Solution of fractional boundary value problems by $ \psi $-shifted operational matrices

Shazia Sadiq; Mujeeb ur Rehman; Shazia Sadiq; Mujeeb ur Rehman

doi:10.3934/math.2022372

AIMS Mathematics

2022, Volume 7, Issue 4: 6669-6693. doi: 10.3934/math.2022372

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Solution of fractional boundary value problems by $ \psi $-shifted operational matrices

Shazia Sadiq ^,,
Mujeeb ur Rehman

Department of Mathematics, School of Natural Sciences, National University of Sciences and Technology, Islamabad, Pakistan

Received: 25 October 2021 Revised: 28 December 2021 Accepted: 05 January 2022 Published: 24 January 2022
MSC : 34A08, 34K10, 34K28

In this paper, a numerical method is presented to solve fractional boundary value problems. In fractional calculus, the modelling of natural phenomenons is best described by fractional differential equations. So, it is important to formulate efficient and accurate numerical techniques to solve fractional differential equations. In this article, first, we introduce $ \psi $-shifted Chebyshev polynomials then project these polynomials to formulate $ \psi $-shifted Chebyshev operational matrices. Finally, these operational matrices are used for the solution of fractional boundary value problems. The convergence is analysed. It is observed that solution of non-integer order differential equation converges to corresponding solution of integer order differential equation. Finally, the efficiency and applicability of method is tested by comparison of the method with some other existing methods.
- fractional differential equations,
- boundary value problems,
- $ \psi $-shifted Chebyshev polynomials,
- operational matrices
Citation: Shazia Sadiq, Mujeeb ur Rehman. Solution of fractional boundary value problems by $ \psi $-shifted operational matrices[J]. AIMS Mathematics, 2022, 7(4): 6669-6693. doi: 10.3934/math.2022372

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Abstract

In this paper, a numerical method is presented to solve fractional boundary value problems. In fractional calculus, the modelling of natural phenomenons is best described by fractional differential equations. So, it is important to formulate efficient and accurate numerical techniques to solve fractional differential equations. In this article, first, we introduce $ \psi $-shifted Chebyshev polynomials then project these polynomials to formulate $ \psi $-shifted Chebyshev operational matrices. Finally, these operational matrices are used for the solution of fractional boundary value problems. The convergence is analysed. It is observed that solution of non-integer order differential equation converges to corresponding solution of integer order differential equation. Finally, the efficiency and applicability of method is tested by comparison of the method with some other existing methods.

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