--[--> Lưu lại<!--]--></span></button></span><!--[--><!--]--></div></div><div class="entity-item-container" data-v-d071913d><a href="/publication/QU%E1%BA%A2NG-C%C3%81O-NH%C6%AF-M%E1%BB%98T-PH%E1%BA%A6N-C%E1%BB%A6A-T%C3%8CNH-HU%E1%BB%90NG-CU%E1%BB%98C-S%E1%BB%90NG-%C4%90%E1%BB%98NG-L%E1%BB%B0C-CHO-VI%E1%BB%86C-D%E1%BA%A0Y-H%E1%BB%8CC-HO%C3%81-H%E1%BB%8CC-V%C3%80-S%E1%BB%B0-PH%E1%BA%A2N-H%E1%BB%92I-V%E1%BB%80-PH%C6%AF%C6%A0NG-PH%C3%81P-D%E1%BA%A0Y-H%E1%BB%8CC-C%E1%BB%A6A-H%E1%BB%8CC-VI%C3%8AN-CAO-H%E1%BB%8CC-T%E1%BA%A0I-TR%C6%AF%E1%BB%9CNG-%C4%90%E1%BA%A0I-H%E1%BB%8CC-S%C6%AF-PH%E1%BA%A0M-TP-HCM-T%E1%BB%AA-TH%C3%81NG-11-%C4%90%E1%BA%BEN-TH%C3%81NG-12-N%C4%82M-2018-PH%E1%BA%A6N-2/49a3eab2-48fa-48f5-a5c1-01447f3eff11" class="entity-title" data-v-d071913d><span data-v-d071913d>QUẢNG CÁO NHƯ MỘT PHẦN CỦA TÌNH HUỐNG CUỘC SỐNG - ĐỘNG LỰC CHO VIỆC DẠY HỌC HOÁ HỌC VÀ SỰ PHẢN HỒI VỀ PHƯƠNG PHÁP DẠY HỌC CỦA HỌC VIÊN CAO HỌC TẠI TRƯỜNG ĐẠI HỌC SƯ PHẠM TP.HCM (TỪ THÁNG 11 ĐẾN THÁNG 12 NĂM 2018) PHẦN 2</span><!----><!----></a><div class="publish-info" data-v-d071913d><span data-v-d071913d><span data-v-d071913d>Tạp chí Khoa học Trường Đại học Sư phạm Thành phố Hồ Chí Minh</span> - </span> Tập 17 Số 8 <span data-v-d071913d> - Trang 1430</span><span data-v-d071913d> - 2020</span></div><!----><div class="abstract-container" data-v-d071913d><div data-v-d071913d><div><span class="">Bài báo trình bày về khả năng vận dụng ditactic ở bậc sau đại học để dánh giá hiểu biết của sinh viên về thái độ của học sinh trong khung đào tạo giáo viên. Bài báo cũng đưa ra khuyến nghị việc dạy học Hoá học định hướng vào học sinh nhiều hơn so với hiện nay. Khái niệm về “dạy học hướng đến học sinh” được xem xét dựa trên những nỗ lực về chính sách về giáo dục ở Việt Nam. Tác giả đã cụ ...</span><span>... <span class="truncate-button"> hiện toàn bộ</span></span></div><!----></div></div><div class="keyword-container" data-v-d071913d><!--[--><span class="el-tag el-tag--primary el-tag--light" style="" data-v-d071913d><span class="el-tag__content"><!--[--> #phương pháp giảng dạy ở bậc đại học<!--]--></span><!--v-if--></span><span class="el-tag el-tag--primary el-tag--light" style="" data-v-d071913d><span class="el-tag__content"><!--[--> #khái niệm dạy học định hướng học sinh<!--]--></span><!--v-if--></span><span class="el-tag el-tag--primary el-tag--light" style="" data-v-d071913d><span class="el-tag__content"><!--[--> #các quá trình học tập của học viên cao học<!--]--></span><!--v-if--></span><span class="el-tag el-tag--primary el-tag--light" style="" data-v-d071913d><span class="el-tag__content"><!--[--> #đào tạo giáo viên sư phạm Hóa học<!--]--></span><!--v-if--></span><span class="el-tag el-tag--primary el-tag--light" style="" data-v-d071913d><span class="el-tag__content"><!--[--> #quảng cáo<!--]--></span><!--v-if--></span><!--]--></div><div class="el-row" style="" data-v-d071913d data-v-5c117f8d><!--[--><div class="el-col el-col-24" style="" data-v-5c117f8d><!--[--><!--[--><span data-v-5c117f8d><!----></span><span data-v-5c117f8d><!----></span><span data-v-5c117f8d><!----></span><!--]--><!--]--></div><!--]--></div><div class="action-container" data-v-d071913d><button ariadisabled="false" type="button" class="el-button el-button--primary el-button--small is-round" style="" data-v-d071913d><!--v-if--><span class=""><!--[--><span class="icon enabled" style="color:inherit !important;cursor:unset;" data-v-d071913d data-v-f4e97fe2><i class="fas fa-quote-left" style="font-size:inherit;color:inherit !important;" data-v-f4e97fe2></i></span> 0<!--]--></span></button><button ariadisabled="false" type="button" class="el-button el-button--primary el-button--small is-plain is-round" style="" data-v-d071913d><!--v-if--><span class=""><!--[--><a href="https://journal.hcmue.edu.vn/index.php/hcmuejos/article/view/2826" target="_blank" rel="nofollow noopener" data-v-d071913d><!--[--><span class="icon enabled" style="color:inherit !important;cursor:unset;" data-v-d071913d data-v-f4e97fe2><i class="fas fa-link" style="font-size:inherit;color:inherit !important;" data-v-f4e97fe2></i></span> Đi đến bài báo <!--]--></a><!--]--></span></button><button ariadisabled="false" type="button" class="el-button el-button--primary el-button--small is-text" style="" data-v-d071913d><!--v-if--><span class=""><!--[--><span class="icon enabled" style="color:inherit !important;cursor:unset;" data-v-d071913d data-v-f4e97fe2><i class="fas fa-quote-right" style="font-size:inherit;color:inherit !important;" data-v-f4e97fe2></i></span> Trích dẫn <!--]--></span></button><span class="save-button-container" data-v-d071913d data-v-c008b161><button ariadisabled="false" type="button" class="el-button el-button--primary el-button--small is-text" style="" data-v-c008b161><i class="el-icon" style=""><!--[--><!--[--><span class="icon enabled" style="color:inherit !important;cursor:unset;" data-v-c008b161 data-v-f4e97fe2><i class="fas fa-bookmark" style="font-size:inherit;color:inherit !important;" data-v-f4e97fe2></i></span><!--]--><!--]--></i><span class=""><!--[--> Lưu lại<!--]--></span></button></span><!--[--><!--]--></div></div><div class="entity-item-container" data-v-d071913d><a href="/publication/S%E1%BB%91-Betti-v%C3%A0-kh%C3%B4ng-gian-c%C3%A1c-%C4%91%E1%BA%A1o-h%C3%A0m-ph%E1%BA%A3n-x%E1%BB%A9ng-c%E1%BB%A7a-c%C3%A1c-%C4%91%E1%BA%A1i-s%E1%BB%91-Lie-to%C3%A0n-ph%C6%B0%C6%A1ng-gi%E1%BA%A3i-%C4%91%C6%B0%E1%BB%A3c-c%C3%B3-s%E1%BB%91-chi%E1%BB%81u-7/b6cd06b1-e647-48b1-a55b-ada58502fe7e" class="entity-title" data-v-d071913d><span data-v-d071913d>Số Betti và không gian các đạo hàm phản xứng của các đại số Lie toàn phương giải được có số chiều  7</span><!----><!----></a><div class="publish-info" data-v-d071913d><span data-v-d071913d><span data-v-d071913d>Tạp chí Khoa học Trường Đại học Sư phạm Thành phố Hồ Chí Minh</span> - </span> Tập 0 Số 5(70) <span data-v-d071913d> - Trang 86</span><span data-v-d071913d> - 2019</span></div><!----><div class="abstract-container" data-v-d071913d><div data-v-d071913d><div><span class="">Trong b&agrave;i b&aacute;o n&agrave;y, ch&uacute;ng t&ocirc;i t&iacute;nh to&aacute;n to&agrave;n bộ số Betti của c&aacute;c đại số Lie to&agrave;n phương giải được c&oacute; số chiều &pound; 7 vừa được ph&acirc;n loại trong [5]. B&ecirc;n cạnh đ&oacute;, kh&ocirc;ng gian c&aacute;c đạo h&agrave;m phản xứng của ch&uacute;ng cũng được m&ocirc; tả tường minh. N...</span><span>... <span class="truncate-button"> hiện toàn bộ</span></span></div><!----></div></div><div class="keyword-container" data-v-d071913d><!--[--><span class="el-tag el-tag--primary el-tag--light" style="" data-v-d071913d><span class="el-tag__content"><!--[--> #đại số Lie<!--]--></span><!--v-if--></span><span class="el-tag el-tag--primary el-tag--light" style="" data-v-d071913d><span class="el-tag__content"><!--[--> #đại số Lie toàn phương<!--]--></span><!--v-if--></span><span class="el-tag el-tag--primary el-tag--light" style="" data-v-d071913d><span class="el-tag__content"><!--[--> #đối đồng điều<!--]--></span><!--v-if--></span><span class="el-tag el-tag--primary el-tag--light" style="" data-v-d071913d><span class="el-tag__content"><!--[--> #đạo hàm phản xứng<!--]--></span><!--v-if--></span><!--]--></div><div class="el-row" style="" data-v-d071913d data-v-5c117f8d><!--[--><div class="el-col el-col-24" style="" data-v-5c117f8d><!--[--><!--[--><span data-v-5c117f8d><!----></span><span data-v-5c117f8d><!----></span><span data-v-5c117f8d><!----></span><!--]--><!--]--></div><!--]--></div><div class="action-container" data-v-d071913d><button ariadisabled="false" type="button" class="el-button el-button--primary el-button--small is-round" style="" data-v-d071913d><!--v-if--><span class=""><!--[--><span class="icon enabled" style="color:inherit !important;cursor:unset;" data-v-d071913d data-v-f4e97fe2><i class="fas fa-quote-left" style="font-size:inherit;color:inherit !important;" data-v-f4e97fe2></i></span> 0<!--]--></span></button><button ariadisabled="false" type="button" class="el-button el-button--primary el-button--small is-plain is-round" style="" data-v-d071913d><!--v-if--><span class=""><!--[--><a href="https://journal.hcmue.edu.vn/index.php/hcmuejos/article/view/531" target="_blank" rel="nofollow noopener" data-v-d071913d><!--[--><span class="icon enabled" style="color:inherit !important;cursor:unset;" data-v-d071913d data-v-f4e97fe2><i class="fas fa-link" style="font-size:inherit;color:inherit !important;" data-v-f4e97fe2></i></span> Đi đến bài báo <!--]--></a><!--]--></span></button><button ariadisabled="false" type="button" class="el-button el-button--primary el-button--small is-text" style="" data-v-d071913d><!--v-if--><span class=""><!--[--><span class="icon enabled" style="color:inherit !important;cursor:unset;" data-v-d071913d data-v-f4e97fe2><i class="fas fa-quote-right" style="font-size:inherit;color:inherit !important;" data-v-f4e97fe2></i></span> Trích dẫn <!--]--></span></button><span class="save-button-container" data-v-d071913d data-v-c008b161><button ariadisabled="false" type="button" class="el-button el-button--primary el-button--small is-text" style="" data-v-c008b161><i class="el-icon" style=""><!--[--><!--[--><span class="icon enabled" style="color:inherit !important;cursor:unset;" data-v-c008b161 data-v-f4e97fe2><i class="fas fa-bookmark" style="font-size:inherit;color:inherit !important;" data-v-f4e97fe2></i></span><!--]--><!--]--></i><span class=""><!--[--> Lưu lại<!--]--></span></button></span><!--[--><!--]--></div></div><div class="entity-item-container" data-v-d071913d><a href="/publication/Existence-uniqueness-and-Ulam-stability-results-for-a-mixed-type-fractional-differential-equations-with-p-Laplacian-operator/8c11afaf-1c4b-4f57-a860-34d2d73ed480" class="entity-title" data-v-d071913d><span data-v-d071913d>Kết quả tồn tại, duy nhất và ổn định Ulam cho các phương trình vi phân phân loại hỗn hợp với toán tử p-Laplacian</span><!----><span class="el-tag el-tag--warning el-tag--small el-tag--light" style="" data-v-d071913d><span class="el-tag__content"><!--[--><span class="icon enabled" style="color:inherit !important;cursor:unset;" data-v-d071913d data-v-f4e97fe2><i class="fas fa-robot" style="font-size:inherit;color:inherit !important;" data-v-f4e97fe2></i></span> Dịch bởi AI <!--]--></span><!--v-if--></span></a><div class="publish-info" data-v-d071913d><span data-v-d071913d><span data-v-d071913d>Arabian Journal of Mathematics</span> - </span> Tập 12 Số 3 <span data-v-d071913d> - Trang 633-645</span><span data-v-d071913d> - 2023</span></div><!----><div class="abstract-container" data-v-d071913d><div data-v-d071913d><div><span class=""><jats:title>Tóm tắt</jats:title><jats:p>Trong bài báo này, chúng tôi nghiên cứu một bài toán giá trị biên phi tuyến với p-Laplacian chứa cả đạo hàm phân số Riemann–Liouville trái và đạo hàm Caputo phải với các điều kiện khởi đầu và điều kiện tích phân. Một số kết quả mới về sự tồn tại và duy nhất của nghiệm cho mô hình được trình bày, cùng với sự ổn định Ulam của các nghiệm. Hai ví dụ được cung cấ...</span><span>... <span class="truncate-button"> hiện toàn bộ</span></span></div><!----></div></div><div class="keyword-container" data-v-d071913d><!--[--><span class="el-tag el-tag--primary el-tag--light" style="" data-v-d071913d><span class="el-tag__content"><!--[--> #phương trình vi phân phân loại<!--]--></span><!--v-if--></span><span class="el-tag el-tag--primary el-tag--light" style="" data-v-d071913d><span class="el-tag__content"><!--[--> #p-Laplacian<!--]--></span><!--v-if--></span><span class="el-tag el-tag--primary el-tag--light" style="" data-v-d071913d><span class="el-tag__content"><!--[--> #ổn định Ulam<!--]--></span><!--v-if--></span><span class="el-tag el-tag--primary el-tag--light" style="" data-v-d071913d><span class="el-tag__content"><!--[--> #đạo hàm phân số<!--]--></span><!--v-if--></span><span class="el-tag el-tag--primary el-tag--light" style="" data-v-d071913d><span class="el-tag__content"><!--[--> #tồn tại và duy nhất của nghiệm<!--]--></span><!--v-if--></span><!--]--></div><div class="el-row" style="" data-v-d071913d data-v-5c117f8d><!--[--><div class="el-col el-col-24" style="" data-v-5c117f8d><!--[--><!--[--><span data-v-5c117f8d><!----></span><span data-v-5c117f8d><!----></span><span data-v-5c117f8d><!----></span><!--]--><!--]--></div><!--]--></div><div class="action-container" data-v-d071913d><button ariadisabled="false" type="button" class="el-button el-button--primary el-button--small is-round" style="" data-v-d071913d><!--v-if--><span class=""><!--[--><span class="icon enabled" style="color:inherit !important;cursor:unset;" data-v-d071913d data-v-f4e97fe2><i class="fas fa-quote-left" style="font-size:inherit;color:inherit !important;" data-v-f4e97fe2></i></span> 0<!--]--></span></button><button ariadisabled="false" type="button" class="el-button el-button--primary el-button--small is-plain is-round" style="" data-v-d071913d><!--v-if--><span class=""><!--[--><a href="https://link.springer.com/10.1007/s40065-023-00436-x" target="_blank" rel="nofollow noopener" data-v-d071913d><!--[--><span class="icon enabled" style="color:inherit !important;cursor:unset;" data-v-d071913d data-v-f4e97fe2><i class="fas fa-link" style="font-size:inherit;color:inherit !important;" data-v-f4e97fe2></i></span> Đi đến bài báo <!--]--></a><!--]--></span></button><button ariadisabled="false" type="button" class="el-button el-button--primary el-button--small is-text" style="" data-v-d071913d><!--v-if--><span class=""><!--[--><span class="icon enabled" style="color:inherit !important;cursor:unset;" data-v-d071913d data-v-f4e97fe2><i class="fas fa-quote-right" style="font-size:inherit;color:inherit !important;" data-v-f4e97fe2></i></span> Trích dẫn <!--]--></span></button><span class="save-button-container" data-v-d071913d data-v-c008b161><button ariadisabled="false" type="button" class="el-button el-button--primary el-button--small is-text" style="" data-v-c008b161><i class="el-icon" style=""><!--[--><!--[--><span class="icon enabled" style="color:inherit !important;cursor:unset;" data-v-c008b161 data-v-f4e97fe2><i class="fas fa-bookmark" style="font-size:inherit;color:inherit !important;" data-v-f4e97fe2></i></span><!--]--><!--]--></i><span class=""><!--[--> Lưu lại<!--]--></span></button></span><!--[--><!--]--></div></div><div class="entity-item-container" data-v-d071913d><a href="/publication/Nh%C3%B3m-%C4%91%E1%BB%91i-%C4%91%E1%BB%93ng-%C4%91i%E1%BB%81u-c%E1%BB%A7a-c%C3%A1c-%C4%91%E1%BA%A1i-s%E1%BB%91-Lie-to%C3%A0n-ph%C6%B0%C6%A1ng-c%C6%A1-b%E1%BA%A3n/233271f4-288c-4bd8-a9e1-98b2c4f270a4" class="entity-title" data-v-d071913d><span data-v-d071913d>Nhóm đối đồng điều của các đại số Lie toàn phương cơ bản</span><!----><!----></a><div class="publish-info" data-v-d071913d><span data-v-d071913d><span data-v-d071913d>Tạp chí Khoa học Trường Đại học Sư phạm Thành phố Hồ Chí Minh</span> - </span> Tập 0 Số 47 <span data-v-d071913d> - Trang 25</span><span data-v-d071913d> - 2019</span></div><!----><div class="abstract-container" data-v-d071913d><div data-v-d071913d><div><span class="">v\:* {behavior:url(#default#VML);} o\:* {behavior:url(#default#VML);} w\:* {behavior:url(#default#VML);} .shape {behavior:url(#default#VML);} Trong b&agrave;i b&aacute;o n&agrave;y, ch&uacute;ng t&ocirc;i m&ocirc; tả nh&oacute;m đối đồng điều&nbsp;v&agrave; t&iacute;nh to&aacute;n số chiều của n&oacute; đối với c&aacute;c đại số Lie to&agrave;n phương cơ bản đ&atilde; được ph&acirc;n ...</span><span>... <span class="truncate-button"> hiện toàn bộ</span></span></div><!----></div></div><div class="keyword-container" data-v-d071913d><!--[--><span class="el-tag el-tag--primary el-tag--light" style="" data-v-d071913d><span class="el-tag__content"><!--[--> #đại số Lie<!--]--></span><!--v-if--></span><span class="el-tag el-tag--primary el-tag--light" style="" data-v-d071913d><span class="el-tag__content"><!--[--> #đại số Lie toàn phương<!--]--></span><!--v-if--></span><span class="el-tag el-tag--primary el-tag--light" style="" data-v-d071913d><span class="el-tag__content"><!--[--> #đối đồng điều<!--]--></span><!--v-if--></span><span class="el-tag el-tag--primary el-tag--light" style="" data-v-d071913d><span class="el-tag__content"><!--[--> #đạo hàm phản xứng<!--]--></span><!--v-if--></span><!--]--></div><div class="el-row" style="" data-v-d071913d data-v-5c117f8d><!--[--><div class="el-col el-col-24" style="" data-v-5c117f8d><!--[--><!--[--><span data-v-5c117f8d><!----></span><span data-v-5c117f8d><!----></span><span data-v-5c117f8d><!----></span><!--]--><!--]--></div><!--]--></div><div class="action-container" data-v-d071913d><button ariadisabled="false" type="button" class="el-button el-button--primary el-button--small is-round" style="" data-v-d071913d><!--v-if--><span class=""><!--[--><span class="icon enabled" style="color:inherit !important;cursor:unset;" data-v-d071913d data-v-f4e97fe2><i class="fas fa-quote-left" style="font-size:inherit;color:inherit !important;" data-v-f4e97fe2></i></span> 0<!--]--></span></button><button ariadisabled="false" type="button" class="el-button el-button--primary el-button--small is-plain is-round" style="" data-v-d071913d><!--v-if--><span class=""><!--[--><a href="https://journal.hcmue.edu.vn/index.php/hcmuejos/article/view/896" target="_blank" rel="nofollow noopener" data-v-d071913d><!--[--><span class="icon enabled" style="color:inherit !important;cursor:unset;" data-v-d071913d data-v-f4e97fe2><i class="fas fa-link" style="font-size:inherit;color:inherit !important;" data-v-f4e97fe2></i></span> Đi đến bài báo <!--]--></a><!--]--></span></button><button ariadisabled="false" type="button" class="el-button el-button--primary el-button--small is-text" style="" data-v-d071913d><!--v-if--><span class=""><!--[--><span class="icon enabled" style="color:inherit !important;cursor:unset;" data-v-d071913d data-v-f4e97fe2><i class="fas fa-quote-right" style="font-size:inherit;color:inherit !important;" data-v-f4e97fe2></i></span> Trích dẫn <!--]--></span></button><span class="save-button-container" data-v-d071913d data-v-c008b161><button ariadisabled="false" type="button" class="el-button el-button--primary el-button--small is-text" style="" data-v-c008b161><i class="el-icon" style=""><!--[--><!--[--><span class="icon enabled" style="color:inherit !important;cursor:unset;" data-v-c008b161 data-v-f4e97fe2><i class="fas fa-bookmark" style="font-size:inherit;color:inherit !important;" data-v-f4e97fe2></i></span><!--]--><!--]--></i><span class=""><!--[--> Lưu lại<!--]--></span></button></span><!--[--><!--]--></div></div><div class="entity-item-container" data-v-d071913d><a href="/publication/Negotiating-Boundaries-of-Power-in-the-Global-Governance-for-Care/d2dcc9fc-e231-498d-92d0-b9377182f321" class="entity-title" data-v-d071913d><span data-v-d071913d>Đàm phán Ranh giới Quyền Lực trong Quản trị Toàn cầu về Chăm sóc</span><!----><span class="el-tag el-tag--warning el-tag--small el-tag--light" style="" data-v-d071913d><span class="el-tag__content"><!--[--><span class="icon enabled" style="color:inherit !important;cursor:unset;" data-v-d071913d data-v-f4e97fe2><i class="fas fa-robot" style="font-size:inherit;color:inherit !important;" data-v-f4e97fe2></i></span> Dịch bởi AI <!--]--></span><!--v-if--></span></a><div class="publish-info" data-v-d071913d><span data-v-d071913d><span data-v-d071913d>Development</span> - </span> Tập 64 <span data-v-d071913d> - Trang 39-47</span><span data-v-d071913d> - 2021</span></div><!----><div class="abstract-container" data-v-d071913d><div data-v-d071913d><div><span class="">Trọng tâm của việc xây dựng các nền kinh tế chăm sóc như một bước cần thiết hướng tới công bằng giới tính đòi hỏi một cuộc đánh giá lại về quản trị kinh tế toàn cầu và chủ nghĩa đa phương tập trung vào nhà nước. Các cấu trúc quyền lực toàn cầu không thể được xem chỉ đơn thuần là vấn đề về biên giới chính trị của các quốc gia, điều này thuộc về phạm vi truyền thống của chính sách đối ngoại. Thay vì...</span><span>... <span class="truncate-button"> hiện toàn bộ</span></span></div><!----></div></div><div class="keyword-container" data-v-d071913d><!--[--><span class="el-tag el-tag--primary el-tag--light" style="" data-v-d071913d><span class="el-tag__content"><!--[--> #kinh tế chăm sóc<!--]--></span><!--v-if--></span><span class="el-tag el-tag--primary el-tag--light" style="" data-v-d071913d><span class="el-tag__content"><!--[--> #công bằng giới tính<!--]--></span><!--v-if--></span><span class="el-tag el-tag--primary el-tag--light" style="" data-v-d071913d><span class="el-tag__content"><!--[--> #quản trị toàn cầu<!--]--></span><!--v-if--></span><span class="el-tag el-tag--primary el-tag--light" style="" data-v-d071913d><span class="el-tag__content"><!--[--> #quyền lực<!--]--></span><!--v-if--></span><span class="el-tag el-tag--primary el-tag--light" style="" data-v-d071913d><span class="el-tag__content"><!--[--> #đạo đức chăm sóc<!--]--></span><!--v-if--></span><!--]--></div><div class="el-row" style="" data-v-d071913d data-v-5c117f8d><!--[--><div class="el-col el-col-24" style="" data-v-5c117f8d><!--[--><!--[--><span data-v-5c117f8d><!----></span><span data-v-5c117f8d><!----></span><span data-v-5c117f8d><!----></span><!--]--><!--]--></div><!--]--></div><div class="action-container" data-v-d071913d><!----><button ariadisabled="false" type="button" class="el-button el-button--primary el-button--small is-plain is-round" style="" data-v-d071913d><!--v-if--><span class=""><!--[--><a href="https://link.springer.com/article/10.1057/s41301-021-00283-3" target="_blank" rel="nofollow noopener" data-v-d071913d><!--[--><span class="icon enabled" style="color:inherit !important;cursor:unset;" data-v-d071913d data-v-f4e97fe2><i class="fas fa-link" style="font-size:inherit;color:inherit !important;" data-v-f4e97fe2></i></span> Đi đến bài báo <!--]--></a><!--]--></span></button><button ariadisabled="false" type="button" class="el-button el-button--primary el-button--small is-text" style="" data-v-d071913d><!--v-if--><span class=""><!--[--><span class="icon enabled" style="color:inherit !important;cursor:unset;" data-v-d071913d data-v-f4e97fe2><i class="fas fa-quote-right" style="font-size:inherit;color:inherit !important;" data-v-f4e97fe2></i></span> Trích dẫn <!--]--></span></button><span class="save-button-container" data-v-d071913d data-v-c008b161><button ariadisabled="false" type="button" class="el-button el-button--primary el-button--small is-text" style="" data-v-c008b161><i class="el-icon" style=""><!--[--><!--[--><span class="icon enabled" style="color:inherit !important;cursor:unset;" data-v-c008b161 data-v-f4e97fe2><i class="fas fa-bookmark" style="font-size:inherit;color:inherit !important;" data-v-f4e97fe2></i></span><!--]--><!--]--></i><span class=""><!--[--> Lưu lại<!--]--></span></button></span><!--[--><!--]--></div></div><div class="entity-item-container" data-v-d071913d><a href="/publication/A-Control-Parameterization-Method-to-Solve-the-Fractional-Order-Optimal-Control-Problem/1a24c699-7fab-4aba-a871-773e80bbef12" class="entity-title" data-v-d071913d><span data-v-d071913d>Phương pháp tham số hóa điều khiển để giải bài toán điều khiển tối ưu bậc phân thức</span><!----><span class="el-tag el-tag--warning el-tag--small el-tag--light" style="" data-v-d071913d><span class="el-tag__content"><!--[--><span class="icon enabled" style="color:inherit !important;cursor:unset;" data-v-d071913d data-v-f4e97fe2><i class="fas fa-robot" style="font-size:inherit;color:inherit !important;" data-v-f4e97fe2></i></span> Dịch bởi AI <!--]--></span><!--v-if--></span></a><div class="publish-info" data-v-d071913d><span data-v-d071913d><span data-v-d071913d>Journal of Optimization Theory and Applications</span> - </span> Tập 187 <span data-v-d071913d> - Trang 234-247</span><span data-v-d071913d> - 2017</span></div><!----><div class="abstract-container" data-v-d071913d><div data-v-d071913d><div><span class="">Bài báo này xem xét một lớp các bài toán điều khiển tối ưu bậc phân thức với các ràng buộc cân bằng và bất cân bằng chuẩn. Một đạo hàm phân thức trong hệ động được định nghĩa theo cảm nghĩa Caputo. Bằng cách sử dụng phương pháp tham số hóa điều khiển, chúng tôi xấp xỉ các bài toán điều khiển tối ưu bậc phân thức bằng một chuỗi các bài toán tối ưu có chiều hữu hạn. Chúng tôi sau đó trình bày công t...</span><span>... <span class="truncate-button"> hiện toàn bộ</span></span></div><!----></div></div><div class="keyword-container" data-v-d071913d><!--[--><span class="el-tag el-tag--primary el-tag--light" style="" data-v-d071913d><span class="el-tag__content"><!--[--> #điều khiển tối ưu<!--]--></span><!--v-if--></span><span class="el-tag el-tag--primary el-tag--light" style="" data-v-d071913d><span class="el-tag__content"><!--[--> #bậc phân thức<!--]--></span><!--v-if--></span><span class="el-tag el-tag--primary el-tag--light" style="" data-v-d071913d><span class="el-tag__content"><!--[--> #phương pháp tham số hóa<!--]--></span><!--v-if--></span><span class="el-tag el-tag--primary el-tag--light" style="" data-v-d071913d><span class="el-tag__content"><!--[--> #đạo hàm phân thức<!--]--></span><!--v-if--></span><span class="el-tag el-tag--primary el-tag--light" style="" data-v-d071913d><span class="el-tag__content"><!--[--> #tối ưu hóa dựa trên gradient<!--]--></span><!--v-if--></span><!--]--></div><div class="el-row" style="" data-v-d071913d data-v-5c117f8d><!--[--><div class="el-col el-col-24" style="" data-v-5c117f8d><!--[--><!--[--><span data-v-5c117f8d><!----></span><span data-v-5c117f8d><!----></span><span data-v-5c117f8d><!----></span><!--]--><!--]--></div><!--]--></div><div class="action-container" data-v-d071913d><!----><button ariadisabled="false" type="button" class="el-button el-button--primary el-button--small is-plain is-round" style="" data-v-d071913d><!--v-if--><span class=""><!--[--><a href="https://link.springer.com/article/10.1007/s10957-017-1163-7" target="_blank" rel="nofollow noopener" data-v-d071913d><!--[--><span class="icon enabled" style="color:inherit !important;cursor:unset;" data-v-d071913d data-v-f4e97fe2><i class="fas fa-link" style="font-size:inherit;color:inherit !important;" data-v-f4e97fe2></i></span> Đi đến bài báo <!--]--></a><!--]--></span></button><button ariadisabled="false" type="button" class="el-button el-button--primary el-button--small is-text" style="" data-v-d071913d><!--v-if--><span class=""><!--[--><span class="icon enabled" style="color:inherit !important;cursor:unset;" data-v-d071913d data-v-f4e97fe2><i class="fas fa-quote-right" style="font-size:inherit;color:inherit !important;" data-v-f4e97fe2></i></span> Trích dẫn <!--]--></span></button><span class="save-button-container" data-v-d071913d data-v-c008b161><button ariadisabled="false" type="button" class="el-button el-button--primary el-button--small is-text" style="" data-v-c008b161><i class="el-icon" style=""><!--[--><!--[--><span class="icon enabled" style="color:inherit !important;cursor:unset;" data-v-c008b161 data-v-f4e97fe2><i class="fas fa-bookmark" style="font-size:inherit;color:inherit !important;" data-v-f4e97fe2></i></span><!--]--><!--]--></i><span class=""><!--[--> Lưu lại<!--]--></span></button></span><!--[--><!--]--></div></div><div class="entity-item-container" data-v-d071913d><a href="/publication/Boundary-value-problems-of-fractional-q-difference-equations-on-the-half-line/0b79909e-36ae-477c-a877-aedbc4a059e4" class="entity-title" data-v-d071913d><span data-v-d071913d>Các bài toán giá trị biên của phương trình vi phân không tuyến tính phân số q trên nửa đoạn</span><!----><span class="el-tag el-tag--warning el-tag--small el-tag--light" style="" data-v-d071913d><span class="el-tag__content"><!--[--><span class="icon enabled" style="color:inherit !important;cursor:unset;" data-v-d071913d data-v-f4e97fe2><i class="fas fa-robot" style="font-size:inherit;color:inherit !important;" data-v-f4e97fe2></i></span> Dịch bởi AI <!--]--></span><!--v-if--></span></a><div class="publish-info" data-v-d071913d><span data-v-d071913d><span data-v-d071913d>Springer Science and Business Media LLC</span> - </span> Tập 2019 <span data-v-d071913d> - Trang 1-16</span><span data-v-d071913d> - 2019</span></div><!----><div class="abstract-container" data-v-d071913d><div data-v-d071913d><div><span class="">Trong bài báo này, chúng tôi xem xét bài toán giá trị biên của một lớp phương trình vi phân không tuyến tính phân số q liên quan đến đạo hàm phân số Riemann–Liouville trên nửa đoạn. Thông qua định lý điểm cố định Schauder và định lý điểm cố định Leggett–Williams, một số kết quả về sự tồn tại và tính nhiều nghiệm của các nghiệm được đưa ra.</span><!----></div><!----></div></div><div class="keyword-container" data-v-d071913d><!--[--><span class="el-tag el-tag--primary el-tag--light" style="" data-v-d071913d><span class="el-tag__content"><!--[--> #phương trình vi phân<!--]--></span><!--v-if--></span><span class="el-tag el-tag--primary el-tag--light" style="" data-v-d071913d><span class="el-tag__content"><!--[--> #phân số<!--]--></span><!--v-if--></span><span class="el-tag el-tag--primary el-tag--light" style="" data-v-d071913d><span class="el-tag__content"><!--[--> #q-đạo hàm<!--]--></span><!--v-if--></span><span class="el-tag el-tag--primary el-tag--light" style="" data-v-d071913d><span class="el-tag__content"><!--[--> #bài toán giá trị biên<!--]--></span><!--v-if--></span><span class="el-tag el-tag--primary el-tag--light" style="" data-v-d071913d><span class="el-tag__content"><!--[--> #điểm cố định<!--]--></span><!--v-if--></span><!--]--></div><div class="el-row" style="" data-v-d071913d data-v-5c117f8d><!--[--><div class="el-col el-col-24" style="" data-v-5c117f8d><!--[--><!--[--><span data-v-5c117f8d><!----></span><span data-v-5c117f8d><!----></span><span data-v-5c117f8d><!----></span><!--]--><!--]--></div><!--]--></div><div class="action-container" data-v-d071913d><!----><button ariadisabled="false" type="button" class="el-button el-button--primary el-button--small is-plain is-round" style="" data-v-d071913d><!--v-if--><span class=""><!--[--><a href="https://boundaryvalueproblems.springeropen.com/articles/10.1186/s13661-019-1159-3" target="_blank" rel="nofollow noopener" data-v-d071913d><!--[--><span class="icon enabled" style="color:inherit !important;cursor:unset;" data-v-d071913d data-v-f4e97fe2><i class="fas fa-link" style="font-size:inherit;color:inherit !important;" data-v-f4e97fe2></i></span> Đi đến bài báo <!--]--></a><!--]--></span></button><button ariadisabled="false" type="button" class="el-button el-button--primary el-button--small is-text" style="" data-v-d071913d><!--v-if--><span class=""><!--[--><span class="icon enabled" style="color:inherit !important;cursor:unset;" data-v-d071913d data-v-f4e97fe2><i class="fas fa-quote-right" style="font-size:inherit;color:inherit !important;" data-v-f4e97fe2></i></span> Trích dẫn <!--]--></span></button><span class="save-button-container" data-v-d071913d data-v-c008b161><button ariadisabled="false" type="button" class="el-button el-button--primary el-button--small is-text" style="" data-v-c008b161><i class="el-icon" style=""><!--[--><!--[--><span class="icon enabled" style="color:inherit !important;cursor:unset;" data-v-c008b161 data-v-f4e97fe2><i class="fas fa-bookmark" style="font-size:inherit;color:inherit !important;" data-v-f4e97fe2></i></span><!--]--><!--]--></i><span class=""><!--[--> Lưu lại<!--]--></span></button></span><!--[--><!--]--></div></div><!--]--></div><div class="el-row is-justify-end" style="" data-v-fa649ea5><!--[--><div class="pagination-container" data-v-fa649ea5><span data-v-fa649ea5> Tổng số: 46</span>    <div class="el-pagination is-background" data-v-fa649ea5><button type="button" class="btn-prev is-first" disabled aria-label="Go to previous page" aria-disabled="true"><i class="el-icon" style=""><!--[--><svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 1024 1024"><path fill="currentColor" d="M609.408 149.376 277.76 489.6a32 32 0 0 0 0 44.672l331.648 340.352a29.12 29.12 0 0 0 41.728 0 30.592 30.592 0 0 0 0-42.752L339.264 511.936l311.872-319.872a30.592 30.592 0 0 0 0-42.688 29.12 29.12 0 0 0-41.728 0z"></path></svg><!--]--></i></button><ul class="el-pager"><li class="is-active number" aria-current="true" aria-label="page 1" tabindex="0"> 1 </li><!--v-if--><!--[--><li class="number" aria-current="false" aria-label="page 2" tabindex="0">2</li><li 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class="post-page-container"><!--[--><div class="entity-item-container"><a href="/topic/capacity development" class=""><span class="el-tag el-tag--primary el-tag--light" style=""><span class="el-tag__content"><!--[-->#capacity development<!--]--></span><!--v-if--></span><br><p>Capacity development là gì? Các công bố khoa học về Capacity development</p></a></div><div class="entity-item-container"><a href="/topic/chitin" class=""><span class="el-tag el-tag--primary el-tag--light" style=""><span class="el-tag__content"><!--[-->#chitin<!--]--></span><!--v-if--></span><br><p>Chitin là gì? Các công bố khoa học về Chitin</p></a></div><div class="entity-item-container"><a href="/topic/nạn nhân" class=""><span class="el-tag el-tag--primary el-tag--light" style=""><span class="el-tag__content"><!--[-->#nạn nhân<!--]--></span><!--v-if--></span><br><p>Nạn nhân là gì? Các công bố khoa học về Nạn nhân</p></a></div><div class="entity-item-container"><a href="/topic/bã mía" class=""><span class="el-tag el-tag--primary el-tag--light" style=""><span class="el-tag__content"><!--[-->#bã mía<!--]--></span><!--v-if--></span><br><p>Bã mía là gì? Các công bố khoa học về Bã mía</p></a></div><div class="entity-item-container"><a href="/topic/phản xạ h" class=""><span class="el-tag el-tag--primary el-tag--light" style=""><span class="el-tag__content"><!--[-->#phản xạ h<!--]--></span><!--v-if--></span><br><p>Phản xạ h là gì? Các công bố khoa học về Phản xạ h</p></a></div><div class="entity-item-container"><a href="/topic/văn hóa truyền thống" class=""><span class="el-tag el-tag--primary el-tag--light" style=""><span class="el-tag__content"><!--[-->#văn hóa truyền thống<!--]--></span><!--v-if--></span><br><p>Văn hóa truyền thống là gì? Các bài báo nghiên cứu liên quan</p></a></div><div class="entity-item-container"><a href="/topic/ước lượng tham số" class=""><span class="el-tag el-tag--primary el-tag--light" style=""><span class="el-tag__content"><!--[-->#ước lượng tham số<!--]--></span><!--v-if--></span><br><p>Ước lượng tham số là gì? Các nghiên cứu khoa học về Ước lượng tham số</p></a></div><div class="entity-item-container"><a href="/topic/chuyển đơn phôi" class=""><span class="el-tag el-tag--primary el-tag--light" style=""><span class="el-tag__content"><!--[-->#chuyển đơn phôi<!--]--></span><!--v-if--></span><br><p>Chuyển đơn phôi là gì? Các công bố khoa học về Chuyển đơn phôi</p></a></div><div class="entity-item-container"><a href="/topic/nghiên cứu thuốc" class=""><span class="el-tag el-tag--primary el-tag--light" style=""><span class="el-tag__content"><!--[-->#nghiên cứu thuốc<!--]--></span><!--v-if--></span><br><p>Nghiên cứu thuốc là gì? Các nghiên cứu khoa học liên quan</p></a></div><div class="entity-item-container"><a href="/topic/quang phổ phát xạ" class=""><span class="el-tag el-tag--primary el-tag--light" style=""><span class="el-tag__content"><!--[-->#quang phổ phát xạ<!--]--></span><!--v-if--></span><br><p>Quang phổ phát xạ là gì? 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hàm phân số","Đạo hàm phân số là gì? 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Nội dung tập trung vào lịch sử phát triển, các định nghĩa Riemann–Liouville và Caputo, tính chất toán học, phương pháp tính toán, ứng dụng và thách thức nghiên cứu.\u003C/p>\u003Cp>Nội dung chi tiết gồm bốn phần đầu: (1) lịch sử hình thành và hoàn thiện lý thuyết đạo hàm phân số từ thế kỷ XVII đến nay, (2) khái niệm cơ bản và quan hệ giữa tích phân phân số và đạo hàm phân số, (3) công thức định nghĩa Riemann–Liouville, (4) công thức định nghĩa Caputo. Phần hai (sẽ trình bày sau) làm rõ tính chất, phương pháp số, ứng dụng vào vật lý, sinh học, điều khiển và triển vọng tương lai.\u003C/p>\u003Cp>Độc giả sau khi theo dõi phần này sẽ hiểu được nguồn gốc khái niệm, cách xây dựng đạo hàm phân số và sự khác biệt giữa hai định nghĩa kinh điển để áp dụng vào bài toán giá trị đầu và mô phỏng động lực học có độ nhớ dài.\u003C/p>\u003Ch2>Định nghĩa và lịch sử phát triển\u003C/h2>\u003Cp>Ý tưởng về đạo hàm bậc phân số bắt đầu khi Gottfried Wilhelm Leibniz đặt câu hỏi năm 1695: “Liệu có thể tính đạo hàm bậc ½ của hàm số hay không?” Dù trả lời chưa rõ ràng, câu hỏi này mở đường cho các nhà toán học thế kỷ XIX như Riemann, Liouville và Grünwald–Letnikov phát triển lý thuyết cơ bản.\u003C/p>\u003Cp>Đến thập niên 1870, Bernhard Riemann đưa ra tích phân phân số đầu tiên, trong khi Joseph Liouville định nghĩa thuật toán tính đạo hàm phân số qua tích phân tường minh. Năm 1867, Grünwald và Letnikov độc lập đề xuất công thức giới hạn:\u003C/p>\u003Cp>\u003Cscript type=\"math/tex\">\\lim_{h\\to 0} \\frac{1}{h^\\alpha}\\sum_{k=0}^n (-1)^k \\binom{\\alpha}{k} f(x - kh) = D^\\alpha f(x).\u003C/script>\u003C/p>\u003Cfigure class=\"table\">\u003Ctable>\u003Cthead>\u003Ctr>\u003Cth>Năm\u003C/th>\u003Cth>Nhà toán học\u003C/th>\u003Cth>Đóng góp\u003C/th>\u003C/tr>\u003C/thead>\u003Ctbody>\u003Ctr>\u003Ctd>1695\u003C/td>\u003Ctd>Leibniz\u003C/td>\u003Ctd>Đề xuất ý tưởng đạo hàm bậc phân số\u003C/td>\u003C/tr>\u003Ctr>\u003Ctd>1867\u003C/td>\u003Ctd>Grünwald–Letnikov\u003C/td>\u003Ctd>Công thức giới hạn tính đạo hàm phân số\u003C/td>\u003C/tr>\u003Ctr>\u003Ctd>1872\u003C/td>\u003Ctd>Riemann\u003C/td>\u003Ctd>Định nghĩa tích phân phân số\u003C/td>\u003C/tr>\u003Ctr>\u003Ctd>1884\u003C/td>\u003Ctd>Liouville\u003C/td>\u003Ctd>Mở rộng tích phân phân số và công thức đạo hàm\u003C/td>\u003C/tr>\u003Ctr>\u003Ctd>1999\u003C/td>\u003Ctd>Podlubny\u003C/td>\u003Ctd>Xây dựng “Fractional Differential Equations” hệ thống hóa\u003C/td>\u003C/tr>\u003C/tbody>\u003C/table>\u003C/figure>\u003Cp>Sự quan tâm gần đây tăng mạnh nhờ khả năng mô tả quá trình nhớ lâu dài (long-range memory) và lan truyền bất thường (anomalous diffusion) trong vật lý, sinh học và kinh tế.\u003C/p>\u003Ch2>Các khái niệm cơ bản về đạo hàm phân số\u003C/h2>\u003Cp>Đạo hàm phân số bậc α (α ∈ ℝ hoặc ℂ) tổng quát hóa phép vi phân bậc n (n ∈ ℕ) và phép tích phân bậc m (m ∈ ℕ), kết hợp qua quan hệ nghịch đảo:\u003C/p>\u003Cp>\u003Cscript type=\"math/tex\">D^\\alpha I^\\alpha f(x) = f(x), \\quad I^\\alpha D^\\alpha f(x) = f(x) - \\sum_{k=0}^{n-1} \\frac{f^{(k)}(a)}{k!}(x-a)^k.\u003C/script>\u003C/p>\u003Cul>\u003Cli>\u003Cstrong>Tuyến tính:\u003C/strong> \\(D^\\alpha[c_1 f + c_2 g] = c_1 D^\\alpha f + c_2 D^\\alpha g\\).\u003C/li>\u003Cli>\u003Cstrong>Không giao hoán:\u003C/strong> \\(D^\\alpha D^\\beta \\neq D^\\beta D^\\alpha\\) nói chung.\u003C/li>\u003Cli>\u003Cstrong>Quy tắc Leibniz tổng quát:\u003C/strong>&nbsp;\u003Cscript type=\"math/tex\">D^\\alpha(uv)=\\sum_{k=0}^\\infty \\binom{\\alpha}{k}D^{\\alpha-k}u\\;D^k v.\u003C/script>\u003C/li>\u003C/ul>\u003Cp>Việc chọn không gian hàm (Hölder, Sobolev) và điều kiện ban đầu/cận biên phù hợp quyết định tính hợp lệ của định nghĩa và tính tồn tại–duy nhất nghiệm bài toán vi phân phân số.\u003C/p>\u003Ch2>Định nghĩa Riemann–Liouville và Caputo\u003C/h2>\u003Cp>Trong hai định nghĩa kinh điển, đạo hàm phân số Riemann–Liouville và Caputo khác nhau ở việc đặt phép vi phân bậc nguyên trước hay sau tích phân:\u003C/p>\u003Cp>\u003Cstrong>Riemann–Liouville:\u003C/strong>\u003C/p>\u003Cp>\u003Cscript type=\"math/tex\">D_{a}^{\\alpha}f(x)=\\frac{1}{\\Gamma(n-\\alpha)}\\frac{d^n}{dx^n}\\int_{a}^{x}(x-t)^{n-\\alpha-1}f(t)\\,dt,\\quad n-1\u003C\\alpha\u003Cn.\u003C/script>\u003C/p>\u003Cp>\u003Cstrong>Caputo:\u003C/strong>\u003C/p>\u003Cp>\u003Cscript type=\"math/tex\">{}^{C}D_{a}^{\\alpha}f(x)=\\frac{1}{\\Gamma(n-\\alpha)}\\int_{a}^{x}(x-t)^{n-\\alpha-1}f^{(n)}(t)\\,dt.\\end{script>\u003C/p> \u003Ctable> \u003Cthead> \u003Ctr>\u003Cth>Tiêu chí\u003C/th>\u003Cth>Riemann–Liouville\u003C/th>\u003Cth>Caputo\u003C/th>\u003C/tr> \u003C/thead> \u003Ctbody> \u003Ctr>\u003Ctd>Vị trí vi phân\u003C/td>\u003Ctd>Sau tích phân\u003C/td>\u003Ctd>Trước tích phân\u003C/td>\u003C/tr> \u003Ctr>\u003Ctd>Điều kiện đầu\u003C/td>\u003Ctd>Cần điều kiện dạng phân số\u003C/td>\u003Ctd>Dễ áp dụng điều kiện cổ điển \\(f^{(k)}(a)\\)\u003C/td>\u003C/tr> \u003Ctr>\u003Ctd>Ứng dụng\u003C/td>\u003Ctd>Phương trình biến tích phân phức tạp\u003C/td>\u003Ctd>Bài toán giá trị đầu thực tế\u003C/td>\u003C/tr> \u003C/tbody> \u003C/table> \u003Cp>Caputo được ưa chuộng hơn khi cần áp dụng điều kiện ban đầu truyền thống vì không yêu cầu tính toán đạo hàm phân số của hàm tại điểm a.\u003C/p> \u003Ch2>Tính chất toán học\u003C/h2> \u003Cp>Đạo hàm phân số kế thừa tính tuyến tính từ đạo hàm thông thường: với \\(c_1,c_2\\in\\mathbb{R}\\) và hàm đủ điều kiện, có \u003Cscript type=\"math/tex\">D^\\alpha[c_1 f + c_2 g] = c_1 D^\\alpha f + c_2 D^\\alpha g.\u003C/script>&nbsp;Tuy nhiên, phép hợp (semigroup) chỉ giữ khi xét trong không gian hàm thích hợp:&nbsp;\u003Cscript type=\"math/tex\">D^\\alpha D^\\beta f = D^{\\alpha+\\beta} f\u003C/script>&nbsp;nếu và chỉ nếu các điều kiện về tính khả tích và khả vi đủ chặt chẽ.\u003C/p>\u003Cp>Quy tắc nhân Leibniz mở rộng cho đạo hàm phân số được biểu diễn qua chuỗi vô hạn:&nbsp;\u003Cscript type=\"math/tex\">D^\\alpha(u\\,v)=\\sum_{k=0}^\\infty \\binom{\\alpha}{k}\\,D^{\\alpha-k}u\\;\\,D^k v,\u003C/script>&nbsp;đem lại khả năng phân tích tương tác giữa hai hàm nhưng đòi hỏi kiểm soát sai số khi áp dụng thực tế.\u003C/p>\u003Cul>\u003Cli>\u003Cstrong>Không giao hoán:\u003C/strong> nói chung \\(D^\\alpha D^\\beta \\neq D^\\beta D^\\alpha\\), phụ thuộc vào điều kiện biên.\u003C/li>\u003Cli>\u003Cstrong>Định lý truyền thống mở rộng:\u003C/strong> nếu \\(f^{(n)}\\) liên tục, có thể kết hợp tích phân và đạo hàm theo công thức nghịch đảo.\u003C/li>\u003C/ul>\u003Ch2>Các phương pháp tính toán\u003C/h2>\u003Cp>Ba cách định nghĩa chính được sử dụng trong tính toán thực nghiệm và lý thuyết:\u003C/p>\u003Col>\u003Cli>\u003Cstrong>Định nghĩa Grünwald–Letnikov:\u003C/strong>&nbsp;\u003Cscript type=\"math/tex\">D^\\alpha f(x)=\\lim_{h\\to 0}\\frac{1}{h^\\alpha}\\sum_{k=0}^{\\lfloor x/h\\rfloor}(-1)^k\\binom{\\alpha}{k}f(x-kh).\u003C/script>Phương pháp này dễ cài đặt nhưng yêu cầu lưu giữ toàn bộ lịch sử giá trị hàm, gây tốn bộ nhớ.\u003C/li>\u003Cli>\u003Cstrong>Định nghĩa Riemann–Liouville:\u003C/strong> sử dụng tích phân convolution với hạt nhân \\((x-t)^{n-\\alpha-1}\\), thuận tiện khi áp dụng Laplace transform cho phương trình tuyến tính.\u003C/li>\u003Cli>\u003Cstrong>Định nghĩa Caputo:\u003C/strong> phù hợp cho các bài toán giá trị đầu vì có thể dùng điều kiện ban đầu cổ điển \\(f^{(k)}(a)\\), thường được sử dụng trong mô hình sinh học và kỹ thuật.\u003C/li>\u003C/ol>\u003Cp>Khi triển khai số, thường kết hợp ba kỹ thuật chính để tính xấp xỉ đạo hàm phân số:\u003C/p>\u003Cfigure class=\"table\">\u003Ctable>\u003Cthead>\u003Ctr>\u003Cth>Phương pháp\u003C/th>\u003Cth>Ưu điểm\u003C/th>\u003Cth>Nhược điểm\u003C/th>\u003C/tr>\u003C/thead>\u003Ctbody>\u003Ctr>\u003Ctd>Chuỗi Grünwald–Letnikov\u003C/td>\u003Ctd>Tri thức trực tiếp từ định nghĩa\u003C/td>\u003Ctd>Bộ nhớ và tính toán cao\u003C/td>\u003C/tr>\u003Ctr>\u003Ctd>Laplace/Fourier transform\u003C/td>\u003Ctd>Giảm bài toán vi phân phân số thành đa thức trong biến phụ\u003C/td>\u003Ctd>Chỉ áp dụng cho hệ tuyến tính và biên xác định\u003C/td>\u003C/tr>\u003Ctr>\u003Ctd>Predictor–Corrector (Adams)\u003C/td>\u003Ctd>Độ chính xác cao, ổn định với bước nhỏ\u003C/td>\u003Ctd>Phức tạp, tốn bộ nhớ lưu lịch sử\u003C/td>\u003C/tr>\u003C/tbody>\u003C/table>\u003C/figure>\u003Ch2>Ứng dụng trong mô hình vật lý và sinh học\u003C/h2>\u003Cp>Trong cơ học và truyền nhiệt, phương trình phân số mô tả quá trình khuếch tán dị thường (anomalous diffusion) với động lực bước Lévy thay vì Brownian motion. Ví dụ, phương trình phân số thời gian:&nbsp;\u003Cscript type=\"math/tex\">\\frac{\\partial^\\alpha u}{\\partial t^\\alpha} = D\\,\\frac{\\partial^2 u}{\\partial x^2}\u003C/script>&nbsp;mô phỏng lan truyền nhiệt trong môi trường xốp hoặc sinh chất nhớ lâu (\u003Ca href=\"https://link.springer.com/article/10.1007/s00498-019-00212-9\">Springer\u003C/a>).\u003C/p>\u003Cp>Trong sinh học, mô hình phân số được áp dụng để miêu tả:\u003C/p>\u003Cul>\u003Cli>Chuyển động vi khuẩn nhớ hướng (chemotaxis) có độ nhớ dài.\u003C/li>\u003Cli>Lan truyền tín hiệu điện trong hệ thần kinh với hiệu ứng tích phân phân số thể hiện sự tích lũy và phân tán tín hiệu.\u003C/li>\u003C/ul>\u003Ch2>Ứng dụng trong xử lý tín hiệu và điều khiển\u003C/h2>\u003Cp>Bộ lọc fractional-order filters ứng dụng đạo hàm phân số giúp giảm méo pha và điều chỉnh đáp ứng tần số linh hoạt hơn so với bộ lọc integer-order truyền thống. Ví dụ, bộ lọc PIαDβ mở rộng phương pháp PID để điều khiển hệ thống có động học phức tạp và nhớ lâu (\u003Ca href=\"https://ieeexplore.ieee.org/document/6413821\">IEEE\u003C/a>).\u003C/p>\u003Cp>Trong robot và tự động hóa, bộ điều khiển fractional-order cho phép tối ưu hóa thời gian đáp ứng và giới hạn quá độ, cải thiện ổn định theo phản hồi trong hệ thống cơ–điện tử.\u003C/p>\u003Ch2>Phương pháp số và giả thiết tính toán\u003C/h2>\u003Cp>Các phương pháp số thường yêu cầu giả thiết hàm đủ mịn và điều kiện biên thích hợp. Ví dụ:\u003C/p>\u003Cul>\u003Cli>\u003Cstrong>Backward Euler phân số:\u003C/strong> Ổn định nhưng chỉ đạt thứ tự chính xác thấp, phù hợp bài toán thời gian dài.\u003C/li>\u003Cli>\u003Cstrong>Predictor–Corrector Adams:\u003C/strong> Độ chính xác cao, cần lưu lịch sử và tính toán nhiều lần để sửa lỗi.\u003C/li>\u003Cli>\u003Cstrong>Spectral methods:\u003C/strong> Hiệu quả với hàm mịn, đạt hội tụ nhanh nhưng kém với hàm không liên tục.\u003C/li>\u003C/ul>\u003Cfigure class=\"table\">\u003Ctable>\u003Cthead>\u003Ctr>\u003Cth>Phương pháp\u003C/th>\u003Cth>Độ chính xác\u003C/th>\u003Cth>Yêu cầu bộ nhớ\u003C/th>\u003C/tr>\u003C/thead>\u003Ctbody>\u003Ctr>\u003Ctd>Backward Euler\u003C/td>\u003Ctd>O(h^1)\u003C/td>\u003Ctd>Thấp\u003C/td>\u003C/tr>\u003Ctr>\u003Ctd>Predictor–Corrector Adams\u003C/td>\u003Ctd>O(h^{2-\\alpha})\u003C/td>\u003Ctd>Cao\u003C/td>\u003C/tr>\u003Ctr>\u003Ctd>Spectral\u003C/td>\u003Ctd>Exponential\u003C/td>\u003Ctd>Trung bình\u003C/td>\u003C/tr>\u003C/tbody>\u003C/table>\u003C/figure>\u003Ch2>Thách thức và vấn đề mở\u003C/h2>\u003Cp>Vẫn còn thiếu những định lý toàn diện về tồn tại và duy nhất nghiệm cho phương trình vi phân phân số phi tuyến. Việc xây dựng không gian hàm phù hợp và điều kiện biên phức tạp (nonlocal boundary conditions) là hướng nghiên cứu cấp thiết.\u003C/p>\u003Cp>Việc xác định chính xác tham số phân số từ dữ liệu thực nghiệm là bài toán nghịch đảo có tính không ổn định cao, đòi hỏi kết hợp regularization và phương pháp thống kê để ước lượng tham số tin cậy.\u003C/p>\u003Ch2>Xu hướng nghiên cứu và triển vọng\u003C/h2>\u003Cul>\u003Cli>Phát triển thuật toán AI và machine learning để ước lượng tham số α từ dữ liệu thời gian thực.\u003C/li>\u003Cli>Tích hợp mô hình phân số với mạng lưới phức hợp và mô hình lan truyền dịch bệnh nhớ lâu (memory epidemic models).\u003C/li>\u003Cli>Nghiên cứu tính ổn định và hội tụ của phương pháp số mới qua lý thuyết error analysis.\u003C/li>\u003Cli>Mở rộng ứng dụng vào tài chính định lượng để mô tả chuyển động giá tài sản có tính phụ thuộc lịch sử dài hạn.\u003C/li>\u003C/ul>\u003Ch2>Tài liệu tham khảo\u003C/h2>\u003Cul>\u003Cli>Podlubny I. \u003Ci>Fractional Differential Equations\u003C/i>. Academic Press; 1999.\u003C/li>\u003Cli>Kilbas AA, Srivastava HM, Trujillo JJ. \u003Ci>Theory and Applications of Fractional Differential Equations\u003C/i>. Elsevier; 2006.\u003C/li>\u003Cli>Machado JAT, Kiryakova V, Mainardi F. “Recent history of fractional calculus.” \u003Ci>Communications in Nonlinear Science and Numerical Simulation\u003C/i>. 2011;16(3):1140–1153. DOI:10.1016/j.cnsns.2010.06.019.\u003C/li>\u003Cli>Diethelm K. \u003Ci>The Analysis of Fractional Differential Equations\u003C/i>. Springer; 2010.\u003C/li>\u003Cli>Metzler R, Klafter J. “The Random Walk’s Guide to Anomalous Diffusion.” \u003Ci>Phys. Rep.\u003C/i> 2000;339(1):1–77. DOI:10.1016/S0370-1573(00)00070-3.\u003C/li>\u003C/ul>",{"id":314,"researcherId":17,"name":315,"imageUrl":316},2633,"Hoài Đặng","https://lh3.googleusercontent.com/a/ACg8ocLVXVafUxrnnC_H60UeQdpLE3UgeKGCSesypz_6Vi59B-n9C41A=s96-c",{"id":318,"researcherId":17,"name":319,"imageUrl":320},2681,"Cuong Ta","https://lh3.googleusercontent.com/a/ACg8ocKFuWh0ZD601Hp0_V_2xAK3kIOs20Kyv3V5ZLBo6fwfjyymJYzE=s96-c",[309],"PUBLISHED","2025-05-27T01:53:06.085+00:00","2025-09-09T06:27:03.839+00:00","2025-06-19T01:36:29.048+00:00",130,[328,331,334,337,340,343,346,349,352,355],{"id":329,"title":330},"capacity development","Capacity development là gì? Các công bố khoa học về Capacity development",{"id":332,"title":333},"chitin","Chitin là gì? Các công bố khoa học về Chitin",{"id":335,"title":336},"nạn nhân","Nạn nhân là gì? Các công bố khoa học về Nạn nhân",{"id":338,"title":339},"bã mía","Bã mía là gì? Các công bố khoa học về Bã mía",{"id":341,"title":342},"phản xạ h","Phản xạ h là gì? Các công bố khoa học về Phản xạ h",{"id":344,"title":345},"văn hóa truyền thống","Văn hóa truyền thống là gì? Các bài báo nghiên cứu liên quan",{"id":347,"title":348},"ước lượng tham số","Ước lượng tham số là gì? Các nghiên cứu khoa học về Ước lượng tham số",{"id":350,"title":351},"chuyển đơn phôi","Chuyển đơn phôi là gì? Các công bố khoa học về Chuyển đơn phôi",{"id":353,"title":354},"nghiên cứu thuốc","Nghiên cứu thuốc là gì? Các nghiên cứu khoa học liên quan",{"id":356,"title":357},"quang phổ phát xạ","Quang phổ phát xạ là gì? Các nghiên cứu khoa học liên quan",{"meta":359,"data":361},{"total":360},"46",[362,621,1014,1388,1512,1581,1851,1910,2027,2287],{"id":363,"createTime":364,"updateTime":365,"relativeEntities":366,"slug":367,"properties":368,"entityType":191,"verifyStatus":383,"verifyTime":364,"verifyNote":384,"syncStatus":385,"languages":386,"translateLanguages":388,"viewCount":22,"primaryUrl":390,"fullTextUrl":17,"authors":391,"publicationType":193,"publisherRelationship":446,"citationCount":532,"citationInfo":533,"publishDate":548,"publishYear":549,"citationAnalyzeStatus":385,"lastCitationAnalyze":17,"indexDatabases":17,"openAccess":17,"references":550,"isForceReanalyzing":24},"a5686c7f-c9e9-439f-b792-c51743636317","2024-09-02T10:54:52.265+00:00","2024-12-26T23:09:35.850+00:00",[],"Self-Consistent-Molecular-Orbital-Methods-XII-Further-Extensions-of-Gaussian-Type-Basis-Sets-for-Use-in-Molecular-Orbital-Studies-of-Organic-Molecules",{"mag":369,"keywords":371,"openalex":373,"abstract":375,"title":378,"doi":381},{"VOID":370},"2046412723",{"VI":372},"quỹ đạo phân tử, hàm cơ sở Gaussian, cacbon, flo, năng lượng tổng, cân bằng hình học, phân tử đa nguyên tử",{"VOID":374},"W2046412723",{"EN":376,"VI":377},"\u003Cjats:p>Two extended basis sets (termed 5–31G and 6–31G) consisting of atomic orbitals expressed as fixed linear combinations of Gaussian functions are presented for the first row atoms carbon to fluorine. These basis functions are similar to the 4–31G set [J. Chem. Phys. 54, 724 (1971)] in that each valence shell is split into inner and outer parts described by three and one Gaussian function, respectively. Inner shells are represented by a single basis function taken as a sum of five (5–31G) or six (6–31G) Gaussians. Studies with a number of polyatomic molecules indicate a substantial lowering of calculated total energies over the 4–31G set. Calculated relative energies and equilibrium geometries do not appear to be altered significantly.\u003C/jats:p>","\u003Cjats:p>Hai bộ cơ sở mở rộng (được gọi là 5–31G và 6–31G) bao gồm các hàm sóng nguyên tử được biểu diễn dưới dạng kết hợp tuyến tính cố định của các hàm Gaussian được trình bày cho các nguyên tố hàng đầu từ cacbon đến flo. Những hàm cơ sở này tương tự như bộ 4–31G [J. Chem. Phys. 54, 724 (1971)] ở chỗ mỗi lớp vỏ hóa trị được chia thành các phần bên trong và ngoài được mô tả tương ứng bằng ba và một hàm Gaussian. Các lớp vỏ bên trong được biểu diễn bởi một hàm cơ sở đơn lẻ, bao gồm tổng của năm (5–31G) hoặc sáu (6–31G) hàm Gaussian. Nghiên cứu với một số phân tử đa nguyên tử cho thấy giảm đáng kể năng lượng tổng tính toán so với bộ 4–31G. Tính toán năng lượng tương đối và hình học cân bằng dường như không thay đổi đáng kể.\u003C/jats:p>",{"EN":379,"VI":380},"Self—Consistent Molecular Orbital Methods. XII. Further Extensions of Gaussian—Type Basis Sets for Use in Molecular Orbital Studies of Organic Molecules","\nPhương pháp quỹ đạo phân tử tự nhất quán. XII. Phát triển bổ sung bộ cơ sở dạng Gaussian cho nghiên cứu quỹ đạo phân tử của các hợp chất hữu cơ",{"VOID":382},"10.1063/1.1677527","VERIFIED","Auto Verify","PENDING",[387],"EN",[389],"VI","https://pubs.aip.org/jcp/article/56/5/2257/908679/Self-Consistent-Molecular-Orbital-Methods-XII",[392,414,431],{"id":393,"sortIndex":34,"researcher":17,"roles":394,"affiliations":395,"properties":407},"604b3225-1ebd-46fe-8632-7495b6d7ec9e",[],[396],{"id":397,"sortIndex":22,"affiliation":398,"properties":17},"fa40d850-5618-412c-a17b-4a1aeb335f57",{"id":399,"createTime":400,"updateTime":401,"relativeEntities":402,"slug":403,"properties":404,"entityType":21,"verifyStatus":385,"verifyTime":17,"verifyNote":17,"syncStatus":385,"languages":17,"translateLanguages":17,"viewCount":22},"09d1f5da-5e1b-41c0-80a2-420ab307729a","2023-12-07T07:52:43.031+00:00","2025-06-11T22:45:07.159+00:00",[],"Department-of-Chemistry-Carnegie-Mellon-University-Pittsburgh-Pennsylvania-15213-",{"title":405},{"VI":406},"Department of Chemistry, Carnegie-Mellon University, Pittsburgh, Pennsylvania 15213.",{"openalex":408,"orcid":410,"title":412},{"VOID":409},"A5088465776",{"VOID":411},"https://orcid.org/0000-0003-3984-1360",{"EN":413},"R. 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Hehre",{"id":432,"sortIndex":47,"researcher":17,"roles":433,"affiliations":434,"properties":441},"d055794c-3783-4def-bc84-5a7a4db25db0",[],[435],{"id":436,"sortIndex":22,"affiliation":437,"properties":17},"f5cc48cb-ad2b-493d-9a42-8c273a32e3f2",{"id":399,"createTime":400,"updateTime":401,"relativeEntities":438,"slug":403,"properties":439,"entityType":21,"verifyStatus":385,"verifyTime":17,"verifyNote":17,"syncStatus":385,"languages":17,"translateLanguages":17,"viewCount":22},[],{"title":440},{"VI":406},{"openalex":442,"title":444},{"VOID":443},"A5038387078",{"EN":445},"John A. Pople",{"url":17,"publisher":447,"properties":525},{"id":448,"createTime":449,"updateTime":450,"relativeEntities":451,"slug":452,"properties":453,"entityType":264,"verifyStatus":385,"verifyTime":17,"verifyNote":17,"syncStatus":385,"languages":17,"translateLanguages":17,"viewCount":22,"subjectFields":464,"manageAffiliations":465,"indexDatabases":466,"url":508,"thumbnailPath":17,"statistic":509,"gsStatistic":17,"type":17,"analyzePriority":17},"b27a65e0-e677-4c05-9546-5a82122191e4","2023-05-29T10:40:18.930+00:00","2025-02-11T05:15:36.972+00:00",[],"Journal-of-Chemical-Physics",{"country":454,"issn":456,"introduce":458,"eissn":460,"title":462},{"VOID":455},"US",{"VOID":457},"10897690",{"EN":459},"The Journal of Chemical Physics publishes quantitative and rigorous science of long-lasting value in methods and applications of chemical physics. The Journal also publishes brief Communications of significant new findings, Perspectives on the latest advances in the field, and Special Topic issues. The Journal focuses on innovative research in experimental and theoretical areas of chemical physics, including spectroscopy, dynamics, kinetics, statistical mechanics, and quantum mechanics. In addition, topical areas such as polymers, soft matter, materials, surfaces/interfaces, and systems of biological relevance are of increasing importance. Topical coverage includes: Theoretical Methods and Algorithms Advanced Experimental Techniques Atoms, Molecules, and Clusters Liquids, Glasses, and Crystals Surfaces, Interfaces, and Materials Polymers and Soft Matter Biological Molecules and Networks.",{"VOID":461},"00219606",{"EN":463},"Journal of Chemical Physics",[],[],[467,488],{"id":468,"indexDatabase":469,"url":483,"indexYears":17,"academicFieldIds":484},"10841295-20fc-4732-836c-b22051fdb593",{"id":470,"createTime":471,"updateTime":472,"relativeEntities":473,"label":474,"description":476,"key":479,"publicationTags":480,"standard":17},"a4921856-b128-4d9f-8f1f-e80813d3bbd4","2023-05-22T09:59:31.026+00:00","2025-01-14T09:58:12.199+00:00",[],{"EN":475,"VI":475},"ISI/SCIE - Science Citation Index Expanded",{"VI":477,"EN":478},"Cơ sở dữ liệu SCIE","SCIE database","scie",[481,482],"SCIE","ISI","https://mjl.clarivate.com/search-results?issn=0021-9606",[485,486,487],"bb53981f-29b6-4b16-89ea-99cf1e4357c7","d6c3d63f-ea74-46da-84a0-f1951ff6d9b8","1f780287-8dee-4820-a6a9-dc25c634f349",{"id":489,"indexDatabase":490,"url":502,"indexYears":503,"academicFieldIds":504},"a54f8fec-e3a1-4a0a-89f7-9c60c0145e2d",{"id":491,"createTime":492,"updateTime":493,"relativeEntities":494,"label":495,"description":497,"key":499,"publicationTags":500,"standard":17},"3c7051d4-eb7d-4c57-a56b-36fc74c5d1e9","2023-05-22T09:57:18.509+00:00","2025-07-10T09:55:58.024+00:00",[],{"EN":496,"VI":496},"Scopus - Elsevier",{"EN":496,"VI":498},"Cơ sở dữ liệu Scopus thuộc Elsevier","scopus",[501],"SCOPUS","https://www.scopus.com/sourceid/28134","1933-2023",[505,506,507],"0c2541d6-7567-4793-aad8-840b3d33f5e0","e89cebf4-fecf-4b74-8ab4-88fca891e4c2","b4efb51f-da98-4990-a79e-a5f531f11b28","https://aip.scitation.org/journal/jcp",{"impactFactor":22,"impactFactorByYear":510,"i10Index":512,"i10IndexLast5Year":22,"totalPublication":512,"totalPublicationByYear":514,"totalCitation":515,"totalCitationByYear":516,"totalCitationPerPublication":522,"totalCitationPerPublicationByYear":523,"hindexLast5Year":512,"hindex":512},{"1995":71,"1996":511,"2015":512,"2016":513},2.33,6,10,{"1994":34,"1995":47,"2001":34,"2002":34,"2014":34},241,{"1994":517,"1995":518,"2001":519,"2002":520,"2014":521},25,132,16,24,44,40.17,{"1994":517,"1995":524,"2001":519,"2002":520,"2014":521},66,{"volume":526,"pages":528,"issue":530},{"VOID":527},"56",{"VOID":529},"2257-2261",{"VOID":531},"5",13956,{"total":532,"publishYear":17,"statisticByYear":534},{"2012":535,"2013":536,"2014":537,"2015":538,"2016":539,"2017":540,"2018":541,"2019":542,"2020":543,"2021":544,"2022":545,"2023":546,"2024":547},524,529,580,592,625,605,609,626,657,728,674,780,428,"1972-03-01",1972,[551,555,558,562,566,570,573,577,581,585,589,593,597,601,605,609,613,617],{"id":17,"text":552,"url":17,"identifiers":553},"1966, J. Chem. Phys., 45, 2593, 10.1063/1.1727979",{"doi":554},"10.1063/1.1727979",{"id":17,"text":556,"url":17,"identifiers":557},"1967, J. Computational Phys., 2, 223",{},{"id":17,"text":559,"url":17,"identifiers":560},"1967, J. Chem. Phys., 46, 4725, 10.1063/1.1840626",{"doi":561},"10.1063/1.1840626",{"id":17,"text":563,"url":17,"identifiers":564},"1967, J. Chem. Phys., 46, 2759, 10.1063/1.1841110",{"doi":565},"10.1063/1.1841110",{"id":17,"text":567,"url":17,"identifiers":568},"1968, J. Chem. Phys., 49, 2056, 10.1063/1.1670367",{"doi":569},"10.1063/1.1670367",{"id":17,"text":571,"url":17,"identifiers":572},"1967, J. Chem. Phys., 47, 201",{},{"id":17,"text":574,"url":17,"identifiers":575},"1968, J. Chem. Phys., 49, 5007, 10.1063/1.1669992",{"doi":576},"10.1063/1.1669992",{"id":17,"text":578,"url":17,"identifiers":579},"1969, J. Am. Chem. Soc., 91, 2189, 10.1021/ja01037a001",{"doi":580},"10.1021/ja01037a001",{"id":17,"text":582,"url":17,"identifiers":583},"1967, J. Chem. 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Letters, 5, 13, 10.1016/0009-2614(70)80116-6",{"doi":612},"10.1016/0009-2614(70)80116-6",{"id":17,"text":614,"url":17,"identifiers":615},"1970, J. Am. Chem. Soc., 92, 4796, 10.1021/ja00719a006",{"doi":616},"10.1021/ja00719a006",{"id":17,"text":618,"url":17,"identifiers":619},"1971, J. Am. Chem. Soc., 93, 289, 10.1021/ja00731a001",{"doi":620},"10.1021/ja00731a001",{"id":622,"createTime":623,"updateTime":624,"relativeEntities":625,"slug":626,"properties":627,"entityType":191,"verifyStatus":383,"verifyTime":624,"verifyNote":384,"syncStatus":385,"languages":642,"translateLanguages":643,"viewCount":22,"primaryUrl":644,"fullTextUrl":17,"authors":645,"publicationType":193,"publisherRelationship":686,"citationCount":755,"citationInfo":795,"publishDate":798,"publishYear":799,"citationAnalyzeStatus":800,"lastCitationAnalyze":801,"indexDatabases":17,"openAccess":17,"references":802,"isForceReanalyzing":24},"e54dd51e-e37d-4c97-9bb4-cc615d72e111","2024-04-13T18:58:47.862+00:00","2025-01-15T08:42:15.736+00:00",[],"On-the-existence-of-solutions-for-a-multi-singular-pointwise-defined-fractional-q-integro-differential-equation",{"mag":628,"keywords":630,"openalex":632,"abstract":634,"title":637,"doi":640},{"VOID":629},"3030948033",{"VI":631},"phương trình phân số, đạo hàm q-phân số, điều kiện biên, tồn tại nghiệm, phương trình integro-diện tích",{"VOID":633},"W3030948033",{"VI":635,"EN":636},"\u003Cjats:title>Tóm tắt\u003C/jats:title>\u003Cjats:p>Bằng cách sử dụng đạo hàm phân số loại Caputo và đạo hàm phân số loại Riemann–Liouville, chúng tôi nghiên cứu sự tồn tại của nghiệm cho một phương trình phân dị tổ hợp q-mẫu đa điểm được xác định tại các điểm với một số điều kiện biên. Trên thực tế, chúng tôi đưa ra một số kết quả bằng cách xem xét các điều kiện khác nhau và sử dụng một số kỹ thuật điểm cố định cổ điển cũng như định lý hội tụ chi phối Lebesgue.\u003C/jats:p>","\u003Cjats:title>Abstract\u003C/jats:title>\u003Cjats:p>By using the Caputo type and the Riemann–Liouville type fractional\u003Cjats:italic>q\u003C/jats:italic>-derivative, we investigate the existence of solutions for a multi-term pointwise defined fractional\u003Cjats:italic>q\u003C/jats:italic>-integro-differential equation with some boundary value conditions. In fact, we give some results by considering different conditions and using some classical fixed point techniques and the Lebesgue dominated convergence theorem.\u003C/jats:p>",{"VI":638,"EN":639},"Về sự tồn tại của các nghiệm cho phương trình phân dị tổ hợp q-mẫu đa điểm được xác định điểm","On the existence of solutions for a multi-singular pointwise defined fractional q-integro-differential 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F.H.: q-Difference equations. 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Model. 57, 1053–1059 (2013)",{"doi":1013},"10.1016/j.mcm.2012.06.024",{"id":1015,"createTime":1016,"updateTime":1017,"relativeEntities":1018,"slug":1019,"properties":1020,"entityType":191,"verifyStatus":383,"verifyTime":1039,"verifyNote":384,"syncStatus":385,"languages":1040,"translateLanguages":1041,"viewCount":22,"primaryUrl":1042,"fullTextUrl":17,"authors":1043,"publicationType":193,"publisherRelationship":1105,"citationCount":520,"citationInfo":1194,"publishDate":1196,"publishYear":1197,"citationAnalyzeStatus":800,"lastCitationAnalyze":1198,"indexDatabases":17,"openAccess":17,"references":1199,"isForceReanalyzing":24},"2ba9e150-dd64-4a58-a871-e5fa632a24a4","2024-04-14T18:06:19.697+00:00","2025-02-15T21:59:32.776+00:00",[],"Theoretical-and-numerical-analysis-for-transmission-dynamics-of-COVID-19-mathematical-model-involving-Caputo-Fabrizio-derivative",{"mag":1021,"keywords":1023,"pmc":1025,"openalex":1027,"abstract":1029,"title":1032,"pm":1035,"doi":1037},{"VOID":1022},"3143946795",{"VI":1024},"COVID-19, mô hình toán học, đạo hàm Caputo–Fabrizio, phương pháp lặp Picard, biến đổi Laplace, phân hoạch Adomian",{"VOID":1026},"7988648",{"VOID":1028},"W3143946795",{"EN":1030,"VI":1031},"\u003Cjats:title>Abstract\u003C/jats:title>\u003Cjats:p>This manuscript is devoted to a study of the existence and uniqueness of solutions to a mathematical model addressing the transmission dynamics of the coronavirus-19 infectious disease (COVID-19). The mentioned model is considered with a nonsingular kernel type derivative given by Caputo–Fabrizo with fractional order. For the required results of the existence and uniqueness of solution to the proposed model, Picard’s iterative method is applied. Furthermore, to investigate approximate solutions to the proposed model, we utilize the Laplace transform and Adomian’s decomposition (LADM). Some graphical presentations are given for different fractional orders for various compartments of the model under consideration.\u003C/jats:p>","\u003Cjats:title>Tóm tắt\u003C/jats:title>\u003Cjats:p>Bài viết này tập trung vào nghiên cứu sự tồn tại và duy nhất của các nghiệm cho một mô hình toán học liên quan đến động lực truyền bệnh truyền nhiễm coronavirus-19 (COVID-19). Mô hình đã đề cập được xem xét với một đạo hàm dạng hạt nhân phi kỳ có chỉ số cấp thấp do Caputo–Fabrizio cung cấp. Để đạt được kết quả cần thiết về sự tồn tại và duy nhất của nghiệm cho mô hình đề xuất, phương pháp lặp Picard đã được áp dụng. Hơn nữa, để điều tra các nghiệm gần đúng cho mô hình đề xuất, chúng tôi sử dụng biến đổi Laplace và phân hoạch Adomian (LADM). Một số biểu diễn đồ họa được cung cấp cho các chỉ số cấp thấp khác nhau cho các thành phần khác nhau của mô hình đang xem xét.\u003C/jats:p>",{"EN":1033,"VI":1034},"Theoretical and numerical analysis for transmission dynamics of COVID-19 mathematical model involving Caputo–Fabrizio derivative","Phân tích lý thuyết và số học cho động lực truyền bệnh COVID-19 dựa trên mô hình toán học liên quan đến đạo hàm Caputo–Fabrizio",{"VOID":1036},"33777126",{"VOID":1038},"10.1186/s13662-021-03316-w","2025-02-05T21:24:37.942+00:00",[387],[389],"https://advancesindifferenceequations.springeropen.com/articles/10.1186/s13662-021-03316-w",[1044,1064,1085],{"id":1045,"sortIndex":47,"researcher":17,"roles":1046,"affiliations":1047,"properties":1057},"70dbdebf-0239-46fe-9028-8c7e03ef2ad7",[],[1048],{"id":17,"sortIndex":22,"affiliation":1049,"properties":17},{"id":1050,"createTime":1051,"updateTime":1051,"relativeEntities":1052,"slug":1053,"properties":1054,"entityType":21,"verifyStatus":385,"verifyTime":17,"verifyNote":17,"syncStatus":385,"languages":17,"translateLanguages":17,"viewCount":22},"eb825299-4a56-430e-8532-de176fa75bd5","2024-04-14T18:06:19.735+00:00",[],"Department-of-Mathematics-University-of-Malakand-Chakdara-Dir-L-KPK-Pakistan",{"title":1055},{"EN":1056},"Department of Mathematics, University of Malakand, Chakdara, Dir(L), KPK, Pakistan",{"openalex":1058,"orcid":1060,"title":1062},{"VOID":1059},"A5073895350",{"VOID":1061},"https://orcid.org/0000-0002-8851-4844",{"EN":1063},"Kamal Shah",{"id":1065,"sortIndex":22,"researcher":17,"roles":1066,"affiliations":1067,"properties":1078},"0509a321-d044-45f2-b6dc-63ac812ce3fd",[],[1068],{"id":17,"sortIndex":22,"affiliation":1069,"properties":17},{"id":1070,"createTime":1071,"updateTime":1072,"relativeEntities":1073,"slug":1074,"properties":1075,"entityType":21,"verifyStatus":385,"verifyTime":17,"verifyNote":17,"syncStatus":385,"languages":17,"translateLanguages":17,"viewCount":22},"dda198c9-8229-4d0b-aac0-8b53fca6657c","2024-04-08T10:53:51.351+00:00","2024-07-24T12:56:25.875+00:00",[],"Department-of-Mathematics-University-of-Aden-Aden-Yemen",{"title":1076},{"VI":1077},"Department of Mathematics, University of Aden, Aden, Yemen",{"openalex":1079,"orcid":1081,"title":1083},{"VOID":1080},"A5023573264",{"VOID":1082},"https://orcid.org/0000-0002-4568-9732",{"EN":1084},"Sabri T. M. Thabet",{"id":1086,"sortIndex":34,"researcher":17,"roles":1087,"affiliations":1088,"properties":1098},"2ba2f383-d474-447c-8280-84ae635bc0af",[],[1089],{"id":17,"sortIndex":22,"affiliation":1090,"properties":17},{"id":1091,"createTime":1092,"updateTime":1092,"relativeEntities":1093,"slug":1094,"properties":1095,"entityType":21,"verifyStatus":385,"verifyTime":17,"verifyNote":17,"syncStatus":385,"languages":17,"translateLanguages":17,"viewCount":22},"42568468-4838-4769-aaf5-051de9488fdc","2024-04-14T18:06:19.722+00:00",[],"Department-of-Mathematics-Hodeidah-University-Hodeidah-Yemen",{"title":1096},{"EN":1097},"Department of Mathematics, Hodeidah University, Hodeidah, Yemen",{"openalex":1099,"orcid":1101,"title":1103},{"VOID":1100},"A5070524846",{"VOID":1102},"https://orcid.org/0000-0001-9085-324X",{"EN":1104},"Mohammed S. Abdo",{"url":17,"publisher":1106,"properties":17},{"id":1107,"createTime":1108,"updateTime":1109,"relativeEntities":1110,"slug":692,"properties":1111,"entityType":264,"verifyStatus":385,"verifyTime":17,"verifyNote":17,"syncStatus":385,"languages":17,"translateLanguages":17,"viewCount":22,"subjectFields":1115,"manageAffiliations":1116,"indexDatabases":1117,"url":17,"thumbnailPath":17,"statistic":1129,"gsStatistic":17,"type":17,"analyzePriority":17},"161338e6-8909-4fe0-8029-847e19a5399f","2024-04-19T02:42:48.933+00:00","2025-01-14T07:19:21.856+00:00",[],{"eissn":1112,"title":1114},{"VOID":1113},"1687-1847",{"EN":699},[],[],[1118],{"id":1119,"indexDatabase":1120,"url":1125,"indexYears":1126,"academicFieldIds":1127},"58f80c7a-2f77-464e-9551-7a61963caa78",{"id":491,"createTime":492,"updateTime":493,"relativeEntities":1121,"label":1122,"description":1123,"key":499,"publicationTags":1124,"standard":17},[],{"EN":496,"VI":496},{"EN":496,"VI":498},[501],"https://www.scopus.com/sourceid/4000149605","2004-2021",[713,714,1128],"f58e9119-c14c-473f-8a7c-5a035e4ab69a",{"impactFactor":22,"impactFactorByYear":1130,"i10Index":1141,"i10IndexLast5Year":1142,"totalPublication":1143,"totalPublicationByYear":1144,"totalCitation":1158,"totalCitationByYear":1159,"totalCitationPerPublication":1177,"totalCitationPerPublicationByYear":1178,"hindexLast5Year":760,"hindex":760},{"2012":1131,"2013":1132,"2014":1133,"2015":1134,"2016":1132,"2017":1135,"2018":1136,"2019":1137,"2020":1138,"2021":1139,"2022":1140,"2023":736},0.42,0.18,0.17,0.09,0.22,0.32,0.71,0.62,0.55,0.74,177,67,2795,{"2004":512,"2005":58,"2006":743,"2007":1145,"2008":797,"2009":519,"2010":1146,"2011":1147,"2012":1148,"2013":1149,"2014":1150,"2015":1151,"2016":1152,"2017":1153,"2018":1154,"2019":1155,"2020":1156,"2021":1157},7,28,57,152,263,215,101,189,246,336,361,454,328,6043,{"2004":519,"2005":1160,"2006":1161,"2007":1162,"2008":1163,"2009":1164,"2010":1165,"2011":1166,"2012":1167,"2013":1168,"2014":1169,"2015":1170,"2016":1171,"2017":1172,"2018":1173,"2019":1174,"2020":1175,"2021":1176},109,50,69,32,43,87,114,259,406,187,490,271,681,925,568,1035,644,2.16,{"2004":1179,"2005":1180,"2006":1181,"2007":1182,"2008":71,"2009":1183,"2010":1184,"2011":47,"2012":1185,"2013":1186,"2014":1187,"2015":1188,"2016":1189,"2017":1190,"2018":791,"2019":1191,"2020":1192,"2021":1193},2.67,36.33,2.94,9.86,2.69,3.11,1.7,1.54,0.87,4.85,1.43,2.77,1.57,2.28,1.96,{"total":520,"publishYear":17,"statisticByYear":1195},{"2021":58,"2022":1145,"2023":513,"2024":58},"2021-12-01",2021,"2024-04-15T08:49:28.558+00:00",[1200,1204,1208,1212,1216,1220,1223,1226,1230,1234,1237,1241,1245,1249,1252,1256,1260,1264,1268,1271,1274,1278,1282,1286,1290,1294,1298,1302,1306,1310,1314,1318,1322,1326,1330,1334,1338,1342,1346,1349,1353,1357,1361,1365,1369,1373,1377,1381,1384],{"id":17,"text":1201,"url":17,"identifiers":1202},"Jeffrey, S.K., Kenneth, M.: History and recent advances in coronavirus discovery. 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Appl. 8(1), 17–25 (2018)",{"doi":1360},"10.11121/ijocta.01.2018.00532",{"id":17,"text":1362,"url":17,"identifiers":1363},"Hussain, A., Baleanu, D., Adeel, M.: Existence of solution and stability for the fractional order novel coronavirus (nCoV-2019) model. Adv. Differ. Equ. 2020, 384 (2020)",{"doi":1364},"10.1186/s13662-020-02845-0",{"id":17,"text":1366,"url":17,"identifiers":1367},"Thabet, S.T.M., Abdo, M.S., Shah, K., Abdeljawad, T.: Study of transmission dynamics of COVID-19 mathematical model under ABC fractional order derivative. Results Phys. 19, 103507 (2020)",{"doi":1368},"10.1016/j.rinp.2020.103507",{"id":17,"text":1370,"url":17,"identifiers":1371},"Shah, N.H., Suthar, A.H., Jayswal, E.N.: Control strategies to curtail transmission of COVID-19. Int. J. Math. Math. Sci. 2020, 12 (2020). https://doi.org/10.1155/2020/2649514",{"doi":1372},"10.1155/2020/2649514",{"id":17,"text":1374,"url":17,"identifiers":1375},"Losada, J., Nieto, J.J.: Properties of a new fractional derivative without singular kernel. Prog. Fract. Differ. Appl. 1(2), 87–92 (2015). https://doi.org/10.12785/pfda/010202",{"doi":1376},"10.12785/pfda/010202",{"id":17,"text":1378,"url":17,"identifiers":1379},"Lyons, R., Vatsala, A., Chiquet, R.: Picard’s iterative method for Caputo fractional differential equations with numerical results. Mathematics 5(4), 65 (2017)",{"doi":1380},"10.3390/math5040065",{"id":17,"text":1382,"url":17,"identifiers":1383},"Rida, S.Z., Arafa, A.A.M., Gaber, Y.A.: Solution of the fractional epidemic model by L-ADM. J. Fract. Calc. Appl. 7(1), 189–195 (2016)",{},{"id":17,"text":1385,"url":17,"identifiers":1386},"Shah, K., Abdeljawad, T., Jarad, F.: On a nonlinear fractional order model of Dengue fever disease under Caputo–Fabrizio derivative. Alex. Eng. J. 59(4), 2305–2313 (2020)",{"doi":1387},"10.1016/j.aej.2020.02.022",{"id":1389,"createTime":1390,"updateTime":1391,"relativeEntities":1392,"slug":1393,"properties":1394,"entityType":191,"verifyStatus":383,"verifyTime":1408,"verifyNote":1409,"syncStatus":385,"languages":17,"translateLanguages":17,"viewCount":22,"primaryUrl":1410,"fullTextUrl":1411,"authors":1412,"publicationType":193,"publisherRelationship":1420,"citationCount":22,"citationInfo":1508,"publishDate":1510,"publishYear":799,"citationAnalyzeStatus":1511,"lastCitationAnalyze":1391,"indexDatabases":17,"openAccess":17,"references":17,"isForceReanalyzing":24},"49a3eab2-48fa-48f5-a5c1-01447f3eff11","2023-07-31T18:35:30.438+00:00","2025-02-14T18:49:24.469+00:00",[],"QU%E1%BA%A2NG-C%C3%81O-NH%C6%AF-M%E1%BB%98T-PH%E1%BA%A6N-C%E1%BB%A6A-T%C3%8CNH-HU%E1%BB%90NG-CU%E1%BB%98C-S%E1%BB%90NG-%C4%90%E1%BB%98NG-L%E1%BB%B0C-CHO-VI%E1%BB%86C-D%E1%BA%A0Y-H%E1%BB%8CC-HO%C3%81-H%E1%BB%8CC-V%C3%80-S%E1%BB%B0-PH%E1%BA%A2N-H%E1%BB%92I-V%E1%BB%80-PH%C6%AF%C6%A0NG-PH%C3%81P-D%E1%BA%A0Y-H%E1%BB%8CC-C%E1%BB%A6A-H%E1%BB%8CC-VI%C3%8AN-CAO-H%E1%BB%8CC-T%E1%BA%A0I-TR%C6%AF%E1%BB%9CNG-%C4%90%E1%BA%A0I-H%E1%BB%8CC-S%C6%AF-PH%E1%BA%A0M-TP-HCM-T%E1%BB%AA-TH%C3%81NG-11-%C4%90%E1%BA%BEN-TH%C3%81NG-12-N%C4%82M-2018-PH%E1%BA%A6N-2",{"gsPaper":1395,"keywords":1397,"abstract":1400,"title":1403,"doi":1406},{"VOID":1396},"18367352089487344567",{"EN":1398,"VI":1399},"Didactics in Higher Education,the conception of pupil-orientation,learning processes of master students,teacher training in Chemistry Education,advertisement","phương pháp giảng dạy ở bậc đại học,khái niệm dạy học định hướng học sinh,các quá trình học tập của học viên cao học,đào tạo giáo viên sư phạm Hóa học,quảng cáo",{"EN":1401,"VI":1402},"     This article describes possibilities of higher education didactic to qualify the students’ professional knowledge about pupil behaviour in the frame of the teacher training. The article is a plea to teach Chemistry more pupil-orientated than presently. The conception “pupil orientation” is considered in actual education policy-initiated efforts in Vietnam. The author concretizes his previous explanations in the “Journal of Science (HCMCUE)” about “pupil-orientation” relating to own experiences of a methodological seminar for master students. At the same time, a Didactic of the Didactic of Chemistry is emphasized. Part 2 analyses and interprets the students' achievements on the background of the seminar concept. Consequences for the teacher training in Methodology are discussed, possible changes are justified. ","     Bài báo trình bày về khả năng vận dụng ditactic ở bậc sau đại học để dánh giá hiểu biết của sinh viên về thái độ của học sinh trong khung đào tạo giáo viên. Bài báo cũng đưa ra khuyến nghị việc dạy học Hoá học định hướng vào học sinh nhiều hơn so với hiện nay. Khái niệm về “dạy học hướng đến học sinh” được xem xét dựa trên những nỗ lực về chính sách về giáo dục ở Việt Nam. Tác giả đã cụ thể hoá những giải thích về khái niệm này dựa trên tổ chức hoạt động thuyết trình về phương pháp dạy học cho học viên cao học trên Tạp chí Khoa học của Trường Đại học Sư phạm Thành phố Hồ Chí Minh. Đồng thời, ditactic về ditactic Hoá học cũng được nhấn mạnh. Phần 2 của bài báo phân tích và diễn giải những kết quả của học viên dựa trên nội dung của bài seminar. Bài báo cũng chỉ ra những hệ quả của việc đào tạo giáo viên về Phương pháp giảng dạy và cũng đề xuất những điều chỉnh có thể thực hiện.   ",{"EN":1404,"VI":1405},"COMMERCIALS AS PART OF DAILY LIFE SITUATIONS - IMPULSES FOR CHEMISTRY TEACHING AND FOR METHODICAL REFLECTIONS OF MS-COURSE STUDENTS OF THE HO CHI MINH CITY UNIVERSITY OF EDUCATION/DEPARTMENT CHEMISTRY IN NOV./DEC.2018 (PART 2)","QUẢNG CÁO NHƯ MỘT PHẦN CỦA TÌNH HUỐNG CUỘC SỐNG - ĐỘNG LỰC CHO VIỆC DẠY HỌC HOÁ HỌC VÀ SỰ PHẢN HỒI VỀ PHƯƠNG PHÁP DẠY HỌC CỦA HỌC VIÊN CAO HỌC TẠI TRƯỜNG ĐẠI HỌC SƯ PHẠM TP.HCM (TỪ THÁNG 11 ĐẾN THÁNG 12 NĂM 2018) PHẦN 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số Lie, đại số Lie toàn phương, đối đồng điều, đạo hàm phản xứng",{"VI":1524}," Trong b&agrave;i b&aacute;o n&agrave;y, ch&uacute;ng t&ocirc;i t&iacute;nh to&aacute;n to&agrave;n bộ số Betti của c&aacute;c đại số Lie to&agrave;n phương giải được c&oacute; số chiều &pound; 7 vừa được ph&acirc;n loại trong [5]. 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Some new results on the existence and uniqueness of a solution for the model are obtained as well as the Ulam stability of the solutions. Two examples are provided to show the applicability of our results.\u003C/jats:p>","\u003Cjats:title>Tóm tắt\u003C/jats:title>\u003Cjats:p>Trong bài báo này, chúng tôi nghiên cứu một bài toán giá trị biên phi tuyến với p-Laplacian chứa cả đạo hàm phân số Riemann–Liouville trái và đạo hàm Caputo phải với các điều kiện khởi đầu và điều kiện tích phân. Một số kết quả mới về sự tồn tại và duy nhất của nghiệm cho mô hình được trình bày, cùng với sự ổn định Ulam của các nghiệm. 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Repositioning CoDesign in the age of platform capitalism: from sharing to caring. International Journal of CoCreation in Design and the Arts 15 (3): 185–191.\nBalakrishnan, Radhika, James Heintz, and Diane Elson. 2016. Rethinking Economic Policy for Social Justice. London and New York: Routledge.\nBidegain Ponte, Nicole, Marina Durano, and Corina Rodriguez Enriquez. 2015. Shifting Responsibilities without Changing the Balance of Power: What Chance of Equality with the Addis Ababa Action Agenda? http://www.unrisd.org/road-to-addis-bidegain-et-al.\nBreen, Keith. 2015. Law beyond command? An evaluation of Arendt’s Understanding of Law. In Hannah Arendt and the Law, Marco Goldoni and Christopher McCorkindale (eds.). New York: Hart Publishing.\nChang, Ha-Joon. 2001. Breaking the Mould: An Institutionalist Political Economy Alternative to the Neoliberal Theory of the Market and the State, UNRISD Social Policy and Development Programme Paper No. 6. Geneva, May.\nCurato, Nicole, John S. Dryzek, Selen A. Ercan, Carolyn M. Hendriks, and Simon Niemeyer. 2017. Twelve Key Findings in Deliberative Democracy Research, Daedalus (Summer): 28–38. https://www.amacad.org/sites/default/files/daedalus/downloads/17_Summer_Daedalus.pdf.\nDavid, Sarah, Catalina Gil Pinzon, Elisa Lorenz, and Antonia Schmidt. 2019. Raising voices: Empowering female farmers in drug crop cultivation areas. Bonn: BMZ and OSF.\nDjelic, Marie-Laure, and Sigrid Quack. 2003. Governing Globalization-Bringing Institutions Back In. In Globalization and Institutions: Redefining the Rules of the Economic Game, Marie-Laure. Djelic and Sigrid Quack (eds.). Cheltenham: Edward Elgar.\nDurano, Marina, and Nicole Bidegain-Ponte. 2016. A Feminist Perspective on the Follow-Up Process for Financing for Development. Development 59(1–2): 32–39. https://doi.org/10.1057/s41301-017-0075-z.\nFischer, Bernice, and Joan C. Tronto. 1990. Toward a Feminist Theory of Care. In Circles of Care: Work and Identity in Women’s Lives, Emily K. Abel and Margaret K. Nelson (eds.). New York: State University of New York Press, SUNY Series on Women and Work.\nFolbre, Nancy, and Erik Olin Wright. 2012. Defining Care. In For Love and Money: Care Provision in the United States, Nancy Folbre (ed.). New York: Russell Sage Foundation.\nFraser, Nancy. 2013. Fortunes of Feminism: From State-managed Capitalism to Neoliberal Crisis. New York: Verso Books.\nGearhart-Serna, Terra L. 2010. Women’s Work, Women’s Knowing: Intellectual Property and the Recognition of Women’s Traditional Knowledge, Yale Journal of Law and Feminism 21(372): 373–404, https://digitalcommons.law.yale.edu/cgi/viewcontent.cgi?referer=https://www.google.com/&httpsredir=1&article=1304&context=yjlf.\nGurumurthy, Anita. 2016. Internet governance as seen from the Right to Development, OpenDemocracy 16 June, https://www.opendemocracy.net/en/internet-governance-as-seen-from-right-to-development/.\nHeintz, James. 2019. The Economy’s Other Half: How Taking Gender Seriously Transforms Macroeconomics. Newcastle-upon-Tyne: Agenda Publishing.\nHeld, Virginia. 2006. The Ethics of Care. New York, NY: Oxford University Press.\nKramm, Matthias, and Ingrid Robeyns. 2020. Limits to wealth in the history of Western philosophy. European Journal of Philosophy. https://doi.org/10.1111/ejop.12535.\nLe, Quéré, Robert B. Corinne, Matthew W. Jackson, Adam J.P. Jones, Sam Abernethy Smith, Robbie M. Andrew, Anthony J. De-Gol, David R. Willis, Yuli Shan, Josep G. Canadell, Pierre Friedlingstein, Felix Creutzig, and Glen P. Peters. 2020. Temporary reduction in global CO2 emissions during the COVID19 forced confinement. Nature Climate Change. https://doi.org/10.1038/s41558-020-0797-x.\nMinaker, Joanne, and Bryan Hogeveen (eds.). 2015. Criminalized Mothers, Criminalized Mothering. Bradford, Ontario: Demeter Press.\nNussbaum, Martha. 2013. Political Emotions: Why Love Matters for Justice. Cambridge: Belknap Press.\nNussbaum, Martha. 2003a. Capabilities as fundamental entitlements: Sen and social justice. Feminist Economics 9 (2–3): 33–59.\nNussbaum, Martha. 2003b. Compassion and terror, Daedalus (Winter 2003): 10–26, https://www.amacad.org/sites/default/files/daedalus/downloads/03_winter_daedalus_articles.pdf.\nRazavi, Shahra. 2007. The Political and Social Economy of Care in a Development Context: Contextual Issues, Research Questions, and Policy Options, Gender and Development Programme Paper 3. United Nations Research Institute for Social Development (June).\nRobeyns, Ingrid. 2017. Having too much. In Wealth—Yearbook of the American society for political and legal philosophy, J. Knight and M. Schwartzberg (ed.), 1–44. New York: New York University Press.\nRobinson, Fiona. 1999. Globalizing Care: Ethics, Feminist Theory, and International Relations. Boulder, CO: Westview Press.\nRocheleau, Dianne, Barbara Thomas-Slatter, and Esther Wangari. 1996. Gender and environment: a feminist political ecology perspective. Abingdon and New York: Routledge.\nRockström, Johan, Will Steffen, Kevin Noone, F. Åsa Persson, I.I.I. Stuart Chapin, Eric F. Lambin, Timothy M. Lenton, Marten Scheffer, Carl Folke, Hans Joachim Schellnhuber, Björn. Nykvist, Cynthia A. de Wit, Terry Hughes, Sander van der Leeuw, Henning Rodhe, Sverker Sörlin, Peter K. Snyder, Robert Costanza, Uno Svedin, Malin Falkenmark, Louise Karlberg, Robert W. Corell, Victoria J. Fabry, James Hansen, Brian Walker, Diana Liverman, Katherine Richardson, Paul Crutzen, and Jonathan A. Foley. 2009. A safe operating space for humanity. Nature 461 (September): 472–475.\nRoy, Arundhati. 2020. The pandemic is a portal, Financial Times, 2 April (online edition).\nSeguino, Stephanie. 2019. Engendering Macroeconomic Theory and Policy. Feminist Economics 26 (2): 27–61.\nSen, Amartya. 1990. Gender and Cooperative Conflicts. In Persistent Inequalities, Irene Tinker (ed.). New York: Oxford University Press.\nSen, Gita, and Marina Durano (eds.). 2014. The Remaking of Social Contracts: Feminists in a Fierce New World. London: Zed Books.\nTandon, Ambika and Aayush. 2020. Digital mediation of reproductive and care work in India: research reflexivity and challenges, https://genderit.org/articles/digital-mediation-of-reproductive-and-care-work. Accessed 30 June 2020.\nTronto, Joan C. 1993. Moral Boundaries: A Political Argument for an Ethic of Care. New York: Routledge.\nUnited Nations. 2010. Guidance Note of the Secretary-General: United Nations Approach to Transitional Justice.\nVon Broemsen, Marlese. 2020. Realizing the New Urban Agenda’s ideal of social inclusion: street vendors’ participation in decision-making about the use of public space. In Law and the New Urban Agenda, Nestor Davidson and Geeta Tewari (eds.). New York: Routledge.",{"VI":1920,"EN":1921},"Trọng tâm của việc xây dựng các nền kinh tế chăm sóc như một bước cần thiết hướng tới công bằng giới tính đòi hỏi một cuộc đánh giá lại về quản trị kinh tế toàn cầu và chủ nghĩa đa phương tập trung vào nhà nước. Các cấu trúc quyền lực toàn cầu không thể được xem chỉ đơn thuần là vấn đề về biên giới chính trị của các quốc gia, điều này thuộc về phạm vi truyền thống của chính sách đối ngoại. Thay vì địa lý, điều cần được xem xét là các cuộc đàm phán về ranh giới quyền lực, từ đó mở ra khả năng vẽ lại biên giới và ranh giới như được biểu hiện trong các mối quan hệ xã hội trong đó các chức năng chăm sóc được thực hiện. Năm lĩnh vực tham gia được xác định và thảo luận. Một ghi chú ngắn về chủ nghĩa giới hạn đặt ra câu hỏi về giá trị của nó trong một nền kinh tế chăm sóc và cách mà đạo đức chăm sóc liên kết với điều đó.","The centrality of building care economies as a necessary step towards gender justice requires a reassessment of global economic governance and state-centred multilateralism. Globalized structures of power can no longer be seen solely as matters of political borders of nation-states, which is the traditional remit of foreign policy. Rather than geography, it is negotiations over the boundaries of power that must be interrogated for the possibility of redrawing borders and boundaries as these are expressed in social relations where care functions are performed. Five spheres of engagement are identified and discussed. A short note on limitarianism raises a question about its value in a care economy and how ethics of care links to it.",{"VI":1923,"EN":1924},"Đàm phán Ranh giới Quyền Lực trong Quản trị Toàn cầu về Chăm sóc","Negotiating Boundaries of Power in the Global Governance for Care",{"VOID":1926},"10.1057/s41301-021-00283-3",{"VI":1928},"kinh tế chăm sóc, công bằng giới tính, quản trị toàn cầu, quyền lực, đạo đức chăm sóc","2024-12-31T07:09:52.692+00:00",[389],"https://link.springer.com/article/10.1057/s41301-021-00283-3",[1933],{"id":1934,"sortIndex":22,"researcher":17,"roles":1935,"affiliations":1937,"properties":1946},"b0ef18eb-a480-452b-af7e-73b9d67cada6",[1936],"AUTHOR",[1938],{"id":17,"sortIndex":22,"affiliation":1939,"properties":17},{"id":1940,"createTime":1941,"updateTime":1941,"relativeEntities":1942,"slug":17,"properties":1943,"entityType":21,"verifyStatus":385,"verifyTime":17,"verifyNote":17,"syncStatus":385,"languages":17,"translateLanguages":17,"viewCount":22},"8baceaa5-f7e0-4950-b7a8-839fcd7a4680","2024-01-18T15:42:18.867+00:00",[],{"title":1944},{"VI":1945},"Open Society Foundations, New York, USA",{"title":1947},{"VI":1948},"Marina Durano",{"url":1931,"publisher":1950,"properties":2021},{"id":1951,"createTime":1952,"updateTime":1953,"relativeEntities":1954,"slug":1955,"properties":1956,"entityType":264,"verifyStatus":385,"verifyTime":17,"verifyNote":17,"syncStatus":385,"languages":17,"translateLanguages":17,"viewCount":22,"subjectFields":1962,"manageAffiliations":1963,"indexDatabases":1964,"url":1977,"thumbnailPath":17,"statistic":1978,"gsStatistic":17,"type":1497,"analyzePriority":17},"be6621f7-7580-4a68-8024-81891091cd08","2024-04-10T02:23:19.994+00:00","2025-01-14T07:19:48.512+00:00",[],"Development",{"issn":1957,"eissn":1959,"title":1961},{"VOID":1958},"1461-7072",{"VOID":1960},"1011-6370",{"EN":1955},[],[],[1965],{"id":1966,"indexDatabase":1967,"url":1972,"indexYears":1973,"academicFieldIds":1974},"c279a27d-3a3d-4b49-a363-0adb1a3d6c78",{"id":491,"createTime":492,"updateTime":493,"relativeEntities":1968,"label":1969,"description":1970,"key":499,"publicationTags":1971,"standard":17},[],{"EN":496,"VI":496},{"EN":496,"VI":498},[501],"https://www.scopus.com/sourceid/14090","1984-1985,1988-2023",[1975,1976],"fd1d0f42-4448-42de-808a-f34275b91cc3","3b6f904f-cf09-4309-b7c4-b92757e00645","https://link.springer.com/journal/41301",{"impactFactor":22,"impactFactorByYear":1979,"i10Index":521,"i10IndexLast5Year":34,"totalPublication":1981,"totalPublicationByYear":1982,"totalCitation":1991,"totalCitationByYear":1992,"totalCitationPerPublication":2001,"totalCitationPerPublicationByYear":2002,"hindexLast5Year":2020,"hindex":2020},{"2012":1134,"2013":733,"2014":733,"2015":1134,"2016":729,"2017":730,"2018":731,"2020":1134,"2021":1980,"2022":735,"2023":1134},0.28,1329,{"1999":1983,"2000":773,"2001":1495,"2002":1984,"2003":1985,"2004":1141,"2005":1142,"2006":1986,"2007":1987,"2008":755,"2009":745,"2010":1988,"2011":1989,"2012":755,"2013":742,"2014":1722,"2015":1164,"2016":520,"2017":745,"2018":1717,"2019":757,"2020":1990,"2021":1462,"2022":1716,"2023":1716},61,79,49,93,82,62,83,35,1944,{"1999":1461,"2000":746,"2001":1142,"2002":1987,"2003":1993,"2004":1994,"2005":1995,"2006":1151,"2007":1476,"2008":1996,"2009":772,"2010":1997,"2011":1998,"2012":1999,"2013":512,"2014":2000,"2015":755,"2016":760,"2017":47,"2018":1470,"2019":757,"2020":769,"2021":512,"2022":512,"2023":58},103,168,77,88,111,198,94,250,1.46,{"1999":2003,"2000":2004,"2001":2005,"2002":2006,"2003":2007,"2004":2008,"2005":2009,"2006":2010,"2007":2011,"2008":2012,"2009":2013,"2010":2014,"2011":2015,"2012":1728,"2013":1731,"2014":2016,"2015":2017,"2016":2018,"2017":1446,"2018":777,"2019":34,"2020":2019,"2021":737,"2022":1730,"2023":733},0.84,18.45,5.58,1.04,2.1,0.95,1.15,1.09,1.1,1.52,0.88,1.79,2.39,5.56,1.35,1.42,1.71,20,{"volume":2022,"pages":2024},{"VOID":2023},"64",{"VOID":2025},"39-47","2021-03-29",{"id":2028,"createTime":2029,"updateTime":2030,"relativeEntities":2031,"slug":2032,"properties":2033,"entityType":191,"verifyStatus":383,"verifyTime":2046,"verifyNote":384,"syncStatus":385,"languages":17,"translateLanguages":2047,"viewCount":22,"primaryUrl":2048,"fullTextUrl":17,"authors":2049,"publicationType":193,"publisherRelationship":2104,"citationCount":17,"citationInfo":17,"publishDate":2285,"publishYear":2286,"citationAnalyzeStatus":385,"lastCitationAnalyze":17,"indexDatabases":17,"openAccess":17,"references":17,"isForceReanalyzing":24},"1a24c699-7fab-4aba-a871-773e80bbef12","2024-02-11T11:39:09.841+00:00","2025-02-13T06:16:00.022+00:00",[],"A-Control-Parameterization-Method-to-Solve-the-Fractional-Order-Optimal-Control-Problem",{"references":2034,"abstract":2036,"title":2039,"doi":2042,"keywords":2044},{"VOID":2035},"Letnikov, A.V.: Theory of differentiation with an arbitrary index. Math. Sb. 3, 1–66 (1868)\nCaputo, M.: Linear models of dissipation whose Q is almost frequency independent II. Geophys. J. Int. 13, 529–539 (1967)\nDiethelm, K.: The Analysis of Fractional Differential Equations. Springer, Berlin (2010)\nSamko, S.G., Kilbas, A.A., Marichev, O.I.: Fractional Integrals and Derivatives: Theory and Applications. Gordon and Breach, Yverdon (1993)\nFreed, A.D., Diethelm, K.: Fractional calculus in biomechanics: a 3D viscoelastic model using regularized fractional derivative kernels with application to the human calcaneal fat pad. Biomech. Model. Mechanobiol. 5, 203–215 (2006)\nRossikhin, Y.A., Shitikova, M.V.: Application of fractional calculus for dynamic problems of solid mechanics: novel trends and recent results. Appl. Mech. Rev. 63, 010801 (2010)\nHilfer, R. (ed.): Applications of Fractional Calculus in Physics. World Scientific, Singapore (2000)\nGutirrez, R.E., Rosrio, J.M., Tenreiro, M.J.: Fractional order calculus: basic concepts and engineering applications. Math. Probl. Eng. 2010, 375858 (2010)\nAgrawal, O.P.: Formulation of Euler–Lagrange equations for fractional variational problems. J. Math. Anal. Appl. 272, 368–379 (2002)\nAgrawal, O.P.: A quadratic numerical scheme for fractional optimal control problems. J. Dyn. Syst. Meas. Control 130, 011010 (2008)\nAgrawal, O.P., Baleanu, D.: A Hamiltonian formulation and a direct numerical scheme for fractional optimal control problems. J. Vib. Control 13, 1269–1281 (2007)\nTricaud, C., Chen, Y.Q.: An approximate method for numerically solving fractional-order optimal control problems of general form. Comput. Math. Appl. 59, 1644–1655 (2010)\nBhrawy, A.H., Doha, E.H., Baleanu, D., Ezz-Eldien, S.S., Abdelkawy, M.A.: An accurate numerical technique for solving fractional optimal control problems. Differ. Equ. 16, 47–54 (2015)\nKeshavarz, E., Ordokhani, Y., Razzaghi, M.: A numerical solution for fractional optimal control problems via Bernoulli polynomials. J. Vib. Control 22, 3889–3903 (2016)\nDoha, E.H., Bhrawy, A.H., Baleanu, D., Ezz-Eldien, S.S., Hafez, R.M.: An efficient numerical scheme based on the shifted orthonormal Jacobi polynomials for solving fractional optimal control problems. Adv. Differ. Equ. 2015, 1–17 (2015)\nSafaie, E., Farahi, M.H., Ardehaie, M.F.: An approximate method for numerically solving multi-dimensional delay fractional optimal control problems by Bernstein polynomials. Comput. Appl. Math. 34, 831–846 (2015)\nLotfi, A., Yousefi, S.A., Dehghan, M.: Numerical solution of a class of fractional optimal control problems via the Legendre orthonormal basis combined with the operational matrix and the Gauss quadrature rule. J. Comput. Appl. Math. 250, 143–160 (2013)\nJafari, H., Tajadodi, H.: Fractional-order optimal control problems via the operational matrices of Bernstein polynomials. UPB Sci. Bull. 76, 115–128 (2014)\nLotfi, A., Dehghan, M., Yousefi, S.A.: A numerical technique for solving fractional optimal control problems. Comput. Math. Appl. 62, 1055–1067 (2011)\nGoh, C.J., Teo, K.L.: Control parametrization: a unified approach to optimal control problems with general constraints. Automatica 24, 3–18 (1988)\nLee, H.W.J.: Control parametrization enhancing technique for time optimal control problems. Dyn. Syst. Appl. 6, 243–262 (1997)\nLin, Q., Loxton, R., Teo, K.L.: Optimal control of nonlinear switched systems: computational methods and applications. J. Oper. Res. Soc. China 1, 275–311 (2013)\nLin, Q., Loxton, R., Teo, K.L.: The control parameterization method for nonlinear optimal control: a survey. J. Ind. Manag. Optim. 10, 275–309 (2014)\nCao, J., Zhou, S., Inman, D.J., Chen, Y.: Chaos in the fractionally damped broadband piezoelectric energy generator. Nonlinear Dyn. 80, 1705–1719 (2015)\nZhou, Y., Ionescu, C., Machado, J.A.T.: Fractional dynamics and its applications. Nonlinear Dyn. 80, 1661–1664 (2015)\nTeo, K.L., Goh, C.J., Wong, K.H.: A Unified Computational Approach to Optimal Control Problems. Longman Scientific and Technical, Essex (1991)\nLiu, C., Gong, Z., Teo, K.L., Sun, J., Caccetta, L.: Robust multi-objective optimal switching control arising in 1, 3-propanediol microbial fed-batch process. Nonlinear Anal. Hybrid Syst. 25, 1–20 (2017)\nLoxton, R.C., Teo, K.L., Rehbock, V.: Robust suboptimal control of nonlinear systems. Appl. Math. Comput. 217, 6566–6576 (2011)\nNocedal, J., Wright, S.: Numerical Optimization. Springer, Berlin (2006)\nLoxton, R.C.: Optimal control problems involving constrained, switched, and delay systems. Doctoral dissertation, Curtin University (2010)\nLoxton, R.C., Lin, Q., Rehbock, V., Teo, K.L.: Control parameterization for optimal control problems with continuous inequality constraints: new convergence results. Numer. Algebra Control Optim. 2, 571–599 (2012)",{"EN":2037,"VI":2038},"This paper considers a class of fractional optimal control problems with canonical equality and inequality constraints. A fractional derivative in the dynamic system is defined in the Caputo sense. Using the control parameterization method, we approximate fractional optimal control problems by a sequence of finite-dimensional optimization problems. We then present the gradient formulae by introducing some auxiliary fractional systems. On this basis, a gradient-based optimization is developed to solve the fractional optimal control problems. Finally, a numerical example is used to test the proposed method.","Bài báo này xem xét một lớp các bài toán điều khiển tối ưu bậc phân thức với các ràng buộc cân bằng và bất cân bằng chuẩn. Một đạo hàm phân thức trong hệ động được định nghĩa theo cảm nghĩa Caputo. Bằng cách sử dụng phương pháp tham số hóa điều khiển, chúng tôi xấp xỉ các bài toán điều khiển tối ưu bậc phân thức bằng một chuỗi các bài toán tối ưu có chiều hữu hạn. Chúng tôi sau đó trình bày công thức gradient bằng cách giới thiệu một số hệ thống phân thức phụ trợ. Trên cơ sở này, một phương pháp tối ưu hóa dựa trên gradient được phát triển để giải các bài toán điều khiển tối ưu bậc phân thức. Cuối cùng, một ví dụ số được sử dụng để kiểm tra phương pháp được đề xuất.",{"EN":2040,"VI":2041},"A Control Parameterization Method to Solve the Fractional-Order Optimal Control Problem","Phương pháp tham số hóa điều khiển để giải bài toán điều khiển tối ưu bậc phân thức",{"VOID":2043},"10.1007/s10957-017-1163-7",{"VI":2045},"điều khiển tối ưu, bậc phân thức, phương pháp tham số hóa, đạo hàm phân thức, tối ưu hóa dựa trên gradient","2025-02-05T19:33:25.907+00:00",[389],"https://link.springer.com/article/10.1007/s10957-017-1163-7",[2050,2077,2092],{"id":2051,"sortIndex":47,"researcher":17,"roles":2052,"affiliations":2053,"properties":2074},"0b8839bd-ac61-43db-a80f-31848829bbef",[1936],[2054,2062],{"id":17,"sortIndex":22,"affiliation":2055,"properties":17},{"id":2056,"createTime":2057,"updateTime":2057,"relativeEntities":2058,"slug":17,"properties":2059,"entityType":21,"verifyStatus":385,"verifyTime":17,"verifyNote":17,"syncStatus":385,"languages":17,"translateLanguages":17,"viewCount":22},"778d5eed-9098-4f14-9f0b-2fb3e3d5a3e8","2023-12-31T14:44:57.421+00:00",[],{"title":2060},{"VI":2061},"School of Mathematics and Information Science, Shandong Institute of Business and Technology, Yantai, People’s Republic of China",{"id":2063,"sortIndex":34,"affiliation":2064,"properties":2073},"6cbec28f-6238-4839-bb92-fa1d79c27db6",{"id":2065,"createTime":2066,"updateTime":2067,"relativeEntities":2068,"slug":2069,"properties":2070,"entityType":21,"verifyStatus":385,"verifyTime":17,"verifyNote":17,"syncStatus":385,"languages":17,"translateLanguages":17,"viewCount":22},"1610f572-02b2-48ae-8f74-97b82f0674e0","2023-12-28T16:56:45.865+00:00","2024-10-01T14:40:35.236+00:00",[],"Department-of-Mathematics-and-Statistics-Curtin-University-Perth-Australia",{"title":2071},{"VI":2072},"Department of Mathematics and Statistics, Curtin University, Perth, Australia",{},{"title":2075},{"VI":2076},"Chongyang Liu",{"id":2078,"sortIndex":22,"researcher":17,"roles":2079,"affiliations":2080,"properties":2089},"9fc3bb21-da34-4718-95e6-bc36f1b3eacc",[1936],[2081],{"id":17,"sortIndex":22,"affiliation":2082,"properties":17},{"id":2083,"createTime":2084,"updateTime":2084,"relativeEntities":2085,"slug":17,"properties":2086,"entityType":21,"verifyStatus":385,"verifyTime":17,"verifyNote":17,"syncStatus":385,"languages":17,"translateLanguages":17,"viewCount":22},"916105b9-c99b-46fe-9799-cf23afdb9c81","2023-12-23T10:24:37.834+00:00",[],{"title":2087},{"VI":2088},"School of Mathematical Sciences, Dalian University of Technology, Dalian, People’s Republic of China",{"title":2090},{"VI":2091},"Pan 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Wang",{"url":2048,"publisher":2105,"properties":2280},{"id":2106,"createTime":2107,"updateTime":2108,"relativeEntities":2109,"slug":2110,"properties":2111,"entityType":264,"verifyStatus":385,"verifyTime":17,"verifyNote":17,"syncStatus":385,"languages":17,"translateLanguages":17,"viewCount":22,"subjectFields":2116,"manageAffiliations":2117,"indexDatabases":2118,"url":17,"thumbnailPath":17,"statistic":2142,"gsStatistic":17,"type":1497,"analyzePriority":17},"58c037fb-bc9a-4445-80d3-c53c462a2a00","2024-04-10T01:52:04.743+00:00","2025-02-24T03:35:00.157+00:00",[],"Journal-of-Optimization-Theory-and-Applications",{"issn":2112,"title":2114},{"VOID":2113},"15732878",{"VOID":2115},"Journal of Optimization Theory and Applications",[],[],[2119,2131],{"id":2120,"indexDatabase":2121,"url":2126,"indexYears":2127,"academicFieldIds":2128},"f9737e17-c3b7-4881-bb49-d17796705583",{"id":491,"createTime":492,"updateTime":493,"relativeEntities":2122,"label":2123,"description":2124,"key":499,"publicationTags":2125,"standard":17},[],{"EN":496,"VI":496},{"EN":496,"VI":498},[501],"https://www.scopus.com/sourceid/23966","1967-2023",[2129,2130,1128],"9257bd7d-2e00-4865-a19e-9796f3a21c3f","7a841ccd-a830-4dea-be9c-bf6d019b6994",{"id":2132,"indexDatabase":2133,"url":2138,"indexYears":17,"academicFieldIds":2139},"9ad660a5-e635-4157-a9a8-861959237321",{"id":470,"createTime":471,"updateTime":472,"relativeEntities":2134,"label":2135,"description":2136,"key":479,"publicationTags":2137,"standard":17},[],{"EN":475,"VI":475},{"VI":477,"EN":478},[481,482],"https://mjl.clarivate.com/search-results?issn=0022-3239",[2140,2141],"ef99e6f3-ace4-49a6-aa47-d5e24f7d2768","0a89cee9-2df6-4b14-8fcb-700fd0044c25",{"impactFactor":22,"impactFactorByYear":2143,"i10Index":2150,"i10IndexLast5Year":741,"totalPublication":2151,"totalPublicationByYear":2152,"totalCitation":2181,"totalCitationByYear":2182,"totalCitationPerPublication":2230,"totalCitationPerPublicationByYear":2231,"hindexLast5Year":2189,"hindex":2189},{"2012":2144,"2013":734,"2014":2145,"2015":2146,"2016":1712,"2017":2147,"2018":2148,"2019":2144,"2020":1731,"2021":2145,"2022":2149,"2023":1136},0.34,0.4,0.3,0.47,0.56,0.31,514,6072,{"1967":513,"1968":2153,"1969":744,"1970":755,"1971":1161,"1972":2154,"1973":1472,"1974":769,"1975":2155,"1976":1160,"1977":2156,"1978":2157,"1979":2158,"1980":1166,"1981":2159,"1982":2160,"1983":1151,"1984":2161,"1985":2162,"1986":2163,"1987":1993,"1988":1160,"1989":2164,"1990":2165,"1991":2165,"1992":2166,"1993":2167,"1994":1997,"1995":2168,"1996":326,"1997":518,"1998":2169,"1999":2170,"2000":1166,"2001":1999,"2002":1997,"2003":2171,"2004":2171,"2005":2172,"2006":2173,"2007":2172,"2008":2158,"2009":2162,"2010":2174,"2011":518,"2012":2175,"2013":1453,"2014":2176,"2015":2177,"2016":1997,"2017":518,"2018":2178,"2019":2179,"2020":2170,"2021":766,"2022":2179,"2023":2176,"2024":2180},27,54,80,104,105,89,92,98,106,96,107,95,127,121,100,125,140,124,102,128,85,84,167,126,129,163,120,36,28220,{"1967":71,"1968":71,"1969":1148,"1970":2183,"1971":2184,"1972":2185,"1973":2186,"1974":1169,"1975":2187,"1976":2188,"1977":2189,"1978":2190,"1979":2191,"1980":2192,"1981":2193,"1982":2194,"1983":2195,"1984":2196,"1985":2197,"1986":2198,"1987":2199,"1988":2200,"1989":2201,"1990":2202,"1991":2203,"1992":2204,"1993":2205,"1994":2206,"1995":2207,"1996":2208,"1997":2209,"1998":2210,"1999":2211,"2000":2212,"2001":2213,"2002":2214,"2003":2215,"2004":2216,"2005":2217,"2006":2218,"2007":2219,"2008":1155,"2009":2220,"2010":2221,"2011":2222,"2012":543,"2013":2223,"2014":2224,"2015":2225,"2016":2226,"2017":2227,"2018":2228,"2019":1457,"2020":2229,"2021":2228,"2022":1161,"2023":1496,"2024":47},122,159,190,866,287,537,73,844,333,249,740,276,1148,165,260,182,869,341,436,269,713,468,2190,422,362,847,671,1878,799,423,312,665,516,638,494,319,375,1585,430,384,1102,547,683,548,439,138,147,4.65,{"1967":2145,"1968":1466,"1969":2232,"1970":2007,"1971":2233,"1972":2234,"1973":2235,"1974":2236,"1975":2237,"1976":2238,"1977":792,"1978":2239,"1979":2240,"1980":2241,"1981":2239,"1982":2242,"1983":2243,"1984":2244,"1985":2245,"1986":1185,"1987":2246,"1988":2247,"1989":2248,"1990":2249,"1991":2250,"1992":776,"1993":2251,"1994":2252,"1995":2253,"1996":2254,"1997":2255,"1998":2256,"1999":2257,"2000":2258,"2001":2259,"2002":2260,"2003":2261,"2004":2262,"2005":2263,"2006":2264,"2007":2265,"2008":2266,"2009":2267,"2010":2268,"2011":2269,"2012":2270,"2013":2271,"2014":2272,"2015":2273,"2016":2274,"2017":2275,"2018":2276,"2019":2277,"2020":2278,"2021":2279,"2022":1131,"2023":728,"2024":731},2.87,3.18,3.52,21.65,3.12,3.59,4.93,8.04,3.74,2.18,2.82,11.37,1.56,2.71,8.44,3.13,4.59,2.12,5.61,21.9,3.8,2.9,6.52,5.08,13.41,6.44,3.71,3.32,5.99,5.06,6.25,3.86,3.75,2.93,4.06,16.51,5.12,2.91,3.93,5.38,4.34,5.29,4.94,3.33,0.85,1.75,1.19,1.12,{"volume":2281,"pages":2283},{"VOID":2282},"187",{"VOID":2284},"234-247","2017-09-01",2017,{"id":2288,"createTime":2289,"updateTime":2290,"relativeEntities":2291,"slug":2292,"properties":2293,"entityType":191,"verifyStatus":383,"verifyTime":2306,"verifyNote":384,"syncStatus":385,"languages":17,"translateLanguages":2307,"viewCount":22,"primaryUrl":2308,"fullTextUrl":17,"authors":2309,"publicationType":193,"publisherRelationship":2351,"citationCount":17,"citationInfo":17,"publishDate":2385,"publishYear":1578,"citationAnalyzeStatus":385,"lastCitationAnalyze":17,"indexDatabases":17,"openAccess":17,"references":17,"isForceReanalyzing":24},"0b79909e-36ae-477c-a877-aedbc4a059e4","2024-01-06T19:19:20.387+00:00","2025-02-16T06:45:38.031+00:00",[],"Boundary-value-problems-of-fractional-q-difference-equations-on-the-half-line",{"references":2294,"abstract":2296,"title":2299,"doi":2302,"keywords":2304},{"VOID":2295},"Jackson, F.: On q-functions and a certain difference operator. Trans. R. Soc. Edinb. 46, 253–281 (1908)\nJackson, F.: On q-definite integrals. Pure Appl. Math. Q. 41, 193–203 (1910)\nKac, V., Cheung, P.: Quantum Calculus. Springer, New York (2002)\nAgarwal, R.: Certain fractional q-integrals and q-derivatives. Proc. Camb. Philos. Soc. 66, 365–370 (1969)\nMiao, F., Liang, S.: Uniqueness of positive solutions for fractional q-difference boundary value problems with p-Laplacian operator. Electron. J. Differ. Equ. 2013, 174, 1–11 (2013)\nGraef, J., Kong, L.: Positive solutions for a class of higher order boundary value problems with fractional q-derivatives. Appl. Math. Comput. 218, 9682–9689 (2012)\nAhmad, B., Nietoa, J., Alsaedi, A., Al-Hutami, H.: Existence of solutions for nonlinear fractional q-difference integral equations with two fractional orders and nonlocal four-point boundary conditions. J. Franklin Inst. 351, 2890–2909 (2014)\nHan, Z.L., Pan, Y.Y., Yang, D.W.: The existence and nonexistence of positive solutions to a discrete fractional boundary value problem with a parameter. Appl. Math. Lett. 36, 1–6 (2014)\nLi, X.H., Han, Z.L., Li, X.C.: Boundary value problems of fractional q-difference Schrodinger equations. Appl. Math. Lett. 46, 100–105 (2015)\nChidouh, A., Torres, D.: Existence of positive solutions to a discrete fractional boundary value problem and corresponding Lyapunov-type inequalities. Opusc. Math. 38(1), 31–40 (2018)\nZhao, X., Ge, W.: Unbounded solutions for a fractional boundary value problems on infinite interval. Acta Appl. Math. 109, 495–505 (2010)\nHe, L., Dong, X., Bai, Z., Chen, B.: Solvability of some two-point fractional boundary value problems under barrier strip conditions. J. Funct. Spaces 2017, Article ID 1465623 (2017)\nLiang, S., Zhang, J.: Existence of three positive solutions of m-point boundary value problems for some nonlinear fractional differential equations on an infinite interval. Comput. Math. Appl. 61, 3343–3354 (2011)\nZhao, X., Zhao, D., Ge, W.: Solvability for fractional multi-point boundary value problems on half line. Ann. Differ. Equ. 27(3), 395–400 (2011)\nLiu, X., Jia, M.: Multiple solutions for fractional differential equations with nonlinear boundary conditions. Comput. Math. Appl. 59, 2880–2886 (2010)\nCui, Y., Ma, W., Sun, Q., Su, X.: New uniqueness results for boundary value problem of fractional differential equation. Nonlinear Anal., Model. Control 23, 31–39 (2018)\nCui, Y., Sun, Q., Su, X.: Monotone iterative technique for nonlinear boundary value problems of fractional order \\(p\\in (2,3]\\). Adv. Differ. Equ. 2017, 248, 1–15 (2017)\nZhang, X., Zhong, Q.: Uniqueness of solution for higher-order fractional differential equations with conjugate type integral conditions. Fract. Calc. Appl. Anal. 20(6), 1471–1484 (2017)\nZhang, X., Zhong, Q.: Triple positive solutions for nonlocal fractional differential equations with singularities both on time and space variables. Appl. Math. Lett. 80, 12–19 (2018)\nXu, M., Han, Z.: Positive solutions for integral boundary value problem of two-term fractional differential equations. Bound. Value Probl. 2018, 100, 1–13 (2018)\nSong, Q., Bai, Z.: Positive solutions of fractional differential equations involving the Riemann-Stieltjes integral boundary condition. Adv. Differ. Equ. 2018, 183 (2018)\nGamze Düzgün, F., Iannizzotto, A.: Three nontrivial solutions for nonlinear fractional Laplacian equations. Adv. Nonlinear Anal. 7(2), 211–226 (2018)\nGiacomoni, J., Mukherjee, T., Sreenadh, K.: Positive solutions of fractional elliptic equation with critical and singular nonlinearity. Adv. Nonlinear Anal. 6(3), 327–354 (2017)\nLyons, J., Neugebauer, J.: Positive solutions of a singular fractional boundary value problem with a fractional boundary condition. Opusc. Math. 37(3), 421–434 (2017)\nAnnaby, M., Mansour, Z.: q-Fractional Calculus and Equations. Springer, New York (2012)\nRajković, P., Marinković, S., Stanković, M.: Fractional integrals and derivatives in q-calculus. Appl. Anal. Discrete Math. 1(1), 311–323 (2007)\nLi, X., Han, Z., Sun, S.: Existence of positive solutions of nonlinear fractional q-difference equation with parameter. Adv. Differ. Equ. 2013, 260 (2013)\nGranans, A., Guenther, P., Lee, J.: Some general existence principle in Carathéodory theory of nonlinear systems. J. Math. Pures Appl. 70, 153–196 (1991)\nSu, X., Zhang, S.: Unbounded solutions to a boundary value problem of fractional order on the half-line. Comput. Math. Appl. 61, 1079–1087 (2011)",{"EN":2297,"VI":2298},"In this paper, we consider the boundary value problem of a class of nonlinear fractional q-difference equations involving the Riemann–Liouville fractional q-derivative on the half-line. By means of Schauder fixed point theorem and Leggett–Williams fixed point theorem, some results on existence and multiplicity of solutions are obtained.","Trong bài báo này, chúng tôi xem xét bài toán giá trị biên của một lớp phương trình vi phân không tuyến tính phân số q liên quan đến đạo hàm phân số Riemann–Liouville trên nửa đoạn. Thông qua định lý điểm cố định Schauder và định lý điểm cố định Leggett–Williams, một số kết quả về sự tồn tại và tính nhiều nghiệm của các nghiệm được đưa ra.",{"EN":2300,"VI":2301},"Boundary value problems of fractional q-difference equations on the half-line","Các bài toán giá trị biên của phương trình vi phân không tuyến tính phân số q trên nửa đoạn",{"VOID":2303},"10.1186/s13661-019-1159-3",{"VI":2305},"phương trình vi phân, phân số, q-đạo hàm, bài toán giá trị biên, điểm cố định","2025-02-05T21:43:12.997+00:00",[389],"https://boundaryvalueproblems.springeropen.com/articles/10.1186/s13661-019-1159-3",[2310,2327,2339],{"id":2311,"sortIndex":47,"researcher":17,"roles":2312,"affiliations":2313,"properties":2324},"206eb885-49d6-4a3a-8e6f-d46812e2adfd",[1936],[2314],{"id":17,"sortIndex":22,"affiliation":2315,"properties":17},{"id":2316,"createTime":2317,"updateTime":2318,"relativeEntities":2319,"slug":2320,"properties":2321,"entityType":21,"verifyStatus":385,"verifyTime":17,"verifyNote":17,"syncStatus":385,"languages":17,"translateLanguages":17,"viewCount":22},"07f86fe6-ba3f-4bca-85c6-5c291ccf1923","2024-01-15T22:59:12.928+00:00","2025-06-11T20:59:11.941+00:00",[],"School-of-Mathematical-Sciences-University-of-Jinan-Jinan-P-R-China",{"title":2322},{"VI":2323},"School of Mathematical Sciences, University of Jinan, Jinan, P.R. 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