A meshless method based on the Laplace transform for multi-term time-space fractional diffusion equation

Zihan Yue; Wei Jiang; Boying Wu; Biao Zhang; Zihan Yue; Wei Jiang; Boying Wu; Biao Zhang

doi:10.3934/math.2024343

AIMS Mathematics

2024, Volume 9, Issue 3: 7040-7062. doi: 10.3934/math.2024343

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Research article

A meshless method based on the Laplace transform for multi-term time-space fractional diffusion equation

1.
Department of Mathematics, Harbin Institute of Technology, Harbin 150001, China
2.
Department of Mathematics, Harbin Institute of Technology at Weihai, Weihai 264209, China

Received: 05 December 2023 Revised: 25 January 2024 Accepted: 26 January 2024 Published: 19 February 2024
MSC : 65M12, 65N50

Multi-term fractional diffusion equations can be regarded as a generalisation of fractional diffusion equations. In this paper, we develop an efficient meshless method for solving the multi-term time-space fractional diffusion equation. First, we use the Laplace transform method to deal with the multi-term time fractional operator, we transform the time into complex frequency domain by Laplace transform. The properties of the Laplace transform with respect to fractional-order operators are exploited to deal with multi-term time fractional-order operators, overcoming the dependence of fractional-order operators with respect to time and giving better results. Second, we proposed a meshless method to deal with space fractional operators on convex region based on quintic Hermite spline functions based on the theory of polynomial functions dense theorem. Meanwhile, the approximate solution of the equation is obtained through theory of the minimum residual approximate solution, and the error analysis are provided. Third, we obtain the numerical solution of the diffusion equation by inverse Laplace transform. Finally, we first experimented with a single space-time fractional-order diffusion equation to verify the validity of our method, and then experimented with a multi-term time equation with different parameters and regions and compared it with the previous method to illustrate the accuracy of our method.
- Laplace transform,
- multi-term time-space fractional diffusion equation,
- quintic Hermite method,
- the minimum residual solution
Citation: Zihan Yue, Wei Jiang, Boying Wu, Biao Zhang. A meshless method based on the Laplace transform for multi-term time-space fractional diffusion equation[J]. AIMS Mathematics, 2024, 9(3): 7040-7062. doi: 10.3934/math.2024343

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Abstract

Multi-term fractional diffusion equations can be regarded as a generalisation of fractional diffusion equations. In this paper, we develop an efficient meshless method for solving the multi-term time-space fractional diffusion equation. First, we use the Laplace transform method to deal with the multi-term time fractional operator, we transform the time into complex frequency domain by Laplace transform. The properties of the Laplace transform with respect to fractional-order operators are exploited to deal with multi-term time fractional-order operators, overcoming the dependence of fractional-order operators with respect to time and giving better results. Second, we proposed a meshless method to deal with space fractional operators on convex region based on quintic Hermite spline functions based on the theory of polynomial functions dense theorem. Meanwhile, the approximate solution of the equation is obtained through theory of the minimum residual approximate solution, and the error analysis are provided. Third, we obtain the numerical solution of the diffusion equation by inverse Laplace transform. Finally, we first experimented with a single space-time fractional-order diffusion equation to verify the validity of our method, and then experimented with a multi-term time equation with different parameters and regions and compared it with the previous method to illustrate the accuracy of our method.

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