Meshfree numerical integration for some challenging multi-term fractional order PDEs

Abdul Samad; Imran Siddique; Fahd Jarad; Abdul Samad; Imran Siddique; Fahd Jarad

doi:10.3934/math.2022785

AIMS Mathematics

2022, Volume 7, Issue 8: 14249-14269. doi: 10.3934/math.2022785

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Meshfree numerical integration for some challenging multi-term fractional order PDEs

1.
School of Mathematics, Northwest University Xi'an, 710127, China
2.
Department of Mathematics, University of Management and Technology, Lahore 54770, Pakistan
3.
Department of Mathematics, Cankaya University, Etimesgut, Ankara, Turkey
4.
Department of Medical Research, China Medical University Hospital, China Medical University, Taichung, Taiwan

Received: 08 March 2022 Revised: 03 May 2022 Accepted: 06 May 2022 Published: 30 May 2022
MSC : 35G31, 35G35, 65D12

Fractional partial differential equations (PDEs) have key role in many physical, chemical, biological and economic problems. Different numerical techniques have been adopted to deal the multi-term FPDEs. In this article, the meshfree numerical scheme, Radial basis function (RBF) is discussed for some time-space fractional PDEs. The meshfree RBF method base on the Gaussian function and is used to test the numerical results of the time-space fractional PDE problems. Riesz fractional derivative and Grünwald-Letnikov fractional derivative techniques are used to deal the space fractional derivative terms while the time-fractional derivatives are iterated by Caputo derivative method. The accuracy of the suggested scheme is analyzed by using $ L_\infty $-norm. Stability and convergence analysis are also discussed.
- multi-term fractional derivatives,
- Caputo and Grünwald-Letnikov derivatives,
- radial basis function method
Citation: Abdul Samad, Imran Siddique, Fahd Jarad. Meshfree numerical integration for some challenging multi-term fractional order PDEs[J]. AIMS Mathematics, 2022, 7(8): 14249-14269. doi: 10.3934/math.2022785

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Abstract

Fractional partial differential equations (PDEs) have key role in many physical, chemical, biological and economic problems. Different numerical techniques have been adopted to deal the multi-term FPDEs. In this article, the meshfree numerical scheme, Radial basis function (RBF) is discussed for some time-space fractional PDEs. The meshfree RBF method base on the Gaussian function and is used to test the numerical results of the time-space fractional PDE problems. Riesz fractional derivative and Grünwald-Letnikov fractional derivative techniques are used to deal the space fractional derivative terms while the time-fractional derivatives are iterated by Caputo derivative method. The accuracy of the suggested scheme is analyzed by using $ L_\infty $-norm. Stability and convergence analysis are also discussed.

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