On the Solutions of the Second Kind Nonlinear Volterra-Fredholm Integral Equations via Homotopy Analysis Method

Rahby, A. S.; Abdou, M. A.; Mosa, G. A.

doi:10.28924/2291-8639-20-2022-35

Int. J. Anal. Appl.

2022

DOI: 10.28924/2291-8639-20-2022-35

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On the Solutions of the Second Kind Nonlinear Volterra-Fredholm Integral Equations via Homotopy Analysis Method

A. S. Rahby¹,

M. A. Abdou²,

G. A. Mosa³

Abstract: In this paper, we discuss the existence and uniqueness of the solution of the second kind nonlinear Volterra-Fredholm integral equations (NV-FIEs) which appear in mathematical modeling of many phenomena, using Picard’s method. In addition, we use Banach fixed point theorem to show the solvability of the first kind NV-FIEs. Moreover, we utilize the homotopy analysis method (HAM) to approximate the solution and the convergence of the method is investigated. Finally, some examples are presented and the numerical … Show more

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2024

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Cited by 2 publications

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Analytical and numerical treatment of a nonlinear Fredholm integral equation in two dimensions

Alahmadi,

Abdou,

Abdel-Aty

2024

J. Appl. Math. Comput.

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Analytical and numerical treatment of a nonlinear Fredholm integral equation in two dimensions

Alahmadi,

Abdou,

Abdel-Aty

2024

J. Appl. Math. Comput.

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Analysis of efficient discretization technique for nonlinear integral equations of Hammerstein type

Bhat,

Mishra,

Mishra

et al. 2024

HFF

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Purpose This study focuses on investigating the numerical solution of second-kind nonlinear Volterra–Fredholm–Hammerstein integral equations (NVFHIEs) by discretization technique. The purpose of this paper is to develop an efficient and accurate method for solving NVFHIEs, which are crucial for modeling systems with memory and cumulative effects, integrating past and present influences with nonlinear interactions. They are widely applied in control theory, population dynamics and physics. These equations are essential for solving complex real-world problems. Design/methodology/approach Demonstrating the solution’s existence and uniqueness in the equation is accomplished by using the Picard iterative method as a key technique. Using the trapezoidal discretization method is the chosen approach for numerically approximating the solution, yielding a nonlinear system of algebraic equations. The trapezoidal method (TM) exhibits quadratic convergence to the solution, supported by the application of a discrete Grönwall inequality. A novel Grönwall inequality is introduced to demonstrate the convergence of the considered method. This approach enables a detailed analysis of the equation’s behavior and facilitates the development of a robust solution method. Findings The numerical results conclusively show that the proposed method is highly efficacious in solving NVFHIEs, significantly reducing computational effort. Numerical examples and comparisons underscore the method’s practicality, effectiveness and reliability, confirming its outstanding performance compared to the referenced method. Originality/value Unlike existing approaches that rely on a combination of methods to tackle different aspects of the complex problems, especially nonlinear integral equations, the current approach presents a significant single-method solution, providing a comprehensive approach to solving the entire problem. Furthermore, the present work introduces the first numerical approaches for the considered integral equation, which has not been previously explored in the existing literature. To the best of the authors’ knowledge, the work is the first to address this equation, providing a foundational contribution for future research and applications. This innovative strategy not only simplifies the computational process but also offers a more comprehensive understanding of the problem’s dynamics.

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On the Solutions of the Second Kind Nonlinear Volterra-Fredholm Integral Equations via Homotopy Analysis Method

Cited by 2 publications

References 20 publications

Analytical and numerical treatment of a nonlinear Fredholm integral equation in two dimensions

Analytical and numerical treatment of a nonlinear Fredholm integral equation in two dimensions

Analysis of efficient discretization technique for nonlinear integral equations of Hammerstein type

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