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# A Pseudo-Spectral Scheme for Systems of Two-Point Boundary Value Problems with Left and Right Sided Fractional Derivatives and Related Integral Equations

I. G. Ameen1, N. A. Elkot2, M. A. Zaky3,*, A. S. Hendy4,5, E. H. Doha2

1 Department of Mathematics, Faculty of Science, Al-Azhar University, Cairo, Egypt
2 Department of Mathematics, Faculty of Science, Cairo University, Giza, 12613, Egypt
3 Department of Applied Mathematics, Physics Division, National Research Centre, Dokki, Cairo, 12622, Egypt
4 Department of Computational Mathematics and Computer Science, Institute of Natural sciences and Mathematics, Ural Federal University, Yekaterinburg, 620002, Russia
5 Department of Mathematics, Faculty of Science, Benha University, Benha, 13511, Egypt

* Corresponding Authors: M. A. Zaky. Email: ;

Computer Modeling in Engineering & Sciences 2021, 128(1), 21-41. https://doi.org/10.32604/cmes.2021.015310

## Abstract

We target here to solve numerically a class of nonlinear fractional two-point boundary value problems involving left- and right-sided fractional derivatives. The main ingredient of the proposed method is to recast the problem into an equivalent system of weakly singular integral equations. Then, a Legendre-based spectral collocation method is developed for solving the transformed system. Therefore, we can make good use of the advantages of the Gauss quadrature rule. We present the construction and analysis of the collocation method. These results can be indirectly applied to solve fractional optimal control problems by considering the corresponding Euler–Lagrange equations. Two numerical examples are given to confirm the convergence analysis and robustness of the scheme.

## Keywords

Ameen, I. G., Elkot, N. A., Zaky, M. A., Hendy, A. S., Doha, E. H. (2021). A Pseudo-Spectral Scheme for Systems of Two-Point Boundary Value Problems with Left and Right Sided Fractional Derivatives and Related Integral Equations. CMES-Computer Modeling in Engineering & Sciences, 128(1), 21–41.

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