Solving time fractional partial differential equations with variable coefficients using a spatio-temporal meshless method

Xiangyun Qiu; Xingxing Yue; Xiangyun Qiu; Xingxing Yue

doi:10.3934/math.20241320

AIMS Mathematics

2024, Volume 9, Issue 10: 27150-27166. doi: 10.3934/math.20241320

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Solving time fractional partial differential equations with variable coefficients using a spatio-temporal meshless method

Xiangyun Qiu ¹,
Xingxing Yue ^{2
,
,}

1.
National Engineering Research Center for Intelligent Electrical Vehicle Power System, College of Mechanical and Electrical Engineering, Qingdao University, Qingdao 266071, China
2.
College of Materials Science and Engineering, Qingdao University, Qingdao 266071, China

Received: 16 July 2024 Revised: 25 August 2024 Accepted: 05 September 2024 Published: 19 September 2024
MSC : 35R11, 65M70, 68W25

This paper presents a novel spatio-temporal meshless method (STMM) for solving the time fractional partial differential equations (TFPDEs) with variable coefficients based on the space-time metric. The main idea of the STMM is to directly approximate the solutions of fractional PDEs by using a multiquadric function with the space-time distance within a space-time scale framework. Compared with the existing methods, the present meshless STMM entirely avoids the difference approximation of fractional temporal derivatives and can be easily applied to complicated irregular geometries. Furthermore, both regular and irregular nodal distribution can be used without loss of accuracy. For these reasons, this new space-time meshless method could be regarded as a competitive alternative to the conventional numerical algorithms based on difference decomposition for solving the TFPDEs with variable coefficients. Numerical experiments confirm the ability and accuracy of the proposed methodology.
- spatio-temporal meshless method,
- time fractional PDEs,
- space-time metric,
- variable coefficients,
- multiquadric function
Citation: Xiangyun Qiu, Xingxing Yue. Solving time fractional partial differential equations with variable coefficients using a spatio-temporal meshless method[J]. AIMS Mathematics, 2024, 9(10): 27150-27166. doi: 10.3934/math.20241320

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Abstract

This paper presents a novel spatio-temporal meshless method (STMM) for solving the time fractional partial differential equations (TFPDEs) with variable coefficients based on the space-time metric. The main idea of the STMM is to directly approximate the solutions of fractional PDEs by using a multiquadric function with the space-time distance within a space-time scale framework. Compared with the existing methods, the present meshless STMM entirely avoids the difference approximation of fractional temporal derivatives and can be easily applied to complicated irregular geometries. Furthermore, both regular and irregular nodal distribution can be used without loss of accuracy. For these reasons, this new space-time meshless method could be regarded as a competitive alternative to the conventional numerical algorithms based on difference decomposition for solving the TFPDEs with variable coefficients. Numerical experiments confirm the ability and accuracy of the proposed methodology.

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