Abstract
This paper presents an enhanced moving least square method for the solution of volterra integro-differential equation: an interpolating polynomial. It is a numerical scheme that utilizes a modified shape function of the conventional Moving Least Square (MLS) method to solve fourth order Integro-differential equations. Smooth orthogonal polynomials have been constructed and used as the basis functions. A robust and unrestricted trigonometric weight function, along with the basis function, drives the shape function and facilitates the convergence of the scheme. The choice of the support size and some controlling parameters ensures the existence of the moment matrix inverse and the MLS solution. Valid explanation and illustration were made for the existence of the inverse linear operator. To overcome problems of near-singularity, the singular value decomposition rule is used to compute the inverse of the moment matrix. Gauss quadrature rule is used to compute the integral at the initial test points when the exact solution is unknown. Some tested problems were solved to show the applicability of the method. The results obtained compare favourable with the exact solutions. Finally, a highly significant interpolating polynomial is obtained and used to reproduce the solutions over the entire problem domain. The negligible magnitude of the error at each evaluation knot demonstrates the reliability and effectiveness of this scheme.
Highlights
Integro—differential equations (IDEs) are equations that take into account both integral and derivatives of an unknown function [30]
Based on the results obtained, the value was given as a function of which accounts for the major difference between the Enhanced Moving Least Square (MLS), MLS method and the popular Least Square Method
We conclude that the proposed Enhanced Moving Least Square method is good for solving the class of equations described in this paper
Summary
Integro—differential equations (IDEs) are equations that take into account both integral and derivatives of an unknown function [30]. Mathematical modeling of real-life problems usually results in functional equations like ordinary or partial differential equations, integral and integro—differential equations, and stochastic equations. Armand and Gouyandeh discussed IDE of the first kind in [3] and nonlinear Fredholm Integral Equations of the second kind were discussed by Borzabadi, Kamyad, and Mehne in [7]. Elaborate work on IDEs was discussed in [8, 10, 13, 16, 19, 22,23,24,25, 31] and in [21] where Maleknejad and Mahmoudi applied Taylor polynomial to high-order nonlinear Volterra Fredholm Integro-differential Equations. Taylor Collocation Method was applied to linear IDEs in [18] by Karamete and Sezer
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