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

Ji Lin, Sergiy Reutskiy, Yuhui Zhang, Yu Sun and Jun Lu ()
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Ji Lin: College of Mechanics and Materials, Hohai University, Nanjing 210098, China
Sergiy Reutskiy: A. Pidhornyi Institute of Mechanical Engineering Problems of NAS of Ukraine, 2/10 Pozharsky Street, 61046 Kharkiv, Ukraine
Yuhui Zhang: College of Mechanics and Materials, Hohai University, Nanjing 210098, China
Yu Sun: Nanjing Hydraulic Research Institute, Nanjing 210029, China
Jun Lu: Nanjing Hydraulic Research Institute, Nanjing 210029, China

Mathematics, 2023, vol. 11, issue 4, 1-26

Abstract: 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.

Keywords: multi-dimensional fractional equations; multi-term fractional equations; meshless method; collocation method; analytic representation (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2023
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