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

Lin, Ji; Reutskiy, Sergiy; Zhang, Yuhui; Sun, Yu; Lu, Jun

doi:10.3390/math11040929

Open AccessArticle

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

by

Ji Lin

^1,†,

Sergiy Reutskiy

^2,†,

Yuhui Zhang

^1,†

,

Yu Sun

^3,† and

Jun Lu

^3,*,†

¹

College of Mechanics and Materials, Hohai University, Nanjing 210098, China

²

A. Pidhornyi Institute of Mechanical Engineering Problems of NAS of Ukraine, 2/10 Pozharsky Street, 61046 Kharkiv, Ukraine

³

Nanjing Hydraulic Research Institute, Nanjing 210029, China

^*

Author to whom correspondence should be addressed.

^†

These authors contributed equally to this work.

Mathematics 2023, 11(4), 929; https://doi.org/10.3390/math11040929

Submission received: 5 January 2023 / Revised: 7 February 2023 / Accepted: 10 February 2023 / Published: 12 February 2023

(This article belongs to the Special Issue Computational Methods and Applications for Numerical Analysis)

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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

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MDPI and ACS Style

Lin, J.; Reutskiy, S.; Zhang, Y.; Sun, Y.; Lu, J. The Novel Analytical–Numerical Method for Multi-Dimensional Multi-Term Time-Fractional Equations with General Boundary Conditions. Mathematics 2023, 11, 929. https://doi.org/10.3390/math11040929

AMA Style

Lin J, Reutskiy S, Zhang Y, Sun Y, Lu J. The Novel Analytical–Numerical Method for Multi-Dimensional Multi-Term Time-Fractional Equations with General Boundary Conditions. Mathematics. 2023; 11(4):929. https://doi.org/10.3390/math11040929

Chicago/Turabian Style

Lin, Ji, Sergiy Reutskiy, Yuhui Zhang, Yu Sun, and Jun Lu. 2023. "The Novel Analytical–Numerical Method for Multi-Dimensional Multi-Term Time-Fractional Equations with General Boundary Conditions" Mathematics 11, no. 4: 929. https://doi.org/10.3390/math11040929

APA Style

Lin, J., Reutskiy, S., Zhang, Y., Sun, Y., & Lu, J. (2023). The Novel Analytical–Numerical Method for Multi-Dimensional Multi-Term Time-Fractional Equations with General Boundary Conditions. Mathematics, 11(4), 929. https://doi.org/10.3390/math11040929

Note that from the first issue of 2016, this journal uses article numbers instead of page numbers. See further details here.

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

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