The Novel Analytical–Numerical Method for Multi-Dimensional Multi-Term Time-Fractional Equations with General Boundary Conditions

• DOI: 10.3390/math11040929
• Publication Date: 2023-02-13T10:59:40Z
• Authors: Ji Lin; Sergiy Reutskiy; Yuhui Zhang; Yu Sun; Jun Lu
• Keywords: analytic representation; collocation method; multi-term fractional equations; QA1-939; meshless method; 0101 mathematics; 01 natural sciences; multi-dimensional fractional equations; Mathematics

This article presents a simple but effective two-step analytical–numerical algorithm for solving multi-dimensional multi-term time-fractional equations. The first step is to derive an analytic representation that satisfies boundary requirements for 1D, 2D, and 3D problems, respectively. The second step is the meshless approximation where the Müntz polynomials are used to form the approximate solution and the unknown parameters are obtained by imposing the approximation for the governing equations. We illustrate first the detailed derivation of the analytic approximation and then the numerical implementation of the solution procedure. Several numerical examples are provided to verify the accuracy, efficiency, and adaptability to problems with general boundary conditions. The numerical results are compared with exact solutions and numerical methods reported in the literature, showing that the algorithm has great potential for multi-dimensional multi-term time-fractional equations with various boundary conditions.