A novel numerical method for solution of fractional partial differential equations involving the $ \psi $-Caputo fractional derivative

Amjid Ali; Teruya Minamoto; Rasool Shah; Kamsing Nonlaopon; Amjid Ali; Teruya Minamoto; Rasool Shah; Kamsing Nonlaopon

doi:10.3934/math.2023110

AIMS Mathematics

2023, Volume 8, Issue 1: 2137-2153. doi: 10.3934/math.2023110

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Research article

A novel numerical method for solution of fractional partial differential equations involving the $ \psi $-Caputo fractional derivative

1.
Faculty of Science and Engineering, Saga University, Saga 840-8502, Japan
2.
Department of Mathematics, Abdul Wali Khan University Mardan, Pakistan
3.
Department of Mathematics, Faculty of Science, Khon Kaen University, Khon Kaen 40002, Thailand

Received: 23 August 2022 Revised: 03 October 2022 Accepted: 06 October 2022 Published: 28 October 2022
MSC : 26A33, 35A20, 35A35, 33B15, 33F05

In this study, the $ \psi $-Haar wavelets operational matrix of integration is derived and used to solve linear $ \psi $-fractional partial differential equations ($ \psi $-FPDEs) with the fractional derivative defined in terms of the $ \psi $-Caputo operator. We approximate the highest order fractional partial derivative of the solution of linear $ \psi $-FPDE using Haar wavelets. By combining the operational matrix and $ \psi $-fractional integration, we approximate the solution and its other $ \psi $-fractional partial derivatives. Then substituting these approximations in the given $ \psi $-FPDEs, we obtained a system of linear algebraic equations. Finally, the approximate solution is obtained by solving this system. The simplicity and effectiveness of the proposed method as a mathematical tool for solving $ \psi $-Fractional partial differential equations is one of its main advantages. The sparse nature of the operational matrices improves the ability of the proposed method to execute with less computation complexity. Numerical examples are provided to show the efficiency and effectiveness of the method.
Citation: Amjid Ali, Teruya Minamoto, Rasool Shah, Kamsing Nonlaopon. A novel numerical method for solution of fractional partial differential equations involving the $ \psi $-Caputo fractional derivative[J]. AIMS Mathematics, 2023, 8(1): 2137-2153. doi: 10.3934/math.2023110

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Abstract

In this study, the $ \psi $-Haar wavelets operational matrix of integration is derived and used to solve linear $ \psi $-fractional partial differential equations ($ \psi $-FPDEs) with the fractional derivative defined in terms of the $ \psi $-Caputo operator. We approximate the highest order fractional partial derivative of the solution of linear $ \psi $-FPDE using Haar wavelets. By combining the operational matrix and $ \psi $-fractional integration, we approximate the solution and its other $ \psi $-fractional partial derivatives. Then substituting these approximations in the given $ \psi $-FPDEs, we obtained a system of linear algebraic equations. Finally, the approximate solution is obtained by solving this system. The simplicity and effectiveness of the proposed method as a mathematical tool for solving $ \psi $-Fractional partial differential equations is one of its main advantages. The sparse nature of the operational matrices improves the ability of the proposed method to execute with less computation complexity. Numerical examples are provided to show the efficiency and effectiveness of the method.

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