Solutions of Linear and Nonlinear Volterra Integral Equations Using Hermite and Chebyshev Polynomials

Abstract

The purpose of this work is to provide a novel numerical approach for the Volterra integral equations based on Galerkin weighted residual approximation. In this method Hermite and Chebyshev piecewise, continuous and differentiable polynomials are exploited as basis functions. A rigorous effective matrix formulation is proposed to solve the linear and nonlinear Volterra integral equations of the first and second kind with regular and singular kernels. The algorithm is simple and can be coded easily. The efficiency of the proposed method is tested on several numerical examples to get the desired and reliable good accuracy. KeywordsLinear and nonlinear Volterra integral equations; Galerkin method; Hermite and Chebyshev polynomials.

Comparative Study Between ADM and MDM for a System of Volterra Integral Equation

“…The Galerkin weighted residual approximation method was applied to obtain a numerical approach to Volterra's integral equations in [15] and in [16] Wazwaz focused on recent developments in approximate methods for solving linear and nonlinear integral equations with applications. The aim of this study is to solve a system of Volterra integral equation of the second kind by two accurate methods, which are the Adomian decomposition method and the modified decomposition method.…”

Numerical Solutions of Volterra Equations Using Galerkin Method with Certain Orthogonal Polynomials

“…Rahman [1] used Galerkin method with Hermite polynomial basis for the numerical solutions of Volterra integral equations of the second kind. Shafigul et al [2] used Galerkin method to explore the solutions of linear and nonlinear Volterra equations using both Hermite and Chebychev polynomial basis. Shahsavaran [3] solved Volterra integral equations of Abel type using Block pulse functions.…”

“…For all examples considered, the solutions obtained by the proposed method are compared with the exact solutions available in the literature. The rate of convergence of each of the Linear Volterra integral equations is composed as solution by the proposed method using the nth degree polynomial approximation and δ varies from 6 10 − for 10 n ≥ (See[2]). Example Consider the linear Volterra integral equation of the first with continuous kernel[2] the derived formula of Equation(15)and solving for 1 n ≥ , we get the approximate solution ( ) 2 u x x =  , which is the exact solution.…”