An effective method for solving nonlinear integral equations involving the Riemann-Liouville fractional operator

Supriya Kumar Paul; Lakshmi Narayan Mishra; Vishnu Narayan Mishra; Dumitru Baleanu; Supriya Kumar Paul; Lakshmi Narayan Mishra; Vishnu Narayan Mishra; Dumitru Baleanu

doi:10.3934/math.2023891

AIMS Mathematics

2023, Volume 8, Issue 8: 17448-17469. doi: 10.3934/math.2023891

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Research article

An effective method for solving nonlinear integral equations involving the Riemann-Liouville fractional operator

1.
Department of Mathematics, School of Advanced Sciences, Vellore Institute of Technology, Vellore, 632 014, Tamil Nadu, India
2.
Department of Mathematics, Indira Gandhi National Tribal University, Lalpur, Amarkantak, Anuppur, 484 887, Madhya Pradesh, India
3.
Department of Mathematics, Faculty of Arts and Sciences, Cankaya University, Ankara, 09790, Turkey
4.
Institute of Space Sciences, 077125 Magurele, Ilfov, Romania
5.
Department of Natural Sciences, School of Arts and Sciences, Lebanese American University, Beirut, 11022801, Lebanon

Received: 22 March 2023 Revised: 27 April 2023 Accepted: 07 May 2023 Published: 19 May 2023
MSC : 26A33, 45M10, 65R20

In this paper, under some conditions in the Banach space $ C ([0, \beta], \mathbb{R}) $, we establish the existence and uniqueness of the solution for the nonlinear integral equations involving the Riemann-Liouville fractional operator (RLFO). To establish the requirements for the existence and uniqueness of solutions, we apply the Leray-Schauder alternative and Banach's fixed point theorem. We analyze Hyers-Ulam-Rassias (H-U-R) and Hyers-Ulam (H-U) stability for the considered integral equations involving the RLFO in the space $ C([0, \beta], \mathbb{R}) $. Also, we propose an effective and efficient computational method based on Laguerre polynomials to get the approximate numerical solutions of integral equations involving the RLFO. Five examples are given to interpret the method.
- Riemann-Liouville fractional integral,
- fixed point theorem,
- Laguerre polynomials,
- Hyers-Ulam stability,
- Hyers-Ulam-Rassias stability
Citation: Supriya Kumar Paul, Lakshmi Narayan Mishra, Vishnu Narayan Mishra, Dumitru Baleanu. An effective method for solving nonlinear integral equations involving the Riemann-Liouville fractional operator[J]. AIMS Mathematics, 2023, 8(8): 17448-17469. doi: 10.3934/math.2023891

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Abstract

In this paper, under some conditions in the Banach space $ C ([0, \beta], \mathbb{R}) $, we establish the existence and uniqueness of the solution for the nonlinear integral equations involving the Riemann-Liouville fractional operator (RLFO). To establish the requirements for the existence and uniqueness of solutions, we apply the Leray-Schauder alternative and Banach's fixed point theorem. We analyze Hyers-Ulam-Rassias (H-U-R) and Hyers-Ulam (H-U) stability for the considered integral equations involving the RLFO in the space $ C([0, \beta], \mathbb{R}) $. Also, we propose an effective and efficient computational method based on Laguerre polynomials to get the approximate numerical solutions of integral equations involving the RLFO. Five examples are given to interpret the method.

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