A new integral operational matrix with applications to multi-order fractional differential equations

Imran Talib; Md. Nur Alam; Dumitru Baleanu; Danish Zaidi; Ammarah Marriyam; Imran Talib; Md. Nur Alam; Dumitru Baleanu; Danish Zaidi; Ammarah Marriyam

doi:10.3934/math.2021508

AIMS Mathematics

2021, Volume 6, Issue 8: 8742-8771. doi: 10.3934/math.2021508

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A new integral operational matrix with applications to multi-order fractional differential equations

1.
Nonlinear Analysis Group (NAG), Mathematics Department, Virtual University of Pakistan, Pakistan
2.
Department of mathematics, Pabna University of science and technology, Pabna-6600, Bangladesh
3.
Cankaya University, Department of Mathematics and Computer Sciences, Ankara
4.
Department of Mathematics, University of Management and Technology, Lahore, Pakistan

Received: 11 March 2021 Accepted: 20 May 2021 Published: 09 June 2021
MSC : 65M99, 26A33, 35R11, 45K05

In this article, we propose a numerical method that is completely based on the operational matrices of fractional integral and derivative operators of fractional Legendre function vectors (FLFVs). The proposed method is independent of the choice of the suitable collocation points and expansion of the residual function as a series of orthogonal polynomials as required for Spectral collocation and Spectral tau methods. Consequently, the high efficient numerical results are obtained as compared to the other methods in the literature. The other novel aspect of our article is the development of the new integral and derivative operational matrices in Riemann-Liouville and Caputo senses respectively. The proposed method is computer-oriented and has the ability to reduce the fractional differential equations (FDEs) into a system of Sylvester types matrix equations that can be solved using MATLAB builtin function lyap(.). As an application of the proposed method, we solve multi-order FDEs with initial conditions. The numerical results obtained otherwise in the literature are also improved in our work.
Citation: Imran Talib, Md. Nur Alam, Dumitru Baleanu, Danish Zaidi, Ammarah Marriyam. A new integral operational matrix with applications to multi-order fractional differential equations[J]. AIMS Mathematics, 2021, 6(8): 8742-8771. doi: 10.3934/math.2021508

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Abstract

In this article, we propose a numerical method that is completely based on the operational matrices of fractional integral and derivative operators of fractional Legendre function vectors (FLFVs). The proposed method is independent of the choice of the suitable collocation points and expansion of the residual function as a series of orthogonal polynomials as required for Spectral collocation and Spectral tau methods. Consequently, the high efficient numerical results are obtained as compared to the other methods in the literature. The other novel aspect of our article is the development of the new integral and derivative operational matrices in Riemann-Liouville and Caputo senses respectively. The proposed method is computer-oriented and has the ability to reduce the fractional differential equations (FDEs) into a system of Sylvester types matrix equations that can be solved using MATLAB builtin function lyap(.). As an application of the proposed method, we solve multi-order FDEs with initial conditions. The numerical results obtained otherwise in the literature are also improved in our work.

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