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Document Type

Original Study

Abstract

The present paper investigates a mathematical technique for determining nonlinear integro-differential equations that is based on a combination of the Bownds and Range-Kutta methods. There will be a discussion of what is meant by this class of nonlinear Fredholm equations, as well as an explanation of the mathematical method for this kind of equation. The Bownds technique is used to produce the differential equations from the integral equation obtained from the integro-differential equations. Then, the approximate solution of the differential equation(s) is derived by applying the Range-Kutta method. Through solving three different examples, we prove that the combination of mathematical methods is efficient and reliable. This makes it a valuable tool for determining nonlinear Fredholm equations with different degenerate kernels quickly and accurately. Finally, it is shown that the importance of my research is that it contributes to providing modern strategies for the numerical solution of this type of equation and it has the potential to lead to more efficient numerical methods for solving nonlinear equations.

Keywords

Bownds method, Runge-Kutta method of fourth order, nonlinear integro-differential equation of Fredholm type, approximate solutions, degenerate kernel, Initial value problem, system of first order differential equations

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