Analysis and numerical solution of a non-standard non-linear integro-differential boundary value problem

The aim of this thesis is the analytical study and the development of a numerical method to solve a non-linear, integro-differential boundary value problem on the half line which is representative of a class of non-standard integral equations where the derivatives of the unknown function are multiplied by an integral term depending on the unknown itself. The problems of this class are related to some real phenomena like plasmas kinetics and population dynamics. This equation is defined on the half line, and its non-standard nature makes the analytical and numerical study rather complicated. As a matter of fact, the knowledge about the existence, uniqueness and other properties of the solution is missing and the numerical methods are still undeveloped. Hence, the first part of this thesis concerns with the theoretical analysis of the integro-differential problem, which provides useful informations about the existence and other qualitative properties of the solution itself and represents an essential preparation for a numerical resolution of the problem itself. The second part of this thesis concerns with the development of the numerical method to solve the integro-differential problem and the study of the convergence of the method itself.