Fibonacci collocation pseudo-spectral method of variable-order space-fractional diffusion equations with error analysis

A. S. Mohamed; A. S. Mohamed

doi:10.3934/math.2022789

AIMS Mathematics

2022, Volume 7, Issue 8: 14323-14337. doi: 10.3934/math.2022789

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Fibonacci collocation pseudo-spectral method of variable-order space-fractional diffusion equations with error analysis

A. S. Mohamed ^,

Department of Mathematics, Faculty of Science, Helwan University, Cairo 11795, Egypt

Received: 05 March 2022 Revised: 27 April 2022 Accepted: 10 May 2022 Published: 01 June 2022
MSC : 11B39, 26A33, 35K57

In this article, we evaluated the approximate solutions of one-dimensional variable-order space-fractional diffusion equations (sFDEs) by using a collocation method. This method depends on operational matrices for fractional derivatives and the integration of generalized Fibonacci polynomials. In this method, a Caputo fractional derivative of variable order is applied. Some properties of these polynomials (using boundary conditions) are presented to simplify and transform sFDEs into a system of equations with the expansion coefficients of the solution. Also, we discuss the convergence and error analysis of the generalized Fibonacci expansion. Finally, we compare the obtained results with those obtained via the other methods.
- fractional diffusion equation,
- variable-order fractional equation,
- collocation method,
- generalized Fibonacci polynomials,
- convergence analysis
Citation: A. S. Mohamed. Fibonacci collocation pseudo-spectral method of variable-order space-fractional diffusion equations with error analysis[J]. AIMS Mathematics, 2022, 7(8): 14323-14337. doi: 10.3934/math.2022789

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Abstract

In this article, we evaluated the approximate solutions of one-dimensional variable-order space-fractional diffusion equations (sFDEs) by using a collocation method. This method depends on operational matrices for fractional derivatives and the integration of generalized Fibonacci polynomials. In this method, a Caputo fractional derivative of variable order is applied. Some properties of these polynomials (using boundary conditions) are presented to simplify and transform sFDEs into a system of equations with the expansion coefficients of the solution. Also, we discuss the convergence and error analysis of the generalized Fibonacci expansion. Finally, we compare the obtained results with those obtained via the other methods.

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