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

2019 ◽  
Vol 38 ◽  
pp. 11-25
Author(s):  
Hasib Uddin Molla ◽  
Goutam Saha
Keyword(s):  
Exact Solution ◽  
Galerkin Method ◽  
Collocation Method ◽  
Exact Analytical Solution ◽  
Polynomial Solution ◽  
Hermite Polynomials ◽  
Research Work ◽  
Basis Functions ◽  
Collocation Methods ◽  
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

scholarly journals Numerical Solution of Mixed Volterra – Fredholm Integral Equation Using the Collocation Method

2020 ◽  
Vol 17 (3) ◽  
pp. 0849
Author(s):  
Nahdh S. M. Al-Saif ◽  
Ameen Sh. Ameen
Keyword(s):  
Integral Equations ◽  
Collocation Method ◽  
Delta Function ◽  
Basis Functions ◽  
Collocation Methods ◽  
Polynomial Basis ◽  
Fredholm Integral Equations ◽  
Collocation Points ◽  
Function Property ◽  
Point Interpolation

Volterra – Fredholm integral equations (VFIEs) have a massive interest from researchers recently. The current study suggests a collocation method for the mixed Volterra - Fredholm integral equations (MVFIEs)."A point interpolation collocation method is considered by combining the radial and polynomial basis functions using collocation points". The main purpose of the radial and polynomial basis functions is to overcome the singularity that could associate with the collocation methods. The obtained interpolation function passes through all Scattered Point in a domain and therefore, the Delta function property is the shape of the functions. The exact solution of selective solutions was compared with the results obtained from the numerical experiments in order to investigate the accuracy and the efficiency of scheme.

scholarly journals INITIAL DATA FOR NUMERICAL RELATIVITY: THE USE OF SPECTRAL METHODS

2011 ◽  
Vol 03 ◽  
pp. 417-427
Author(s):  
M. A. ALCOFORADO ◽  
H. P. DE OLIVEIRA ◽  
E. L. RODRIGUES
Keyword(s):  
Initial Data ◽  
Parabolic Equations ◽  
Numerical Relativity ◽  
Considerable Improvement ◽  
Basis Functions ◽  
Collocation Methods ◽  
New Approach ◽  
Constraint Equations ◽  
Galerkin And Collocation 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.

On Projective Methods of Approximate Solution of Singular Integral Equations

1996 ◽  
Vol 3 (5) ◽  
pp. 457-474
Author(s):  
A. Jishkariani ◽  
G. Khvedelidze
Keyword(s):  
Approximate Solution ◽  
Rate Of Convergence ◽  
Integral Equations ◽  
Singular Integral ◽  
Singular Integral Equations ◽  
Collocation Methods ◽  
Projective Methods ◽  
Galerkin And Collocation Methods

Abstract The estimate for the rate of convergence of approximate projective methods with one iteration is established for one class of singular integral equations. The Bubnov–Galerkin and collocation methods are investigated.

scholarly journals A Galerkin method ofO(h2)for singular boundary value problems

2002 ◽  
Vol 29 (6) ◽  
pp. 361-369
Author(s):  
G. K. Beg ◽  
M. A. El-Gebeily
Keyword(s):  
Galerkin Method ◽  
Boundary Value Problems ◽  
Boundary Value ◽  
Basis Functions ◽  
Singular Boundary ◽  
Point Boundary ◽  
Singular Boundary Value Problems ◽  
Special Basis

We describe a Galerkin method with special basis functions for a class of singular two-point boundary value problems. The convergence is shown which is ofO(h2)for a certain subclass of the problems.

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