On the Numerical Discretization of the Fractional Advection-Diffusion Equation with Generalized Caputo-Type Derivatives on Non-uniform Meshes

doi:10.1007/s42967-024-00416-8

Abstract

Abstract: A broad framework for time-fractional advection-diffusion equations is considered by incorporating a general class of Caputo-type fractional derivatives that includes many well-known fractional derivatives as particular cases. Within this general context, in this paper, we propose a numerical approach to provide approximate solutions to time-fractional advection-diffusion models that involve fractional derivatives of the generalized class. The developed approach is based on discretizing the studied models with respect to spatio-temporal domains using non-uniform meshes. Moreover, we discuss the stability of the proposed approach, which does not require solving large systems of linear equations. Numerical results and 3D graphics are presented for some time-fractional advection-diffusion models using several fractional derivatives. The feasibility and the reliability of the proposed approach are clearly demonstrated by comparing the numerical solutions of the studied models with the exact solutions in cases with a known solution.

Key words: Advection-diffusion, Caputo derivative, Generalized fractional derivative, Finite difference method, Predictor-corrector method, Numerical solution

CLC Number:

Zaid Odibat. On the Numerical Discretization of the Fractional Advection-Diffusion Equation with Generalized Caputo-Type Derivatives on Non-uniform Meshes[J]. Communications on Applied Mathematics and Computation, 2026, 8(1): 130-148.

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References

[1] Almeida, R.: A Caputo fractional derivative of a function with respect to another function. Commun. Nonlinear Sci. Numer. Simul. 44, 460-481 (2017)
[2] Atangana, A., Baleanu, D.: New fractional derivatives with nonlocal and non-singular kernel: theory and application to heat transfer model. Therm. Sci. 20, 763-769 (2016)
[3] Caputo, M., Fabrizio, M.: A new definition of fractional derivative without singular kernel. Prog. Fract. Differ. Appl. 1, 73-85 (2015)
[4] Diethelm, K., Ford, N.J., Freed, A.D.: A predictor-corrector approach for the numerical solution of fractional differential equations. Nonlinear Dyn. 29(1/2/3/4), 3-22 (2002)
[5] Hajaj, R., Odibat, Z.: Numerical solutions of fractional models with generalized Caputo-type derivatives. Phys. Scr. 98(4), 045206 (2023)
[6] Herrmann, R.: Fractional Calculus: an Introduction for Physicists. World Scientific Publishing Co. Pte. Ltd, Singapore (2014)
[7] Hilfer, R.: Applications of Fractional Calculus in Physics. World Scientific Publishing Co. Pte. Ltd, Singapore (2000)
[8] Jannelli, A.: Adaptive numerical solutions of time-fractional advection-diffusion-reaction equations. Commun. Nonlinear Sci. Numer. Simul. 105, 106073 (2022)
[9] Jannelli, A., Ruggieri, M., Speciale, M.P.: Numerical solutions of space-fractional advection-diffusion equations with nonlinear source term. Appl. Numer. Math. 155, 93-102 (2020)
[10] Jarad, F., Abdeljawad, T.: Generalized fractional derivatives and Laplace transform. Discrete Contin. Dyn. Syst. 13(3), 709-722 (2020)
[11] Kilbas, A., Srivastava, H., Trujillo, J.: Theory and Applications of Fractional Differential Equations. Elsevier, Amsterdam (2006)
[12] Maji, S., Natesan, S.: Analytical and numerical solutions of time-fractional advection-diffusion-reaction equation. Appl. Numer. Math. 185, 549-570 (2023)
[13] Miller, K.S., Ross, B.: An Introduction to the Fractional Calculus and Fractional Differential Equations. John Wiley and Sons Inc, New York (1993)
[14] Odibat, Z.: A universal predictor-corrector algorithm for numerical simulation of generalized fractional differential equations. Nonlinear Dyn. 105, 2363-2374 (2021)
[15] Odibat, Z.: Numerical solutions of linear time-fractional advection-diffusion equations with modified Mittag-Leffler operator in a bounded domain. Phys. Scr. 99(1), 015205 (2024)
[16] Odibat, Z., Baleanu, D.: A linearization-based approach of homotopy analysis method for non-linear time-fractional parabolic PDEs. Math. Methods Appl. Sci. 42(18), 7222-7232 (2019)
[17] Odibat, Z., Baleanu, D.: Numerical simulation of initial value problems with generalized Caputo-type fractional derivatives. Appl. Numer. Math. 165, 94-105 (2020)
[18] Odibat, Z., Baleanu, D.: On a new modification of the Erdélyi-Kober fractional derivative. Fractal Fract. 5(3), 121 (2021)
[19] Odibat, Z., Baleanu, D.: Nonlinear dynamics and chaos in fractional differential equations with a new generalized Caputo fractional derivative. Chin. J. Phys. 77, 1003-1014 (2022)
[20] Odibat, Z., Baleanu, D.: A new fractional derivative operator with generalized cardinal sine kernel: numerical simulation. Math. Comput. Simul. 212, 224-233 (2023)
[21] Odibat, Z., Baleanu, D.: New solutions of the fractional differential equations with modified Mittag-Leffler kernel. J. Comput. Nonlinear Dyn. 18(9), 091007 (2023)
[22] Odibat, Z., Erturk, V.S., Kumar, P., Govindaraj, V.: Dynamics of generalized Caputo type delay fractional differential equations using a modified predictor-corrector scheme. Phys. Scr. 96(12), 125213 (2021)
[23] Odibat, Z., Shawagfeh, N.: An optimized linearization-based predictor-corrector algorithm for the numerical simulation of nonlinear FDEs. Phys. Scr. 95(6), 065202 (2020)
[24] Oldham, K.B., Spanier, J.: The Fractional Calculus. Academic Press, New York (1974)
[25] Osler, T.J.: Leibniz rule for fractional derivatives generalized and an application to infinite series. SIAM J. Appl. Math. 18(3), 658-674 (1970)
[26] Pandey, R.K., Singh, O.P., Baranwal, V.K., Tripathi, M.P.: An analytic solution for the space-time fractional advection-dispersion equation using the optimal homotopy asymptotic method. Comput. Phys. Commun. 183(10), 2098-2106 (2012)
[27] Samko, G., Kilbas, A., Marichev, O.: Fractional Integrals and Derivatives: Theory and Applications. Gordon and Breach, Amsterdam (1993)
[28] Soori, Z., Aminataei, A.: A new approximation to Caputo-type fractional diffusion and advection equations on non-uniform meshes. Appl. Numer. Math. 144, 21-41 (2019)
[29] Sousa, E.: An explicit high order method for fractional advection diffusion equations. J. Comput. Phys. 278, 257-274 (2014)
[30] West, B.J.: Fractional Calculus View of Complexity: Tomorrow’s Science. Taylor and Francis, UK (2015)
[31] Zerari, A., Odibat, Z., Shawagfeh, N.: On the formulation of a predictor-corrector method to model IVPs with variable-order Liouville-Caputo-type derivatives. Math. Methods Appl. Sci. 46(18), 19100-19114 (2023)
[32] Zhang, F., Gao, X., Xie, Z.: Difference numerical solutions for time-space fractional advection diffusion equation. Bound. Value Probl. 2019, 14 (2019). https://doi.org/10.1186/s13661-019-1120-5
[33] Zhu, X.G., Nie, Y.F., Zhang, W.: An efficient differential quadrature method for fractional advection-diffusion equation. Nonlinear Dyn. 90, 1807-1827 (2017)