Mixed problems for some integro-differential equations

Abstract

Partial differential equations and partial integro-differential equations with nonlocal boundary conditions occur increasingly in many fields of science and engineering. In this thesis, we study two classes of problems with a nonlocal boundary condition: the first one concerning a semilinear and nonlinear parabolic equation and the second one related to a parabolic integro-differential equation. Here we have to mention that the presence of integral conditions complicates the use of standard numerical methods. To overcome these difficulties, we have developed a new and very easy implementable numerical algorithm for computations. This latter is based on a suitable linearization in time and on the principle of linear superposition. The stability analysis has been performed and the optimal error estimates have been derived. The obtained approximate solutions of these problems are compared with the exact solutions and with the solutions obtained by alternative techniques when available.