Nội dung được dịch bởi AI, chỉ mang tính chất tham khảo
Về việc xác định hệ số khuếch tán phụ thuộc thời gian cho phương trình khuếch tán phân số không gian
Tóm tắt
Một bài toán ngược về việc phục hồi một hệ số khuếch tán phụ thuộc vào thời gian cho một phương trình khuếch tán phân số không gian đã được xem xét. Đạo hàm phân số không gian bậc $$1<\alpha < 2$$ được định nghĩa theo nghĩa Caputo. Do điều kiện xác định quá mức loại tích phân, chúng tôi xây dựng một ánh xạ. Dưới những điều kiện nhất định về dữ liệu đã cho và việc áp dụng định lý điểm cố định Banach đảm bảo sự tồn tại duy nhất cục bộ của nghiệm, hơn nữa nghiệm cục bộ được chứng minh là nghiệm cổ điển. Sự tồn tại toàn cục của nghiệm cho bài toán ngược được chỉ ra bằng cách sử dụng định lý điểm cố định Schauder. Các ví dụ cũng được cung cấp để hỗ trợ phân tích của chúng tôi.
Từ khóa
#phương trình khuếch tán phân số #hệ số khuếch tán #bài toán ngược #định lý điểm cố định #tồn tại nghiệmTài liệu tham khảo
Sokolov, I.M., Klafter, J.: From diffusion to anomalous diffusion: a century after Einstein’s Brownian motion. Chaos 15, 026103 (2005)
Fa, K.S.: Fractal and generalized Fokker–Planck equations: description of the characterization of anomalous diffusion in magnetic resonance imaging. J. Stat. Mech. Theo. Exp. 033207 (2017)
Fedotov, S., Korabel, N.: Subdiffusion in an external potential: anomalous effects hiding behind normal behavior. Phys. Rev. E 91, 042112 (2015)
Kang, P.K., Dentz, M., Le Borgne, T., Lee, S., Juanes, R.: Anomalous transport in disordered fracture networks: spatial Markov model for dispersion with variable injection models. Adv. Water Resour. 106, 80–94 (2017)
Metzler, R., Klafter, J.: The restaurant at the end of the random walk: recent developments in the description of anomalous transport by fractional dynamics. J. Phys. A Math. Gen. 37, 161–208 (2004)
Povstenko, Y.: Linear fractional diffusion-wave equation for scientists and engineers. Springer, Switzerland (2015)
Bagley, R.L., Torvik, P.J.: A theoretical basis for the application of fractional calculus to viscoelasticity. J. Rheol. 27, 201–210 (1983)
Mainardi, F.: Fractional calculus and waves in linear viscoelaticity. Imperial College Press, London (2010)
Ionescu, C., Lopes, A., Copot, D., Machado, J.A.T., Bates, J.H.T.: The role of fractional calculus in modelling biological phenomena: a review. Commun. Nonlinear Sci. Numer. Simul. 51, 141–159 (2017)
Höfling, F., Franosch, T.: Anomalous transport in the crowded world of biological cells. Rep. Prog. Phys. 76, 046602 (2013)
Hilfer, R.: Applications of Fractional Calculus in Physics. World Scientific, Singapore (2000)
Tarasov, V.E.: Fractional Dynamics: Applications of Fractional Calculus to Dynamics of Particles. Springer, New York (2010)
Machado, J.A.T., Lopes, A.M.: Relative fractional dynamics of stock markets. Nonlinear Dyn. 86, 1613–1619 (2016)
Caputo, M., Carcione, J.M., Botelho, M.A.B.: Modelling extreme-event precursors with the fractional diffusion equation. Fract. Calc. Appl. Anal. 18, 208–222 (2015)
Hasanov, A., Tatar, S.: An inversion method for identification of elasto-plastic properties of a beam from torsional experiment. Int. J. Non-Linear Mech. 45, 562–571 (2010)
Ali, M., Malik, S.A.: An inverse problem for a family of time fractional diffusion equations. Inverse Prob. Sci. Eng. 25, 1299–1322 (2017)
Malik, S.A., Aziz, S.: An inverse source problem for a two parameter anomalous diffusion equation with nonlocal boundary conditions. Comput. Math. Appl. 73, 2548–2560 (2017)
Tuan, N.H., Kirane, M., Hoan, L.V.C., Long, L.D.: Identification and regularization for unknown source for a time-fractional diffusion equation. Comput. Math. Appl. 73, 931–950 (2017)
Tuan, N.H., Long, L.D., Nguyen, V.T., Tran, T.: On a final value problem for the time-fractional diffusion equation with inhomogeneous source. Inverse Prob. Sci. Eng. 25, 1367–1395 (2017)
Al-Jamal, M.F.: A backward problem for the time-fractional diffusion equation. Math. Meth. Appl. Sci. 40, 2466–2474 (2017)
Lopushanska, H., Rapita, V.: Inverse coefficient problem for the semi-linear fractional telegraph equation. Elect. J. Differ. Equ. 153, 1–13 (2015)
Šišková, K., Slodička, M.: Recognition of a time-dependent source in a time-fractional wave equation. Appl. Numer. Math. 121, 1–17 (2017)
Wen, J., Cheng, J.F.: The method of fundamental solution for the inverse source problem for the space-fractional diffusion equation. Inverse Prob. Sci. Eng. 26, 925–941 (2018)
Tatar, S., Tinaztepe, R., Zeki, M.: Numerical solutions of direct and inverse problems for a time fractional viscoelastoplastic equation. J. Eng. Mech. 143, 1–9 (2017)
Tatar, S.: Monotonicity of input–output mapping related to inverse elastoplastic torsional problem. Appl. Math. Model. 37, 9552–9561 (2013)
Janno, J., Kinash, N.: Reconstruction of an order of derivative and a source term in a fractional diffusion equation from final measurments. Inverse Prob. (2018). https://doi.org/10.1088/1361-6420/aaa0f0
Kawamoto, A.: Lipschitz stability estimates in inverse source problems for a fractional diffusion equation of half order in time by Carleman estimates. J. Inverse Ill-Posed Probl. (2018). https://doi.org/10.1515/jiip-2016-0029
Chi, G., Li, G.: Numerical inversions for diffusion coefficients in two-dimensional space fractional diffusion equation. Inverse Prob. Sci. Eng. 26, 996–1018 (2017)
Aziz, S., Malik, S.A.: Identification of an unknown source term for a time fractional fourth-order parabolic equation. Elect. J. Differ. Equ. 293, 1–28 (2016)
Tatar, S., Ulusoy, S.: An inverse source problem for a one dimensional space-time fractional diffusion equation. Appl. Anal. 94, 2233–2244 (2015)
Tatar, S., Ulusoy, S.: An inverse problem for a nonlinear diffusion equation with time-fractional derivative. J. Inverse Ill-posed Prob. (2015). https://doi.org/10.1515/jiip-2015-0100
Tatar, S., Ulusoy, S.: Analysis of direct and inverse problems for a fractional elastoplasticity model. Filomat 31(3), 699–708 (2017)
Feng, P., Karimov, E.T.: Inverse source problems for time-fractional mixed parabolic-hyperbolic-type equations. J. Inverse Ill-Posed Probl. 23, 339–353 (2015)
Tatar, S., Tinaztepe, R., Ulusoy, S.: Simultaneous inversion for the exponents of the fractional time and space derivatives in the space-time fractional diffusion equation. Appl. Anal. 95, 1–23 (2016)
Tatar, S., Ulusoy, S.: A uniqueness result for an inverse problem in a space-time fractional diffusion equation. Elect. J. Differ. Equ. 2013(258), 1–9 (2013)
Ali, M., Aziz, S., Malik, S.A.: Inverse source problem for a space-time fractional diffusion equation. Fract. Calc. Appl. Anal. 21, 844–863 (2018)
Jia, J., Peng, J., Yang, J.: Harnack’s inequality for a space-time fractional diffusion equation and application to an inverse source problem. J. Differ. Equ. 262, 4415–4450 (2017)
Ali, M., Aziz, S., Malik, S.A.: Inverse problem for a space-time fractional diffusion equation: application of fractional Sturm–Liouville operator. Math. Methods Appl. Sci. 41, 2733–2744 (2018)
Gara, R., Gorenflo, R., Polito, F., Tomovski, Z.: Hilfer–Prabhakar derivatives and some applications. Appl. Math. Comput. 242(2014), 576–589 (2014)
Gorenflo, R., Kilbas, A.A., Mainardi, F., Rogosin, S.V.: Mittag-Leffler Functions Related Topics and Applications. Springer, New York (2014)
Podlubny, I.: Fractional differential equations. Academic Press, San Diego (1999)
Kilbas, A.A., Srivastava, H.M., Trujillo, J.J.: Theory and Applications of Fractional Differential Equations. Elsevier, Amsterdam (2006)
Aleroev, T.S., Kirane, M., Tang, Y.F.: The boundary-value problem for a differential operator of fractional order. J. Math. Sci. 194, 499–512 (2013)
Samko, G.S., Kilbas, A.A., Marichev, D.I.: Fractional Integrals and Derivatives: Theory and Applications. Gordon and Breach Science Publishers, London (1993)