Résolution des équations différentielles fractionnaires.

Abstract

In this work we have studied the existence and non-existence of solutions of nonlinear fractional differential equations. For the study of the existence and uniqueness the fixed point theory has been applied and for non-existence the weak formulations and the test function technique have been used. Our task in this thesis is to detail the demonstrations of certain articles that treat the subject in order to make them clearer for the readers. For this, we organized this work as follows: Chapter 1: In this chapter, we present some basic tools on Laplace and Fourier transforms and some definitions of special functions useful throughout our thesis such as: Euler gamma function, beta function, function of Mittag-Leffler with examples and some interesting properties. Chapter 2: Chapter 2 is devoted to the definitions of derivatives and fractional integrals in the sense of Riemann-Liouville, Grünwald-Letnikov and Caputo and the links between these derivatives with some examples and some complementary properties as well as their Laplace transforms. Chapter 3: This chapter is dedicated to the explicit resolution of some fractional differential and integral equations, for a good understanding and a good manipulation of the fractional operational calculus. Chapter 4 : The purpose of this chapter is to study the existence and uniqueness of solutions of some nonlinear fractional differential equations with integral boundary conditions, using fixed-point techniques. Chapter 5: This chapter is devoted to the study of the global non-existence of solutions of nonlinear hyperbolic fractional equations and fractional differential systems based on the test function method in search of Fujuta-type critical exponents.