Solutions of the Volterra integro-differential equation.

Abstract

Integro-di erential equations has found extensive applications in the eld of engineering, sciences and mathematical modelling of various physical and biological phenomena. In this thesis we focus on the Volterra type integro-di erential equation which has been used to model biological species co-existing, heat di usion, electromagnetic theory etc. In recent years much research has focused on nding approximate solutions of the integro-di erential equation by polynomial methods, speci cally focusing on the Lagrange collocation and piecewise cubic Hermite collocation methods. A further aspect to the thesis will be on analytical methods, mainly the applications of Lie group theory to the Volterra type equation. Lie group theory is one of the most powerful methods applied to obtain solutions of di erential equations. We will present the linear independent symmetries of the Volterra type equation of the rst and second kind. In addition, we shall apply the Laplace transform and it's inverse to determine general solutions for selected forms of kernel, speci cally those with convolution integrands.