Spectral tau solution of the linearized time-fractional KdV-Type equations

Waleed Mohamed Abd-Elhameed; Youssri Hassan Youssri; Waleed Mohamed Abd-Elhameed; Youssri Hassan Youssri

doi:10.3934/math.2022830

AIMS Mathematics

2022, Volume 7, Issue 8: 15138-15158. doi: 10.3934/math.2022830

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Spectral tau solution of the linearized time-fractional KdV-Type equations

Waleed Mohamed Abd-Elhameed ^{1,2
,
,},
Youssri Hassan Youssri ²

1.
Department of Mathematics, College of Science, University of Jeddah, Jeddah 23218, Saudi Arabia
2.
Department of Mathematics, Faculty of Science, Cairo University, Giza 12613, Egypt

Received: 06 April 2022 Revised: 03 June 2022 Accepted: 13 June 2022 Published: 16 June 2022
MSC : 65XX, 65M70, 33C45

The principal objective of the current paper is to propose a numerical algorithm for treating the linearized time-fractional KdV equation based on selecting two different sets of basis functions. The members of the first set are selected to be suitable combinations of the Chebyshev polynomials of the second kind and also to be compatible with the governing boundary conditions of the problem, while the members of the second set are selected to be the shifted second-kind Chebyshev polynomials. After expressing the approximate solutions as a double expansion of the two selected basis functions, the spectral tau method is applied to convert the equation with its underlying conditions into a linear system of algebraic equations that can be treated numerically with suitable standard procedures. The convergence analysis of the double series solution is carefully tested. Some numerical examples accompanied with comparisons with some other methods in the literature are displayed aiming to demonstrate the applicability and accuracy of the presented algorithm.
- tau method,
- second-kind Chebyshev polynomials,
- KdV equations,
- fractional differential equations,
- convergence analysis
Citation: Waleed Mohamed Abd-Elhameed, Youssri Hassan Youssri. Spectral tau solution of the linearized time-fractional KdV-Type equations[J]. AIMS Mathematics, 2022, 7(8): 15138-15158. doi: 10.3934/math.2022830

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Abstract

The principal objective of the current paper is to propose a numerical algorithm for treating the linearized time-fractional KdV equation based on selecting two different sets of basis functions. The members of the first set are selected to be suitable combinations of the Chebyshev polynomials of the second kind and also to be compatible with the governing boundary conditions of the problem, while the members of the second set are selected to be the shifted second-kind Chebyshev polynomials. After expressing the approximate solutions as a double expansion of the two selected basis functions, the spectral tau method is applied to convert the equation with its underlying conditions into a linear system of algebraic equations that can be treated numerically with suitable standard procedures. The convergence analysis of the double series solution is carefully tested. Some numerical examples accompanied with comparisons with some other methods in the literature are displayed aiming to demonstrate the applicability and accuracy of the presented algorithm.

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