Shifted Legendre Polynomials For Solving Second Kind Fredholm Integral Equations

2021 ◽  
Vol 30 (1) ◽  
pp. 76-83
Author(s):  
Shoukralla, E.. S ◽  
Elgohary, . H ◽  
Morgan. , M
Keyword(s):  
Integral Equations ◽  
Legendre Polynomials ◽  
Fredholm Integral Equations ◽  
Shifted Legendre Polynomials

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.

scholarly journals Approximate Solution Technique for Singular Fredholm Integral Equations of the First Kind with Oscillatory Kernels

2019 ◽  
pp. 1-9
Author(s):  
Vivian Ndfutu Nfor ◽  
George Emese Okecka
Keyword(s):  
Approximate Solution ◽  
Integral Equations ◽  
Legendre Polynomials ◽  
Lagrange Interpolation ◽  
Interpolation Formula ◽  
Verification And Validation ◽  
Solution Technique ◽  
Fredholm Integral Equations ◽  
Lagrange Interpolation Formula ◽  
Interpolation Nodes

An efficient quadrature formula was developed for evaluating numerically certain singular Fredholm integral equations of the first kind with oscillatory trigonometric kernels.  The method is based on the Lagrange interpolation formula and the orthogonal polynomial considered are the Legendre polynomials whose zeros served as interpolation nodes. A test example was provided for the verification and validation of the rule developed. The results showed the convergence of the solution and can be improved by increasing n.

scholarly journals On integral equations with Weakly Singular kernel by using Taylor series and Legendre polynomials

Open Physics ◽  
2015 ◽  
Vol 13 (1) ◽  
Author(s):  
Esmail Babolian ◽  
Danial Hamedzadeh ◽  
Hossein Jafari ◽  
Asghar Arzhang Hajikandi ◽  
Dumitru Baleanu
Keyword(s):  
Numerical Solution ◽  
Integral Equations ◽  
Taylor Series ◽  
Unknown Function ◽  
Legendre Polynomials ◽  
Second Order ◽  
Singular Kernel ◽  
Fredholm Integral Equations ◽  
Weakly Singular Kernel ◽  
Weakly Singular

AbstractThis paper is concerned with the numerical solution for a class of weakly singular Fredholm integral equations of the second kind. The Taylor series of the unknown function, is used to remove the singularity and the truncated Taylor series to second order of k(x, y) about the point (x

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