Numerical Approximation of Fredholm Integral Equation (FIE) of 2nd Kind using Galerkin and Collocation Methods

Abstract

In this research work, Galerkin and collocation methods have been introduced for approximating the solution of FIE of 2nd kind using LH (product of Laguerre and Hermite) polynomials which are considered as basis functions. Also, a comparison has been done between the solutions of Galerkin and collocation method with the exact solution. Both of these methods show the outcome in terms of the approximate polynomial which is a linear combination of basis functions. Results reveal that performance of collocation method is better than Galerkin method. Moreover, five different polynomials such as Legendre, Laguerre, Hermite, Chebyshev 1st kind and Bernstein are also considered as a basis functions. And it is found that all these approximate solutions converge to same polynomial solution and then a comparison has been made with the exact solution. In addition, five different set of collocation points are also being considered and then the approximate results are compared with the exact analytical solution. It is observed that collocation method performed well compared to Galerkin method.
 GANIT J. Bangladesh Math. Soc.Vol. 38 (2018) 11-25

Построение Обобщенных Итерационных Методов, Используемых Для Решения Интегрального Уравнения Фредгольма

“…A great number of works is devoted to the search for the best possible numerical solution of the equation ( 1) by inventing new methods or by granulating and improving previously known methods. Let us mention here only a few of them:wavelet methods [2], Galerkin [10,11], collocation [12,13,14], quadrature [12], Chebyshev and Legendre collocation method [15], Rayleigh-Ritz method [16], deep learning [17], Ten-non polynomial cubic splines method [18], Gaussian process regression [19] and Taylor expansion [20].…”

Galerkin Approximations for the Solution of Fredholm Volterra Integral Equation of Second Kind

“…As far our knowledge, BF, CH, CL, FL, LLE polynomials are not introduced before to find the approximate solutions of FVIE of second kind. In recent times, Molla and Saha [8] used LH polynomials as basis function in Galerkin method for finding approximate solution of FIE of 2nd kind within a very short description. A brief discussion about the performance of above mentioned polynomials for the solution of FVIE of 2nd kind is presented in this paper.…”