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.
Highlights
Most mathematical models used in many applied problems of physics, biology, chemistry, engineering, and in other areas are transformed into integral equations, namely linear and nonlinear Volterra integral equations of the first or second kind
In this paper, we provide a numerical approach for the Volterra integral equations based on Hermite and Chebyshev piecewise polynomials [17] basis by the technique of Galerkin weighted residual method [18]
To verify the proposed method, we consider some linear and nonlinear Volterra integral equations with regular and weakly singular kernels, because the exact solutions for these problems are available in the literature
Summary
Most mathematical models used in many applied problems of physics, biology, chemistry, engineering, and in other areas are transformed into integral equations, namely linear and nonlinear Volterra integral equations of the first or second kind. Various techniques [1, 2] have been presented for solving Volterra integral equations such as Adomian's decomposition method, series solution method, Laplace transform method and successive substitution method. These methods generally cover the analytic closed form solution of such equations. In this paper, we provide a numerical approach for the Volterra integral equations based on Hermite and Chebyshev piecewise polynomials [17] basis by the technique of Galerkin weighted residual method [18]. We drive a matrix formulation for general linear problems by the technique of Galerkin method while nonlinear case is given through numerical examples
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