Analytical and Numerical Solutions of Volterra Integral Equation of the Second Kind

In this thesis we focus on the analytical and numerical aspects of the Volterra integral equation of the second kind. This equation has wide range of applications in physics and engineering such as potential theory, Dirichlet problems, electrostatics, the particle transport problems of astrophysics, reactor theory, contact problems, diffusion problems and heat transfer problems. After introducing the types of integral equations, we will investigate some analytical and numerical methods for solving the Volterra integral equation of the second kind. These analytical methods include: the Adomian decomposition method, the modified decomposition method, the method of successive approximations, the series solution method and the conversion to initial value problem. For the numerical treatment of the Volterra integral equation we will implement the following numerical methods: Quadrature methods (Trapezoidal rule, Runge-Kutta method of order two, the fourth order Runge-Kutta method), Projection methods including collocation method and Galerkin method and the Block method. The mathematical framework of these numerical methods together with their convergence properties will be presented. These numerical methods will be illustrated by some numerical examples. Comparisons between these methods will be drawn. Numerical results show that the Trapezoidal rule has proved to be the most efficient method in comparison to the other numerical methods.