Spline Approximation for Solving Fredholm Integro­ Differential Equations of n-th Order

Abstract

This thesis, which consists of four chapters, lists of references and one appendix, is concerned with a procedure which uses spline functions for solving Fredholm integro-differential equation of n-th order.

Chapter I, contains a brief account of some definitions and important theorems related to the approximate solution for solving Fredholm integro-differential equations with deviating argument by spline functions.

Chapter II, contains a method for approximating the solution of the

Fredholm integro-differential equations of n-th order of the form

y<•>(x) = f(X,y(x),fk(x,t,y(t)dt), a$; X$; b

y <i> (a) -y<il ,

. - 0,I, ...,n- I ,

We use the spline functions which are not necessarily polynomials for finding the approximate solution. It is a one-step method o(h...,.,,) in y<il(x); assuming that f E c([a, b] x R 2 ), k E C([a, b] x [a,b] x R) and the

modulus of continuity of y<"1(x) is o{ha ), 0 <a-;;:: 1 and m is an arbitrary positive integer equal to the number of iterations used in the method. It is also shown that the method is stable.

Chapter III, contains a method for approximating the solution of the

Fredholm delay integro-differential equations of n-th order of the form