Chiến lược polynôme Chelyshkov trong việc giải các phương trình tích phân Volterra phi tuyến hai chiều của loại đầu tiên

Springer Science and Business Media LLC - Tập 41 - Trang 1-13 - 2022
A. M. S. Mahdy1,2, D. Shokry1, Kh. Lotfy1,3
1Department of Mathematics, Faculty of Science, Zagazig University, Zagazig, Egypt
2Department of Mathematics and Statistics, College of Science, Taif University, Taif, Saudi Arabia
3Department of Mathematics, Faculty of Science, Taibah University, Madinah, Saudi Arabia

Tóm tắt

Trong bài báo này, chúng tôi xây dựng một kỹ thuật hiệu quả dựa trên các đa thức Chelyshkov hai chiều (2D-CPs) để mở rộng một giải pháp ước lượng cho một loại phương trình tích phân Volterra phi tuyến hai chiều (2D-NVIEs) thuộc loại đầu tiên. Vấn đề này được chuyển đổi thành một 2D-NVIE thuộc loại thứ hai và kỹ thuật này biến đổi phương trình tích phân thành một hệ phương trình đại số phi tuyến thông qua các điểm tụ. Cuối cùng, một số ví dụ số được đưa ra để làm rõ hiệu quả của chiến lược đã được trình bày.

Từ khóa

#Polynôme Chelyshkov #phương trình tích phân Volterra phi tuyến #phương trình đại số phi tuyến #điểm tụ #kỹ thuật giải

Tài liệu tham khảo

Amer YA, Mahdy A M S, Youssef E S M (2018) Solving fractional integro-differential equations by using Sumudu transform method and Hermite spectral collocation method. CMC Comput Mater Continua 54(2):161–180 Ardabili JS, Talaei, Y (2018) Chelyshkov collocation method for solving the two-dimensional Fredholm–Volterra integral equations. Int J Appl Comput Math 4(1):1–15 Babolian E, Masouri Z (2008) Direct method to solve Volterra integral equation of the first kind using operational matrix with block-pulse functions. J Comput Appl Math 220:51–57 Chelyshkov VS (2006) Alternative orthogonal polynomials and quadratures. Electron Trans Numer Anal 25:17–26 Essah WA, Delves LM (1989) The numerical solution of first kind integral equations. J Comput Appl Math 27:363–387 Hamid M, Usman M, Haq RU, Wang W (2020) A Chelyshkov polynomial based algorithm to analyze the transport dynamics and anomalous diffusion in fractional model. Physica A Stat Mech Appl 551:1–15 Hosseininia M, Heydari MH, Avazzadeh Z (2020) The numerical treatment of nonlinear fractal-fractional 2d Emden–Fowler equation utilizing 2d Chelyshkov polynomials. Fractals 28(8):1–12 Hosseininia M, Heydari MH, Ghaini FMM (2021) A numerical method for variable-order fractional version of the coupled 2D Burgers equations by the 2D Chelyshkov polynomials. Math Methods Appl Sci 44:6482–6499 Kythe PK, Puri P (2002) Computational methods for linear integral equations. Birkhäuser, Boston Mahdy AMS (2018) Numerical studies for solving fractional integro-differential equations. J Ocean Eng Sci 3(2):127–132 Mahdy AMS (2022) A numerical method for solving the nonlinear equations of Emden-Fowler models. J Ocean Eng Sci. https://doi.org/10.1016/j.joes.2022.04.019 Mahdy AMS, Mohamed EMH, Marai GMA (2016) Numerical solution of fractional integro-differential equations by least squares method and shifted Chebyshev polynomials of the third kind method. Theor Math Appl 6(4):87–101 Maleknejad K, Sohrabi S, Baranji B (2010) Application of 2D-BPFs to nonlinear integral equations. Commun Nonlinear Sci Numer Simul 15:527–535 Mckee S, Tang T, Diogo T (2000) An Euler-type method for two-dimensional Volterra integral equations of the first kind. IMA J Numer Anal 20:423–440 Mirzaee F, Samadyar N (2018) Convergence of 2D-orthonormal Bernstein collocation method for solving 2D-mixed Volterra–Fredholm integral equations. Trans A. Razmadze Math Inst 172:631–641 Mohamed DSh (2019) Chelyshkov’s collocation method for solving three-dimensional linear Fredholm integral equations. MathLAB J 4:163–171 Mohammadi F, Hassani H (2022) Numerical solution of time-fractional Telegraph equation by using a new class of orthogonal polynomials. Bol Soc Paran Mat 40:1–13 Nemati S, Ordokhanib Y (2015) Solving nonlinear two-dimensional Volterra integral equations of the first-kind using the bivariate shifted Legendre functions. Int J Math Model Comput 05(03):219–230 Pachpatte BG (2011) Multidimensional integral equations and inequalities. Atlantis Press, Amsterdam Rasty M, Hadizadeh M (2010) A Product integration approach on new orthogonal polynomials for nonlinear weakly singular integral equations. Acta Appl Math 109:861–873 Shali JA, Darania P, Akbarfam AA (2012) Collocation method for nonlinear Volterra–Fredholm integral equations. J Appl Sci 2:115–121 Soori Z, Aminataei A (2012) The spectral method for solving Sine Gordon equation using a new orthogonal polynomial. ISRN Appl Math 1–12 Talaei Y, Asgari M (2017) An operational matrix based on Chelyshkov polynomials for solving multi-order fractional differential equations. Neural Comput Appl 1369–1376 Tari A (2014) A numerical solution for the two-dimensional Volterra integral equations of the first kind. In: The \(45{\rm th}\) annual Iranian mathematics conference, pp 1–4 Tari A, Shahmorad S (2012) Numerical solution of a class of two-dimensional nonlinear Volterra integral equations of the first kind. J Appl Math Inf 30:463–475 Tari A, Shahmorad S (2012) A computational method for solving two-dimensional linear Volterra integral equations of the first kind. Sci Iran D 19:829–835 Torabi SM, Tari A (2016) Numerical solution of two-dimensional integral equations of the first kind by multi-step methods. Comput Methods Differ Equ 4(2):128–138