This paper presents the inverse problem (IP) for the fractional order two-dimensional parabolic diffusion equation (FOTDPDE) formulated to depend on a initial-boundary value problem (IBVP) with homogeneous Dirichlet boundary conditions (DBC). The model involves a fractional-order Caputo derivative (FOCD) and an inverse time-dependent source term. A Crank-Nicholson finite difference scheme (CN-FDS) is constructed, and stability inequalities and a theorem in the discrete $ L^2 $ norm are proved to ensure unconditional stability of the proposed scheme. Results calculated by using finite difference methods (FDM) have a temporal convergence rate of $ O(\tau^{2-\alpha}) $ and second-order spatial accuracy. Numerical examples are tested to confirm the theoretical stability results and to represent the effectiveness and the accuracy of the method for solving IP for FOTDPDE depending on BVP.
Citation: Mousa J. Huntul, Mahmut Modanli. Numerical approach for solving the inverse problem: A two-dimensional time-fractional boundary value problem[J]. AIMS Mathematics, 2026, 11(3): 7078-7097. doi: 10.3934/math.2026291
This paper presents the inverse problem (IP) for the fractional order two-dimensional parabolic diffusion equation (FOTDPDE) formulated to depend on a initial-boundary value problem (IBVP) with homogeneous Dirichlet boundary conditions (DBC). The model involves a fractional-order Caputo derivative (FOCD) and an inverse time-dependent source term. A Crank-Nicholson finite difference scheme (CN-FDS) is constructed, and stability inequalities and a theorem in the discrete $ L^2 $ norm are proved to ensure unconditional stability of the proposed scheme. Results calculated by using finite difference methods (FDM) have a temporal convergence rate of $ O(\tau^{2-\alpha}) $ and second-order spatial accuracy. Numerical examples are tested to confirm the theoretical stability results and to represent the effectiveness and the accuracy of the method for solving IP for FOTDPDE depending on BVP.
| [1] |
S. N. Antontsev, S. E. Aitzhanov, G. R. Ashurova, An inverse problem for the pseudo-parabolic equation with $p$-Laplacian, Evol. Equ. Control The., 11 (2022), 399–414. http://doi.org/10.3934/eect.2021005 doi: 10.3934/eect.2021005
|
| [2] |
K. Khompysh, M. Ruzhansky, Inverse source problems for time-fractional nonlinear pseudoparabolic equations with $p$-Laplacian, Fract. Calc. Appl. Anal., 28 (2025), 1353–1383. https://doi.org/10.1007/s13540-025-00404-6 doi: 10.1007/s13540-025-00404-6
|
| [3] |
F. Huang, F. Liu, The time fractional diffusion equation and the advection-dispersion equation, ANZ1AM J., 46 (2005), 317–330. hhttps://doi.org/10.1017/S1446181100008282 doi: 10.1017/S1446181100008282
|
| [4] |
N. H. Sweilam, H. Moharram, N. K. A. Moniem, S. Ahmed, A parallel Crank–Nicolson finite difference method for time-fractional parabolic equation, J. Numer. Math., 22 (2014), 363–382. https://doi.org/10.1515/jnma-2014-0016 doi: 10.1515/jnma-2014-0016
|
| [5] |
A. Akgül, M. Modanli, Crank–Nicholson difference method and reproducing kernel function for third order fractional differential equations in the sense of Atangana–Baleanu Caputo derivative, Chaos Soliton. Fract., 127 (2019), 10–16. https://doi.org/10.1016/j.chaos.2019.06.011 doi: 10.1016/j.chaos.2019.06.011
|
| [6] |
M. Modanli, K. Karadag, S. T. Abdulazeez, Solutions of the mobile-immobile advection-dispersion model based on the fractional operators using the Crank–Nicholson difference scheme, Chaos Soliton. Fract., 167 (2023), 113114. https://doi.org/10.1016/j.chaos.2023.113114 doi: 10.1016/j.chaos.2023.113114
|
| [7] |
K. Oishi, Y. Hashizume, T. Nakao, K. Kashima, Extraction of implicit field cost via inverse optimal Schrödinger bridge, SICE Journal of Control, Measurement, and System Integration, 18 (2025), 2490332. https://doi.org/10.1080/18824889.2025.2490332 doi: 10.1080/18824889.2025.2490332
|
| [8] |
S. E. Chorfi, A. Hasanov, R. Morales, Identification of source terms in the Schrödinger equation with dynamic boundary conditions from final data, Z. Angew. Math. Phys., 76 (2025), 127. https://doi.org/10.1007/s00033-025-02505-x doi: 10.1007/s00033-025-02505-x
|
| [9] |
E. Shivanian, A. Jafarabadi, M. J. Huntul, A local meshless technique for recovering dual forms of time-varying sources in the nonlocal inverse heat equation, Results in Applied Mathematics, 28 (2025), 100673. https://doi.org/10.1016/j.rinam.2025.100673 doi: 10.1016/j.rinam.2025.100673
|
| [10] |
S. S. Liu, L. X. Feng, An inverse problem for a two‐dimensional time‐fractional sideways heat equation, Math. Probl. Eng., 2020 (2020), 5865971. https://doi.org/10.1155/2020/5865971 doi: 10.1155/2020/5865971
|
| [11] |
T. Liu, F. Soleymani, M. Z. Ullah, Solving multi-dimensional European option pricing problems by integrals of the inverse quadratic radial basis function on non-uniform meshes, Chaos Soliton. Fract., 185 (2024), 115156. https://doi.org/10.1016/j.chaos.2024.115156 doi: 10.1016/j.chaos.2024.115156
|
| [12] |
T. Liu, Parameter estimation with the multigrid-homotopy method for a nonlinear diffusion equation, J. Comput. Appl. Math., 413 (2022), 114393. https://doi.org/10.1016/j.cam.2022.114393 doi: 10.1016/j.cam.2022.114393
|
| [13] |
W. Hu, Z. J. Fu, Z. C. Tang, Y. Gu, A meshless collocation method for solving the inverse Cauchy problem associated with the variable-order fractional heat conduction model under functionally graded materials, Eng. Anal. Bound. Elem., 140 (2022), 132–144. https://doi.org/10.1016/j.enganabound.2022.04.007 doi: 10.1016/j.enganabound.2022.04.007
|
| [14] |
Z. J. Fu, L. W. Yang, Q. Xi, C. S. Liu, A boundary collocation method for anomalous heat conduction analysis in functionally graded materials, Comput. Math. Appl., 88 (2021), 91–109. https://doi.org/10.1016/j.camwa.2020.02.023 doi: 10.1016/j.camwa.2020.02.023
|
| [15] |
L. D. Long, B. P. Moghaddam, Y. Gurefe, Iterative weighted Tikhonov regularization technique for inverse problems in time-fractional diffusion-wave equations within cylindrical domains, Comp. Appl. Math., 44 (2025), 215. https://doi.org/10.1007/s40314-025-03176-0 doi: 10.1007/s40314-025-03176-0
|
| [16] |
R. Brociek, A. Wajda, C. Napoli, G. Capizzi, D. Słota, An inverse problem for a fractional space-time diffusion equation with fractional boundary condition, Entropy, 28 (2026), 81. https://doi.org/10.3390/e28010081 doi: 10.3390/e28010081
|
| [17] |
H. M. Zhu, J. Zheng, Z. Y. Zhang, Approximate symmetry of time-fractional partial differential equations with a small parameter, Commun. Nonlinear Sci., 125 (2023), 107404. https://doi.org/10.1016/j.cnsns.2023.107404 doi: 10.1016/j.cnsns.2023.107404
|
| [18] | I. Podlubny, Fractional differential equations: An introduction to fractional derivatives, fractional differential equations, to methods of their solution and some of their applications, San Diego: Academic Press, 1998. |
| [19] |
S. T. Abdulazeez, M. Modanli, Solutions of fractional order pseudo-hyperbolic telegraph partial differential equations using finite difference method, Alex. Eng. J., 61 (2022), 12443–12451. https://doi.org/10.1016/j.aej.2022.06.027 doi: 10.1016/j.aej.2022.06.027
|
| [20] |
M. Nitiema, T. Tindano, W. Some, Crank–Nicolson method for the advection-diffusion equation involving a fractional Laplace operator, Abstr. Appl. Anal., 2025 (2025), 6642234. https://doi.org/10.1155/aaa/6642234 doi: 10.1155/aaa/6642234
|
| [21] | M. Abdalla, F. Bourse, A. De Caro, D. Pointcheval, Simple functional encryption schemes for inner products, In: Public-Key Cryptography–PKC 2015, Berlin: Springer, 2015,733–751. https://doi.org/10.1007/978-3-662-46447-2_33 |
Figures(12) / Tables(4)
Mousa J. Huntul, Mahmut Modanli. Numerical approach for solving the inverse problem: A two-dimensional time-fractional boundary value problem[J]. AIMS Mathematics, 2026, 11(3): 7078-7097. doi: 10.3934/math.2026291
DownLoad: