L'application des méthodes numériques aux équations différentielles d’ordre fractionnaire

Abstract

In this thesis, we use a novel approach to study the existence, uniqueness, and stability of solutions to a Cauchy-type problem of nonlinear fractional differential equations of variable order with finite and infinite delay. Contrary to the techniques taken in the literature, which were centered on the usage of the concept of generalized intervals and the idea of piecewise constant functions, our approach is straightforward and based on a novel fractional operator that is more appropriate and demonstrates the solvability and stability of the main problem under less restrictive presumptions. All results are achieved by using fixed point theory. In all chapters of this work, we have illustrated our theoretical study with numerical applications to approximate the solution to all our proposed problems, and we have used different methods. One of these methods is the finite difference method. The finite difference method is a numerical technique used to solve differential equations by approximating derivatives with finite differences. It is widely used in physics, engineering, and other fields where differential equations need to be solved numerically. Essentially, it discretizes space and time into small intervals and approximates derivatives with finite differences to obtain a numerical solution. The second method is the Euler discretization method, also known as the Euler method, which is one of the simplest numerical methods for solving ordinary differential equations (ODEs). It belongs to the family of finite difference methods and applies to first-order equations. The basic idea of this method is to approximate the solution of a differential equation step by step, using a time interval and starting from an initial condition. In this work, comparisons were made between these two methods to confirm the theoretical results founded in this thesis.