A novel numerical manner for two‐dimensional space fractional diffusion equation arising in transport phenomena

• Published in: Numerical methods for partial differential equations Vol. 37; no. 2; pp. 1397 - 1406
• Main Authors: Tuan, Nguyen H., Aghdam, Yones Esmaeelzadeh, Jafari, Hossein, Mesgarani, Hamid
• Format: Journal Article
• Language: English
• Published: Hoboken, USA John Wiley & Sons, Inc 01.03.2021; Wiley Subscription Services, Inc
• Subjects: Boundary conditions; Chebyshev approximation; Chebyshev collocation method of the second kind; Convergence; Diffusion; Exact solutions; fractional derivative; Mathematical analysis; Polynomials; stability; Transport phenomena; two‐dimensional space fractional diffusion equation
• ISSN: 0749-159X; 1098-2426
• DOI: 10.1002/num.22586

Fractional diffusion equations include a consistent and efficient explanation of transport phenomena that manifest abnormal diffusion, that cannot be often represented by second‐order diffusion equations. In this article, a two‐dimensional space fractional diffusion equation (SFDE‐2D) with nonhomogeneous and homogeneous boundary conditions is considered in Caputo derivative sense. An instant and nevertheless accurate scheme is obtained by the finite‐difference discretization to get the semidiscrete in temporal derivative with convergence order O(δτ2). Moreover, space fractional derivative can be approximated based on the Chebyshev polynomials of second kind which are powerful methods for basing the operational matrix. The convergence and stability of the proposed scheme are discussed theoretically in detail. Finally, two numerical problems with an exact solution are given that numerical results show the effectiveness of the new techniques. These schemes can be simply extended to three spatial dimensions, which will be the subject of our subsequent research.