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Abstract
The main objective of this thesis is to develop new effective approximate and numerical methods and supporting analysis for solving initial value problems of integral and integro-differential equations, boundary value problems of integral and integro-differential equations, a class of integral and integro-differential equations.
This thesis consists of five chapters:
Chapter one:
In this chapter, we introduce some definitions, lemmas and important theorems, with out proof, which are needed and used throughout this thesis.
Chapter two:
we present a numerical method to solve the integro-differential equations (IDEs). The proposed method uses the Legendre cardinal functions to express the approximate solution as a finite series. Then, we used Legendre collocation method to solve numerically the Fredholm-Hammerstein integral equations. This method is based on replacement of the unknown function by truncated series of well known Legendre expansion of functions. Finally, we used shifted Legendre expansion to solve linear and non-linear equations for integro-differential equation and system of integro-differential equations. An approximate formula of the integer derivative is introduced. This methods converts the proposed equation by means of
collocation points to system of algebraic equations with shifted Legendre coefficients. Several examples are used to illustrate the proposed concept.
Chapter three:
In this chapter, we implement a numerical technique for solving linear and non-linear integro-differential equations (IDEs) and we comparing some example with finite difference method. Then, we use this technique for solving linear and non-linear system of integro-differential equations. This technique is based on using matrix operator expressions which applies to the differential terms. This method is replaced of the unknown function by truncated series of well known Chebyshev expansion of functions.
Chapter four:
This chapter applies the homotopy analysis method (HAM) to obtain analytical solutions of integro-differential equations and system of integro-differential equations. The applications of the HAM were
extended to derive analytical solutions in the form of a series with easily computed terms for these generalized integro-differential equations. Series solutions of the problem under consideration are
developed by means of HAM and the recurrence relations are given explicitly. The initial approximation can be freely chosen.
Chapter five:
In this chapter, we introduce a modification of the Taylor matrix method using Padé approximation to obtain an accurate solutions of the linear Fredholm integro-differential equations (FIDEs) and system of integro-differential equation. This modification is based on, first, taking truncated Taylor series of
the functions in equation and then substituting their matrix forms into the given equation. Thereby the equation reduces to a matrix equation, which corresponds to a system of linear algebraic
equations with unknown Taylor coefficients. Finally, we use Padé approximation to obtain an accurate numerical solution of the proposed problem.