On the Discrete Galerkin Method for Fredholm Integral Equations of the Second Kind

1989 ◽  
Vol 9 (3) ◽  
pp. 385-403 ◽  
Author(s):  
K. E. ATKINSON ◽  
F. A. POTRA
Keyword(s):  
Galerkin Method ◽  
Integral Equations ◽  
Fredholm Integral Equations ◽  
Discrete Galerkin Method

scholarly journals Numerical Solutions of Fredholm Integral Equations Using Bernstein Polynomials

2010 ◽  
Vol 2 (2) ◽  
pp. 264-272 ◽  
Author(s):  
A. Shirin ◽  
M. S. Islam
Keyword(s):  
Integral Equation ◽  
Galerkin Method ◽  
Integral Equations ◽  
Fredholm Integral Equation ◽  
Numerical Solutions ◽  
Bernstein Polynomials ◽  
Fredholm Integral Equations ◽  
Matrix Formulation ◽  
Piecewise Polynomials ◽  
The Galerkin Method

In this paper, Bernstein piecewise polynomials are used to solve the integral equations numerically. A matrix formulation is given for a non-singular linear Fredholm Integral Equation by the technique of Galerkin method. In the Galerkin method, the Bernstein polynomials are used as the approximation of basis functions. Examples are considered to verify the effectiveness of the proposed derivations, and the numerical solutions guarantee the desired accuracy.  Keywords: Fredholm integral equation; Galerkin method; Bernstein polynomials. © 2010 JSR Publications. ISSN: 2070-0237 (Print); 2070-0245 (Online). All rights reserved. DOI: 10.3329/jsr.v2i2.4483               J. Sci. Res. 2 (2), 264-272 (2010) 

scholarly journals An Adaptive Approach to Solutions of Fredholm Integral Equations of the Second Kind

2014 ◽  
Vol 2014 ◽  
pp. 1-13
Author(s):  
Nebiye Korkmaz ◽  
Zekeriya Güney
Keyword(s):  
Galerkin Method ◽  
Integral Equations ◽  
Iteration Method ◽  
Legendre Polynomials ◽  
Optimization Problem ◽  
Approximate Solutions ◽  
Basis Functions ◽  
Fredholm Integral Equations ◽  
Coarse Mesh ◽  
Fine Mesh

As an approach to approximate solutions of Fredholm integral equations of the second kind, adaptive hp-refinement is used firstly together with Galerkin method and with Sloan iteration method which is applied to Galerkin method solution. The linear hat functions and modified integrated Legendre polynomials are used as basis functions for the approximations. The most appropriate refinement is determined by an optimization problem given by Demkowicz, 2007. During the calculationsL2-projections of approximate solutions on four different meshes which could occur between coarse mesh and fine mesh are calculated. Depending on the error values, these procedures could be repeated consecutively or different meshes could be used in order to decrease the error values.

Supplement to The Discrete Galerkin Method for Integral Equations

1987 ◽  
Vol 48 (178) ◽  
pp. S11
Author(s):  
Kendall Atkinson ◽  
Alex Bogomolny
Keyword(s):  
Galerkin Method ◽  
Integral Equations ◽  
Discrete Galerkin Method

scholarly journals A fully discrete Galerkin method for Abel-type integral equations

2018 ◽  
Vol 44 (5) ◽  
pp. 1601-1626 ◽  
Author(s):  
Urs Vögeli ◽  
Khadijeh Nedaiasl ◽  
Stefan A. Sauter
Keyword(s):  
Galerkin Method ◽  
Integral Equations ◽  
Fully Discrete ◽  
Type Integral ◽  
Discrete Galerkin Method

scholarly journals The discrete Galerkin method for integral equations

1987 ◽  
Vol 48 (178) ◽  
pp. 595 ◽  
Author(s):  
Kendall Atkinson ◽  
Alex Bogomolny
Keyword(s):  
Galerkin Method ◽  
Integral Equations ◽  
Discrete Galerkin Method

scholarly journals Iterated Petrov–Galerkin Method with Regular Pairs for Solving Fredholm Integral Equations of the Second Kind

2018 ◽  
Vol 23 (4) ◽  
pp. 73
Author(s):  
Silvia Alejandra Seminara ◽  
María Inés Troparevsky
Keyword(s):  
Galerkin Method ◽  
Integral Equations ◽  
Correlation Matrix ◽  
Approximation Scheme ◽  
Approximate Solutions ◽  
Orthogonal Basis ◽  
Fredholm Integral Equations ◽  
Interesting Phenomenon ◽  
Weakly Singular ◽  
Selection Of

In this work we obtain approximate solutions for Fredholm integral equations of the second kind by means of Petrov–Galerkin method, choosing “regular pairs” of subspaces, Xn,Yn, which are simply characterized by the positive definitiveness of a correlation matrix. This choice guarantees the solvability and numerical stability of the approximation scheme in an easy way, and the selection of orthogonal basis for the subspaces make the calculations quite simple. Afterwards, we explore an interesting phenomenon called “superconvergence”, observed in the 1970s by Sloan: once the approximations un∈Xn to the solution of the operator equation u-Ku=g are obtained, the convergence can be notably improved by means of an iteration of the method, un*=g+Kun. We illustrate both procedures of approximation by means of two numerical examples: one for a continuous kernel, and the other for a weakly singular one.

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