Efficient method for solving nonlinear weakly singular kernel fractional integro-differential equations

Ismail Gad Ameen; Dumitru Baleanu; Hussien Shafei Hussien; Ismail Gad Ameen; Dumitru Baleanu; Hussien Shafei Hussien

doi:10.3934/math.2024764

AIMS Mathematics

2024, Volume 9, Issue 6: 15819-15836. doi: 10.3934/math.2024764

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Efficient method for solving nonlinear weakly singular kernel fractional integro-differential equations

1.
Department of Mathematics, Faculty of Science, South Valley University, Qena 83523, Egypt
2.
Department of Computer Science and Mathematics, Lebanese American University, Beirut, Lebanon

Received: 20 January 2024 Revised: 03 March 2024 Accepted: 25 March 2024 Published: 06 May 2024
MSC : 33E12, 47G20, 65K10, 65G99, 65R20

This paper introduced an efficient method to obtain the solution of linear and nonlinear weakly singular kernel fractional integro-differential equations (WSKFIDEs). It used Riemann-Liouville fractional integration (R-LFI) to remove singularities and approximated the regularized problem with a combined approach using the generalized fractional step-Mittag-Leffler function (GFSMLF) and operational integral fractional Mittag matrix (OIFMM) method. The resulting algebraic equations were turned into an optimization problem. We also proved the method's accuracy in approximating any function, as well as its fractional differentiation and integration within WSKFIDEs. The proposed method was performed on some attractive examples in order to show how their solutions behave at various values of the fractional order $ \digamma $. The paper provided a valuable contribution to the field of fractional calculus (FC) by presenting a novel method for solving WSKFIDEs. Additionally, the accuracy of this method was verified by comparing its results with those obtained using other methods.
- singular integro-differential equations,
- generalized fractional Mittag-Leffler function,
- step-function,
- operational integral matrix,
- optimization method,
- error analysis
Citation: Ismail Gad Ameen, Dumitru Baleanu, Hussien Shafei Hussien. Efficient method for solving nonlinear weakly singular kernel fractional integro-differential equations[J]. AIMS Mathematics, 2024, 9(6): 15819-15836. doi: 10.3934/math.2024764

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Abstract

This paper introduced an efficient method to obtain the solution of linear and nonlinear weakly singular kernel fractional integro-differential equations (WSKFIDEs). It used Riemann-Liouville fractional integration (R-LFI) to remove singularities and approximated the regularized problem with a combined approach using the generalized fractional step-Mittag-Leffler function (GFSMLF) and operational integral fractional Mittag matrix (OIFMM) method. The resulting algebraic equations were turned into an optimization problem. We also proved the method's accuracy in approximating any function, as well as its fractional differentiation and integration within WSKFIDEs. The proposed method was performed on some attractive examples in order to show how their solutions behave at various values of the fractional order $ \digamma $. The paper provided a valuable contribution to the field of fractional calculus (FC) by presenting a novel method for solving WSKFIDEs. Additionally, the accuracy of this method was verified by comparing its results with those obtained using other methods.

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