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

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

INITIAL DATA FOR NUMERICAL RELATIVITY: THE USE OF SPECTRAL METHODS

The determination of physical initial data is an important task in numerical relativity. In this direction we have applied the Galerkin and collocation methods to solve the Hamiltonian constraints resulting from the Cauchy formulation in the cases of spacetimes containing black holes as described by Ref. 1. We have shown that a considerable improvement in the accuracy is obtained if the basis functions are chosen such that the boundary conditions are satisfied. We have also introduced a new approach to solve numerically the constraint equations which consists in transforming them into parabolic equations after introducing fictitious diffusion terms. As a consequence, the application of Galerkin or collocation methods produces a dynamical system whose stationary solution corresponds to the initial data.