Numerical Solution of Infinite and Finite Boundary Integral Equations by Polynomials

The Fredholm Integral equation of the form g(s)=f(s) + ∫_a^b▒k(s,t)g(t)dt where k(s,t) is a regular kernel on D ={(s,t): a ~ s,t ~ b} and /(s) is a continuous function defined on [a,b] and the unknown function g(s) is to be determined, are encountered in many problems of ordinary dif...

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Main Author:	Abdul Sathar, Mohammad Hasan
Format:	Thesis
Language:	English English
Published:	2010
Subjects:	Mathematical physics Infinite Finite element method
Online Access:	http://psasir.upm.edu.my/id/eprint/19688/1/IPM_2010_9_F.pdf
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spelling	my-upm-ir.196882013-05-27T08:02:51Z Numerical Solution of Infinite and Finite Boundary Integral Equations by Polynomials 2010-11 Abdul Sathar, Mohammad Hasan The Fredholm Integral equation of the form g(s)=f(s) + ∫_a^b▒k(s,t)g(t)dt where k(s,t) is a regular kernel on D ={(s,t): a ~ s,t ~ b} and /(s) is a continuous function defined on [a,b] and the unknown function g(s) is to be determined, are encountered in many problems of ordinary differential equations and mathematical physics. Unfortunately in many cases the equation (I) cannot be solved in the closed form. Therefore, numerical method is applied to solve the integral equation (I). To approximate the integral equation (IE) (I), we usually choose a finite dimensional family of functions that is believed to contain a function g, (s) close to the true solutiong g(s). The desired approximate solution g,(s) is selected by forcing it to satisfy equation (I). There are various means in which g,(s) can be said to satisfy equation (I) approximately, and this leads to different type of methods. The most popular and powerful tools are Collocation and Galerkin methods. If the limit of integration in (I) is infinite and corresponding functions belong to certain class of functions then equation (I) is called infinite boundary integral equations (!BIEs). Many problems of electromagnetic, scattering problems and boundary integral equations lead to !BIEs of the second kind, g(s) =/(s) +4rk(s,t)g(t)dt, (2) g(s) =/(s) + 4 I k(s,t)g(t)dt. (3) In this thesis, we have developed Galerkin method with Laguerre polynomials for Eq. (2) on the interval [0,(0) and Galekin method with Hermite polynomials for Eq. (3) on the intetval (-00,00) to get the approximate solution. We have also solved equation (I) numerically using Collocation method based on Legendre polynomials on the interval [-1,1]. Existence of the solution for Eq. (2) and (3) and exactness of the approximate method for Eq. (I) are given. These developments gave good fit (even for small n) to the exact solution g(s) for finite and !BIEs (1)-(3). Maple software I I is used to obtain the approximate solution for Eq. (I )-(3) and the results show good convergence for finite and infinite interval for small n. Mathematical physics Infinite Finite element method 2010-11 Thesis http://psasir.upm.edu.my/id/eprint/19688/ http://psasir.upm.edu.my/id/eprint/19688/1/IPM_2010_9_F.pdf application/pdf en public masters Universiti Putra Malaysia Mathematical physics Infinite Finite element method Institute for Mathematical Research English
institution	Universiti Putra Malaysia
collection	PSAS Institutional Repository
language	English English
topic	Mathematical physics Infinite Finite element method
spellingShingle	Mathematical physics Infinite Finite element method Abdul Sathar, Mohammad Hasan Numerical Solution of Infinite and Finite Boundary Integral Equations by Polynomials
description	The Fredholm Integral equation of the form g(s)=f(s) + ∫_a^b▒k(s,t)g(t)dt where k(s,t) is a regular kernel on D ={(s,t): a ~ s,t ~ b} and /(s) is a continuous function defined on [a,b] and the unknown function g(s) is to be determined, are encountered in many problems of ordinary differential equations and mathematical physics. Unfortunately in many cases the equation (I) cannot be solved in the closed form. Therefore, numerical method is applied to solve the integral equation (I). To approximate the integral equation (IE) (I), we usually choose a finite dimensional family of functions that is believed to contain a function g, (s) close to the true solutiong g(s). The desired approximate solution g,(s) is selected by forcing it to satisfy equation (I). There are various means in which g,(s) can be said to satisfy equation (I) approximately, and this leads to different type of methods. The most popular and powerful tools are Collocation and Galerkin methods. If the limit of integration in (I) is infinite and corresponding functions belong to certain class of functions then equation (I) is called infinite boundary integral equations (!BIEs). Many problems of electromagnetic, scattering problems and boundary integral equations lead to !BIEs of the second kind, g(s) =/(s) +4rk(s,t)g(t)dt, (2) g(s) =/(s) + 4 I k(s,t)g(t)dt. (3) In this thesis, we have developed Galerkin method with Laguerre polynomials for Eq. (2) on the interval [0,(0) and Galekin method with Hermite polynomials for Eq. (3) on the intetval (-00,00) to get the approximate solution. We have also solved equation (I) numerically using Collocation method based on Legendre polynomials on the interval [-1,1]. Existence of the solution for Eq. (2) and (3) and exactness of the approximate method for Eq. (I) are given. These developments gave good fit (even for small n) to the exact solution g(s) for finite and !BIEs (1)-(3). Maple software I I is used to obtain the approximate solution for Eq. (I )-(3) and the results show good convergence for finite and infinite interval for small n.
format	Thesis
qualification_level	Master's degree
author	Abdul Sathar, Mohammad Hasan
author_facet	Abdul Sathar, Mohammad Hasan
author_sort	Abdul Sathar, Mohammad Hasan
title	Numerical Solution of Infinite and Finite Boundary Integral Equations by Polynomials
title_short	Numerical Solution of Infinite and Finite Boundary Integral Equations by Polynomials
title_full	Numerical Solution of Infinite and Finite Boundary Integral Equations by Polynomials
title_fullStr	Numerical Solution of Infinite and Finite Boundary Integral Equations by Polynomials
title_full_unstemmed	Numerical Solution of Infinite and Finite Boundary Integral Equations by Polynomials
title_sort	numerical solution of infinite and finite boundary integral equations by polynomials
granting_institution	Universiti Putra Malaysia
granting_department	Institute for Mathematical Research
publishDate	2010
url	http://psasir.upm.edu.my/id/eprint/19688/1/IPM_2010_9_F.pdf
_version_	1747811443091177472

Numerical Solution of Infinite and Finite Boundary Integral Equations by Polynomials

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