Solution of Integral Equations by Operational matrix Method

Abstract

Many researchers and scientist were attracted towards a special kind of newlinefunctional equation which is known as integral equations. An integral equation is newlinedefined as unknown function appears under the sign of integration. Integral equation newlineplays important role in mathematics. It can be represented in the form of mathematical newlineproblem for different physical conditions. Different methods have adopted for solving newlineintegral equations which also present efficacy and accuracy of adopted techniques. In newlinethis work, different numerical methods based on different polynomials such as newlineBernoulli polynomials and Euler polynomials have been developed to solve different newlinetypes of linear integral equations. newlinePreliminary concept of integral equations, Bernoulli polynomial and Euler newlinepolynomials have been presented in Chapter 1. Also, classifications of integral newlineequations, origin of integral equation, properties of integral equations and solution of newlineintegral equation have been briefly discussed. Further, properties of Bernoulli and newlineEuler polynomials, relation between them and application of integral equations have newlinepresented. At last, operational matrix, approximation of function, gram-schmidt newlineorthonormalization and literature survey have been discussed. newlineIn chapter 2, orthonormal polynomials have been constructed of the Bernoulli newlinepolynomials with gram- schmidt orthonormalization. An integration operator has been newlineused on these orthonormal polynomials and obtained the operational matrix of newlineintegration. With the help of this operational matrix, many Abel-type integral newlineequations have been solved, and compared the error between numerical solutions and newlineexact solutions of these integral equations. newlineIn Chapter 3, a new collocation method is employed using Euler polynomials to newlineobtain numerical solution of singular weakly linear Volterra - integro-differential newlineequations. The main feature of this method is that approximate solution will be newlineobtained in the form of algebraic equations. Error analysis is shown for Euler series newlinesolution of weakly linear Volterra- integro-differential equations. At last, few newlinenumerical problems are solved with the help of presented method to show the high newlinex newlineaccuracy and excellent behavior of suggested method in comparison with some other newlinewell-known methods. newlineIn chapter 4, an effective matrix method has been introduced for solving the newlinesystem of second kind linear Volterra integral equation with variable coefficients. newlineWith the help of Bernoulli polynomials and collocation points, the system of Volterra newlineintegral equations reduces into matrix equation which transform to a system of linear newlineequations with the different Bernoulli coefficients. Also, this method generates newlineanalytic solution of numerical example. Some numerical examples are presented to newlineconfirm the reliability of the technique. newlineIn chapter 5, a new method has been developed for solving the mixed Volterra newlineFredholm integral equations (VFIE s) of the second kind numerically. A Bernoulli newlinematrix approach is implemented for solving mixed VFIE s integral equations. The newlinemain characteristic behind this approach is that it reduces such problem to those of newlinesolving a system of algebraic equations. Introducing our purposed method; we used it newlineto convert integral equation into algebraic equation with the help of Bernoulli matrix newlineequation. Finally some numerical results are given to illustrate the efficiency and newlineexactness of this method newline