A New Numerical Approach Using Chebyshev Third Kind Polynomial for Solving Integrodifferential Equations of Higher Order

Abstract

There are several classifications of linear Integral Equations. Some of them include; Voltera Integral Equations, Fredholm Linear Integral Equations, Fredholm-Voltera Integrodifferential. In the past, solutions of higher-order Fredholm-Volterra Integrodifferential Equations [FVIE] have been presented. However, this work uses a computational techniques premised on the third kind Chebyshev polynomials method. The performance of the results for distinctive degrees of approximation (M) of the trial solution is cautiously studied and comparisons have been additionally made between the approximate/estimated and exact/definite solution at different intervals of the problems under consideration. Modelled Problems have been provided to illustrate the performance and relevance of the techniques. However, it turned out that as M increases, the outcomes received after every iteration get closer to the exact solution in all of the problems considered. The results of the experiments are therefore visible from the tables of errors and the graphical representation presented in this work.

Vieta-Lucas Polynomial Computational Tecnique for Volterra Integro-Differential Equations

“…Example 1 [14]: Consider fourth-order Volterra integro-differential equation This study investigated three instances. Examples 1 and 2 were resolved through the approach presented in [14], utilizing Chebyshev third-kind polynomials as the underlying basis.…”

“…Example 1 [14]: Consider fourth-order Volterra integro-differential equation This study investigated three instances. Examples 1 and 2 were resolved through the approach presented in [14], utilizing Chebyshev third-kind polynomials as the underlying basis. A comparison of the outcomes in tables 1 and 2 reveals the superiority of the suggested method employing Vieta Lucas polynomials over the results presented in [14].…”

“…Examples 1 and 2 were resolved through the approach presented in [14], utilizing Chebyshev third-kind polynomials as the underlying basis. A comparison of the outcomes in tables 1 and 2 reveals the superiority of the suggested method employing Vieta Lucas polynomials over the results presented in [14]. Furthermore, example 3 was tackled by [6] employing Euler polynomials combined with the least squares technique.…”

“…In [12], the Lucas polynomial is employed to solve nonlinear differential equations with variable delays. The use of third-kind Chebyshev polynomials for solving IDEs was examined in [13] and [14]. In [15] and [16], a Computational algorithm is used to find the solution of fractional Fredholm IDEs and Volterra-Fredholm IDEs.…”

“…Based on the works mentioned above, this study proposes a computational algorithm that utilizes shifted Legendre polynomials. This technique is inspired by previous research and aims to enhance the outcomes achieved by [14] and [6]. This work considers the Volterra integro-differential equation, which is represented in the following form:…”