New algorithms for solving nonlinear mixed integral equations

R. T. Matoog; M. A. Abdou; M. A. Abdel-Aty; R. T. Matoog; M. A. Abdou; M. A. Abdel-Aty

doi:10.3934/math.20231406

AIMS Mathematics

2023, Volume 8, Issue 11: 27488-27512. doi: 10.3934/math.20231406

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New algorithms for solving nonlinear mixed integral equations

1.
Department of Mathematics, Faculty of Applied Sciences, Umm Al–Qura University, Makkah, Saudi Arabia
2.
Department of Mathematics, Faculty of Education, Alexandria University, Alexandria 21511, Egypt
3.
Department of Mathematics, Faculty of Science, Benha University, Benha 13518, Egypt

Received: 22 August 2023 Revised: 18 September 2023 Accepted: 21 September 2023 Published: 27 September 2023
MSC : 30K05, 45G15, 45L05, 65R20

In this article, the existence and unique solution of the nonlinear Volterra-Fredholm integral equation (NVFIE) of the second kind is discussed. We also prove the solvability of the second kind of the NVFIE using the Banach fixed point theorem. Using quadrature method, the NVFIE leads to a system of nonlinear Fredholm integral equations (NFIEs). The existence and unique numerical solution of this system is discussed. Then, the modified Taylor's method was applied to transform the system of NFIEs into nonlinear algebraic systems (NAS). The existence and uniqueness of the nonlinear algebraic system's solution are discussed using Banach's fixed point theorem. Also, the stability of the modified error is presented. Some numerical examples are performed to show the efficiency and simplicity of the presented method, and all results are obtained using Wolfram Mathematica 11.
Citation: R. T. Matoog, M. A. Abdou, M. A. Abdel-Aty. New algorithms for solving nonlinear mixed integral equations[J]. AIMS Mathematics, 2023, 8(11): 27488-27512. doi: 10.3934/math.20231406

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Abstract

In this article, the existence and unique solution of the nonlinear Volterra-Fredholm integral equation (NVFIE) of the second kind is discussed. We also prove the solvability of the second kind of the NVFIE using the Banach fixed point theorem. Using quadrature method, the NVFIE leads to a system of nonlinear Fredholm integral equations (NFIEs). The existence and unique numerical solution of this system is discussed. Then, the modified Taylor's method was applied to transform the system of NFIEs into nonlinear algebraic systems (NAS). The existence and uniqueness of the nonlinear algebraic system's solution are discussed using Banach's fixed point theorem. Also, the stability of the modified error is presented. Some numerical examples are performed to show the efficiency and simplicity of the presented method, and all results are obtained using Wolfram Mathematica 11.

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