Abstract: In this paper, the first boundary value problem for the Aller equation of fractional time order with generalized memory functions was considered. For the numerical solution of the problem, two difference schemes of an increased order of approximation are constructed. In the case of variable coefficients, a second-order difference scheme of approximation is proposed, both in time and in space. A compact difference scheme of the fourth order of approximation in space and the second order in time for the generalized Aller equation with constant coefficients is proposed. A priori estimates for solutions of the mentioned difference schemes are obtained by the method of energy inequalities. Their unconditional stability and convergence are proved. It is shown that the convergence rate coincides with the order of approximation error in the case of a sufficiently smooth solution of the original problem. On the basis of the proposed algorithms, numerical calculations of test problems were carried out, confirming the obtained theoretical results. All calculations were performed using the Julia v1.5.1 programming language.
Keywords: fractional derivative, generalized memory kernel, a priori estimates, fractional order diffusion equation, difference schemes, stability, convergence
For citation: Alikhanov, A. A., Apekov, A. M. and Khibiev, A. Kh. Higher-Order Approximation Difference Scheme for the Generalized Aller Equation of Fractional Order, Vladikavkaz Math. J., 2021, vol. 23, no. 3, pp. 5-15. DOI 10.46698/p3608-5250-8760-g
1. Samko, S. G, Kilbas, A. A. and Marichev, O. I. Fractional Integrals
and Derivatives: Theory and Applications , Florida, CRC Press, 1993, 1006 p.
2. Oldham, K. B. and Spanier, J. The Fractional Calculus, New York, Academic Press, 1974, 322 p.
3. Podlubny, I. Fractional Differential Equations, San Diego, Academic Press, 1999, 340 p.
4. Hilfer, R. Applications of Fractional Calculus in Physics, Singapore, World Scientific, 2000, 472 p.
5. Kilbas, A. A., Srivastava, H. M. and Trujillo, J. J. Theory and Applications of Fractional Differential Equations, Amsterdam, Elsevier, 2006, 540 p.
6. Alikhanov, A. A. Boundary Value Problems for the Diffusion Equation of the Variable Order in Differential and Difference Settings, Applied Mathematics and Computation, 2012, vol. 219, no. 8, pp. 3938-3946. DOI: 10.1016/j.amc.2012.10.029.
7. Alikhanov, A. A. Numerical Methods of Solutions of Boundary Value Problems for the Multi-Term Variable-Distributed Order Diffusion Equation, Applied Mathematics and Computation, 2015, vol. 268, pp. 12-22. DOI: 10.1016/j.amc.2015.06.045.
8. Sandev, T., Chechkin, A., Kantz, H. and Metzler, R. Diffusion and Fokker-Planck-Smoluchowski Equations with Generalized Memory Kernel, Fractional Calculus and Applied Analysis, 2015, vol. 18, no. 4, pp. 1006-1038. DOI: 10.1515/fca-2015-0059.
9. Alikhanov, A. A. A Priori Estimates for Solutions of Boundary Value Problems for Fractional-Order Equations, Differential Equations, 2011, vol. 46, no. 5, pp. 660-666. DOI: 10.1134/S0012266110050058.
10. Alikhanov, A. A. A New Difference Scheme for the Time Fractional Diffusion Equation, Journal of Computational Physics, 2015, vol. 280, pp. 424-438. DOI: 10.1016/j.jcp.2014.09.031.
11. Alikhanov, A. A. Stability and Convergence of Difference Schemes for Boundary Value Problems for the Fractional-Order Diffusion Equation, Computational Mathematics and Mathematical Physics, 2016, vol. 56, no. 4, pp. 561-575. DOI: 10.1134/S0965542516040035.
12. Alikhanov, A. A. A Time-Fractional Diffusion Equation with Generalized Memory Kernel in Differential and Difference Settings with Smooth Solutions, Computational Methods in Applied Mathematics, 2017, vol. 17, no. 4, pp. 647-660. DOI: 10.1515/cmam-2017-0035.
13. Alikhanov, A., Beshtokov, M. and Mehra, M. The Crank-Nicolson Type Compact Difference Scheme for a Loaded Time-Fractional Hallaire'S Equation, Fractional Calculus and Applied Analysis , 2021, vol. 24, no. 4, pp. 1231-1256. DOI: 10.1515/fca-2021-0053.
14. Gao, G. H., Alikhanov, A. A. and Sun, Z. Z. The Temporal Second Order Difference Schemes Based on the Interpolation Approximation for Solving the Time Multi-Term and Distributed-Order Fractional Sub-Diffusion Equations, Journal of Scientific Computing, 2017, vol. 73, no. 1, pp. 93-121. DOI: 10.1007/s10915-017-0407-x.
15. Khibiev, A. Kh. Stability and Convergence of Difference Schemes for the Multi-Term Time-Fractional Diffusion Equation with Generalized Memory Kernels, Vestn. Samar. Gos. Tekhn. Univ., Ser. Fiz.-Mat. Nauki [Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences], 2019, vol. 23, no. 3, pp. 582--597 (in Russian). DOI: 10.14498/vsgtu1690.
16. Khibiev, A., Alikhanov, A. and Huang, C. A Second Order Difference Scheme for Time Fractional Diffusion Equation with Generalized Memory Kernel, 2021. arXiv:2108.10596 [cs, math].
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