Haar wavelet method for solution of variable order linear fractional integro-differential equations

Rohul Amin; Kamal Shah; Hijaz Ahmad; Abdul Hamid Ganie; Abdel-Haleem Abdel-Aty; Thongchai Botmart; Rohul Amin; Kamal Shah; Hijaz Ahmad; Abdul Hamid Ganie; Abdel-Haleem Abdel-Aty; Thongchai Botmart

doi:10.3934/math.2022301

AIMS Mathematics

2022, Volume 7, Issue 4: 5431-5443. doi: 10.3934/math.2022301

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Haar wavelet method for solution of variable order linear fractional integro-differential equations

1.
Department of Mathematics, University of Peshawar, 25120, Pakistan
2.
Department of Mathematics, University of Malakand, Pakistan
3.
Department of Mathematics and General Sciences, Prince Sultan University, Riyadh, Saudi Arabia
4.
Information Technology Application and Research Center, Istanbul Ticaret University, 34445, Istanbul, Turkey
5.
Department of Mathematics, Faculty of Humanities and Social Sciences, Istanbul Ticaret University, 34445, Istanbul, Turkey
6.
Basic Science department, College of Science and Theoretical Studies, Saudi Electronic University-Abha Male 61421, Saudi Arabia
7.
Department of Physics, College of Sciences, University of Bisha, Bisha 61922, Saudi Arabia
8.
Physics Department, Faculty of Science, Al-Azhar University, Assiut 71524, Egypt
9.
Department of Mathematics, Faculty of Science, Khon Kaen University, Khon Kaen, 40002, Thailand

Received: 22 October 2021 Revised: 14 December 2021 Accepted: 20 December 2021 Published: 06 January 2022
MSC : 34K05, 34K30

In this paper, we developed a computational Haar collocation scheme for the solution of fractional linear integro-differential equations of variable order. Fractional derivatives of variable order is described in the Caputo sense. The given problem is transformed into a system of algebraic equations using the proposed Haar technique. The results are obtained by solving this system with the Gauss elimination algorithm. Some examples are given to demonstrate the convergence of Haar collocation technique. For different collocation points, maximum absolute and mean square root errors are computed. The results demonstrate that the Haar approach is efficient for solving these equations.
- variable-order fractional calculus,
- fixed-point theory,
- Gauss elimination method,
- Haar wavelet,
- numerical approximation
Citation: Rohul Amin, Kamal Shah, Hijaz Ahmad, Abdul Hamid Ganie, Abdel-Haleem Abdel-Aty, Thongchai Botmart. Haar wavelet method for solution of variable order linear fractional integro-differential equations[J]. AIMS Mathematics, 2022, 7(4): 5431-5443. doi: 10.3934/math.2022301

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Abstract

In this paper, we developed a computational Haar collocation scheme for the solution of fractional linear integro-differential equations of variable order. Fractional derivatives of variable order is described in the Caputo sense. The given problem is transformed into a system of algebraic equations using the proposed Haar technique. The results are obtained by solving this system with the Gauss elimination algorithm. Some examples are given to demonstrate the convergence of Haar collocation technique. For different collocation points, maximum absolute and mean square root errors are computed. The results demonstrate that the Haar approach is efficient for solving these equations.

References

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