Journal of Advanced Mathematical Modeling

Journal of Advanced Mathematical Modeling

A hybrid method for time fractional partial differential equations in an unbounded domain

Document Type : Original Paper

Authors
Mohammad Hossein Heydari 1
Seyede Elham Hosseini 2
Hadis Azin 3
1 Department of Mathematics, Shiraz University of Technology, Shiraz, Iran.
2 Department of Mathematics, Shiraz University of Technology,, Shiraz, Iran.
3 Department of Mathematics,, Hakim Sabzevari University,, Sabzevar, Iran.
10.22055/jamm.2025.44730.2198
Abstract
‎In this article‎, ‎two models of time-fractional partial differential equations‎, ‎i.e‎. ‎Sobolov and Cattaneo in the Caputo derivative type are considered in an unbounded space domain‎. ‎To solve these equations‎, ‎the appropriate variable change in Jacobi polynomials is used to construct orthogonal basis functions called Tanh-Jacobi functions in an unbounded domain‎. ‎The shifted orthogonal Jacobi polynomials and Tanh-Jacobi functions are applied for discretization and approximation of time and space variables‎, ‎respectively‎. ‎Also‎, ‎explicit formulas are calculated to compute classical and Caputo fractional derivatives in the numerical solution of these equations‎. ‎The proposed method turns the solution of the proposed problems into the solution of a system of algebraic equations‎, ‎which are obtained by calculating the unknowns of this algebraic system‎, ‎the numerical solution of the problems‎. ‎Finally‎, ‎the accuracy and efficiency of the proposed approach are studied by solving some numerical examples‎.
Keywords
Tanh-Jacobi functions
Shifted Jacobi polynomials
Sobolev equation
Cattaneo equation
Unbounded domain
Caputo fractional derivative‎
Subjects
[1] Avazzadeh Z., Nikan O., Tenreiro M. J., and Rasoulizadeh M. N., 2022. Numerical analysis of timefractional Sobolev equation for fluid-driven processes in impermeable rocks. Advances in Continuous
and Discrete Models, 2022(1), pp.48. doi:10.1186/s13662-022-03720-w
[2] Bird, R. B., 2002. Transport phenomena. Appl. Mech. Rev., 55(1), pp.R1-R4. doi:10.1115/1.1424298
[3] Ford N. J., Xiao J. and Yan Y., 2011. A finite element method for time fractional partial differential
equations. Fractional Calculus and Applied Analysis, 14, pp.454-474. doi:10.2478/s13540-011-0028-
2
[4] Heydari M. H., Azin H. and Hosseini. S. E., 2024. An operational matrix strategy for time fractional fokker-planck equation in an unbounded space domain. Physica Scripta, 99(6), pp.065250.
doi:10.1088/1402-4896/ad4a9f
[5] Li C., and Chen A., 2018. Numerical methods for fractional partial differential equations. International
Journal of Computer Mathematics, 95(6-7), pp.1048-1099. doi:10.1080/00207160.2017.1343941
[6] Li C., and Zeng F., 2012. Finite difference methods for fractional differential equations. International
Journal of Bifurcation and Chaos, 22(04), pp.1230014. doi:10.1142/S0218127412300145
[7] Mainardi F., Luchko Y., and Pagnini G., 2007. The fundamental solution of the space-time fractional
diffusion equation. arXiv preprint cond-mat/0702419, 4(2), pp.153-192. doi:10.48550/arXiv.condmat/0702419
[8] Metzler R., and Klafter J., 2000. The random walk’s guide to anomalous diffusion: a fractional dynamics approach. Physics reports, 339(1), pp.1-77. doi:10.1016/S0370-1573(00)00070-3
[9] Monje C. A., Chen Y. Q., Vinagre B. M., Xue D., and Feliu-Batlle V., 2010. Fractional-order systems
and controls: fundamentals and applications.Springer Science & Business Media, doi:10.1007/978-1-
84996-335-0
[10] Mostafa D., Zaky M. A., Hafez R. M., Hend, Ahmed S., Abdelkawy M. A. and Aldraiweesh A. A.,
2023. Tanh Jacobi spectral collocation method for the numerical simulation of nonlinear Schrödinger
equations on unbounded domain. Mathematical Methods in the Applied Sciences, 46(1), pp.656-674.
doi:10.1002/mma.8538
[11] Poedts S., 1999. Plasma Physics: an introduction to the theory of astrophysical, geophysical and
laboratory plasmas. Space science reviews, doi:10.1017/CBO9781139170598
[12] Scalas E., Gorenflo R. and Mainardi F., 2000. Fractional calculus and continuous-time finance.
Physica A: Statistical Mechanics and its Applications, 284(1-4), pp.376-384. doi:10.1016/S0378-
4371(00)00255-7
[13] Shen J., Tang T., and Wang L. L., 2011. Spectral methods: algorithms, analysis and applications.
Springer Science & Business Media, 41, doi:10.1007/978-3-540-71041-7
[14] Wang M., Yang N. and Guo Z. Y., 2011. Non-Fourier heat conductions in nanomaterials. Journal of
Applied Physics,110(6), doi:10.1063/1.3634078

  • Receive Date 16 March 2024
  • Revise Date 24 August 2024
  • Accept Date 25 September 2025