jzs-10829

Numerical investigation of the two– dimensional time–dependent diffusion equation using Radial basis functions

Hamid Mesgarani1, Masoud Bakhshandeh1, and Yones Esmaeelzade Aghdam*1

1 Faculty of Science, Shahid Rajaee Teacher Training University, Lavizan, Tehran, 16785-163

*Corresponding author’s e-mail: yonesesmaeelzade@gmail.com 

Original: 21 February 2020        Revised: 17 August 2020        Accepted: 26 September 2020        Published online: 20 December 2020  

Doi Link:

Abstract

This paper develops a numerical method for solving the partial differential equation in terms of Caputo derivatives with Dirichlet boundary conditions. The approach is based on the two-dimensional Chebyshev wavelet of the second kind with the operational matrix of the collocation method. Furthermore, the convergence and error bound of the proposed method are investigated. For the illustration of the effects of the proposed method, we solve four examples by the presented technique. The obtained results are compared with the results obtained via other numerical methods in which our results are much more accurate than others.

Key Words: The partial differential equation, The two-dimensional Chebyshev wavelet, The theory of fractional derivatives, The convergence and error bound 

References

[1]  Aghdam, Y.E, Mesgrani, H, Javidi, M, Nikan, O.  "A computational approach for the space-time fractional advection diffusion equation arising in contaminant transport through porous media". Engineering with Computers. Vol. 39, pp. 1-6. (2020).

[2]  Henry B, Wearne S.  "Existence of Turing instabilities in a two-species fractional reaction–diffusion system". SIAM Journal on Applied Mathematics. Vol. 62, pp. 870-887. (2002).

[3]  Metzler R, Klafter J. "The random walk's guide to anomalous diffusion: a fractional dynamics approach". Physics Reports. Vol. 339, pp. 1-77. (2000).

[4]  Safdari, H, Mesgarani, H, Javidi, M, Aghdam, Y.E.  "Convergence analysis of the space fractional-order diffusion equation based on the compact finite difference scheme'. Computational and Applied Mathematics. Vol. 39, nO. 2, pp. 1-15. (2020).

[5]  Kilbas A. A, Srivastava H. M, Trujillo J. J. "Theory and applications of fractional differential equations". Amestardam, North-Holland: Elsevier. (2006).

[6]  Machado J. T, Kiryakova V, Mainardi F.  "Recent history of fractional calculus". Commun. Nonlinear Sci. Numer. Simul. Vol. 3, pp. 1140-1153. (2011).

[7] Song L, Xu S, Yang J. "Dynamical models of happiness with fractional order". Communications in Nonlinear Science and Numerical Simulation. Vol. 15, pp. 616-628. (2010).

[8]  Chen S, Liu F. "Finite difference approximations for the fractional Fokker-Planck equation". Appl Math Model. Vol. 33, pp. 256-273. (2009).

[9]  Chen W, Ye L, Sun H. "Fractional diffusion equations by the Kansa method". Comput Math Appl Vol. 59, pp. 1614-1620. (2010).

[10] Fu WCZJ, Yang H, Sun H. "Boundary particle method for laplace transformed time fractional diffusion equations". J Comput Phys. Vol. 235, pp. 52-62. (2013).

[11] Sun H. G, Chen W, Li C, Chen YQ.  "Finite difference schemes for variable-order time fractional diffusion equation". Int J. Bifurcation Chaos. Vol. 22, No. 4, 1250085. (2012).

[12] Heydari M. H, Hooshmandasl M. R, Maalek F. M, Fereidouni F. "Two-dimensional Legendre wavelets for solving fractional Poisson equation with dirichlet boundary conditions". Engineering Analysis with Boundary Elements. Vol. 37, pp. 1331-1338.  (2013).

[13] Ray S. S, Gupta A. K. "Two-dimensional Legendre wavelet method for travelling wave solutions of time-fractional generalized seventh order KdV equation". Computers Mathematics with Applications. Vol. 6, pp. 1118-1133. (2017).

[14] Gupata A. K, Ray S. S. "Numerical treatment for the solution of fractional fifth-order Sawada-Kotera equation using second kind Chebyshev wavelet method". Appl. Math. Modelling. Vol. 17, pp. 5121-5130. (2015).

[15]  Lepik U. "Solving PDEs with the aid of two-dimensional Haar wavelets". Comput Math Appl. Vol. 61, pp. 1873-1879. (2011).

[16]  Castro L. M. S, Ferreira A, Bertoluzza S, Patra R, Reddy J.  "A wavelet collocation method for the static analysis of sandwich plates using a layerwise theory". Compos Struct. Vol. 92, pp. 1786-1792. (2010).

[17]  Rehman M and Khan R. A. "The Legendre wavelet method for solving fractional differential equations". Commun. Nonlinear Sci. Numer. Simul. Vol. 16, pp. 4163-4173. (2011).

[18]  Odibat Z, Momani S.  "A generalized differential transform method for linear partial differential equations of fractional order". Appl. Math. Lett 2008;  21: 194-199.

[19] Yi M, Huang J, Wei J. "Block pulse operational matrix method for solving fractional partial differential equation". Applied Mathematics and Computation. Vol. 221, pp. 121-131. (2013).

[20] Gupta A. K , Ray S. S.  "An investigation with Hermite Wavelets for accurate solution of fractional Jaulent-Miodek equation associated with energy-dependent Schrödinger potential". Appl. Math. Comp. Vol. 270, pp. 458-471. (2015).

[21]   Wei L, He Y, Zhang A. "Numerical analysis of the fractional seventh-order KDV equation using an implicit fully discrete local discontinuous Galerkin method". International Journal of Numerical Analysis and Modeling. Vol. 10, pp. 430-444. (2013).

[22]   Podlubny I.  "Fractional differential equations". San Diego, USA: Academic Press. (1999).

[23]   Samko S. G, Kilbas A. A, Marichev O. I.  "Fractional integrals and derivatives: Theory and Applications". Taylor and Francis, London. (1993).

[24]   Caputo Me. "Linear model of dissipation whose Q is almost frequency independent". The Geophysical journal of Royal Astronomical Society. Vol. 13, pp. 529-539. (1967).

[25]   Sohrabi S. "Comparison Chebyshev wavelets method with BPFs method for solving Abel’s integral equation". Ain Shams Engineering Journal. Vol. 2, pp. 249- 254. (2011).

View Full Article