A new operational approach for solving weakly singular integro--differential equations

Abstract

Based on Jacobi polynomials, an operational method is proposed to solve weakly singular integro-differential equations. These equations appear in various fields of science such as physics and engineering, the motion of a plate in a viscous fluid under the action of external forces, problems of heat transfer, and surface waves. To solve the weakly singular integro-differential equations, a fast algorithm is used for simplifying the problem under study. The Laplace transform and Jacobi collocation methods are merged, and thus, a novel approach is presented. Some theorems are given and established to theoretically support the computational simplifications which reduce costs. In order to show the efficiency and accuracy of the proposed method some numerical results are provided. It is found that the proposed method has lesser computational size compared to other common methods, such as Adomian decomposition, Taylor expansion, and Bernstein operational methods. It is further found that the absolute errors are almost constant in the studied interval.

Application of Laplace Transform Method for Solving Weakly-Singular Integro-Differential Equations

“…So a variety works focused its interest to tackle the equations of this form which are usually numerical methods, including an operational method [7], Bernstein series [8], Block boundary value method [9], Partition of the interval and introduction of additional parameters [10], Smoothing transformation and spline collocation [11], The asymptotic estimations of the solution [12], Product integration [13], collocations methods as Spline, Piecewise Polynomial, and Spectral respectively [14][15][16], but it is well known that the results of numerical methods have an error rate.…”