Numerical Solutions of Volterra Integral Equation of the Second Kind

Abstract

This thesis is concerned with the numerical solution of integral equations of the second kind Volterra form. Volterra integral equations I arc essentially " initial value " problems for ordinary differential

I equations. In reality any of such problems can be reformulated as a Volterra equations. Examples of these include problems of a preferential I direction (e.g. time, energy, etc.).

Methods of solution of Volterra integral equation can be classified into two categories; analytical solutions and numerical solutions. I Analytical solutions include methods such as differentiation, Laplace transformation and resolvent kernel. But these methods solve Volterra I integral equation whose kernel k(x, t) is dependent solely on the

I difference (x-t). So complicated form of the kernel will complicate the analytical method used. ln practical applications, it is almost invariably I necessary to solve the integral equations numerically. It is a subject, which has expanded very rapidly during the computer era. Volterra equations can be approximated in a straightforward way by means of I quadrature formulae. A quadrature rule is generic name given to any numerical method for evaluating an approximation to an integral I function.

In this thesis the quadrature methods are adapted to usc on a I graded nodes. The equal spaced nodes can be considered as a special case from the graded nodes. The value of error when using quadrature I method on graded nodes is reduced compared with the case of equal I spaced nodes (uniform node technique).