Analytical and Numerical Study of Certain Fractional Boundary Problems

Abstract

The main objective of this thesis is to present an analytical and numerical contribution of certain fractional boundary problems according to different approaches. Original results ensuring the existence and uniqueness/existence as well as stability of solutions are discussed for some new problems involving fractional order operators. In addition, an approach for solving a type of fractional linear problems with boundary conditions is developed and some applications are presented, where the validity and accuracy of this scheme are shown. The analytical results of this thesis focus on the application of some fixed point theorems and certain types of Ulam stability to address two proposed fractional problems. The first problem concerns the Van de Pol Duffing (VDPD)-Jerk oscillator, while the second one involves the pantograph type equation, utilizing the Caputo-Hadamard approach. Illustrative examples will be provided to demonstrate the validity of the results. We devote a final part of our project to numerical results, where an approach is developed to approximate the solutions of a class of fractional linear boundary value problems and some applications are presented in this context.