Etude des équations intégro-différentielles linéaire de seconde espèce: Approche analytique et numérique

Abstract

The objective of this thesis is to deal with linear integro-differential Fredholm equations especially as these kinds of equations play an important role in modeling of different problems in various fields such as physics and biology. Therefore, we are going to study them in the analytically and numerically sense. We study these equations in Banach space C1[a, b] with two cases of continuous and weakly singular kernels. In the continuous case we construct three methods based on the Nyström, Collocation and Kantorovich methods in order to find the best approximation of our solution. In the weakly singularcase, we construct two methods which are b-spline collocation and product integration in an essential reason which is a good precision and acceleration in the calculations. We analysis our equation analytically in Sobolev spaces W1,p[a, b], p 2 [1,+1[. We give a sufficient condition that shows the existence and uniqueness of the solution in these mathematical spaces. We have constructed in both spaces: W1,1[a, b] and H1[a, b] two projection methods based on Galerkin and Kantorovich.