An RBF-FD Method for Numerical Solutions of 2D Diffusion-Wave and Diffusion Equations of Distributed Fractional Order

• Published in: Journal of nonlinear mathematical physics Vol. 30; no. 4; pp. 1357 - 1374
• Main Authors: Taghipour, Fatemeh, Shirzadi, Ahmad, Safarpoor, Mansour
• Format: Journal Article
• Language: English
• Published: Dordrecht Springer Netherlands 01.12.2023; Springer Nature B.V
• Subjects: Accuracy; Algorithms; Approximation; Boundary conditions; Complex systems; Complexity; Derivatives; Differential equations; Discretization; Finite difference method; Fractional calculus; Integrals; Mathematical Physics; Mathematics; Mathematics and Statistics; Methods; Numerical analysis; Quadratures; Radial basis function; Research Article; Wave equations; Local meshless methods; 65D12; Finite differences; 35R11; Distributed-order fractional differential equations; RBF-FD; 65N99
• ISSN: 1776-0852; 1402-9251; 1776-0852
• DOI: 10.1007/s44198-023-00153-1

The subject of this paper is to propose a numerical algorithm for solving 2D diffusion and diffusion-wave equations of distributed order fractional derivatives. Such equations arise in modelling complex systems and have many important applications. Existence of integral term over the order of fractional derivative causes the high complexity of these equations and so their numerical solutions needs special cares. Using Gauss quadrature approach for discretizing the integral term of fractional derivative converts the distributed equation into a multi-term fractional differential equation. Then, the time variable is discretized with a suitable finite difference approach. The resultant semi-discretized equations are fully discretized by a radial basis function-generated finite difference based method. Convergence of the method are studied numerically. Various kind of test problems are considered for a comprehensive numerical study and the results confirm the efficiency of the method.