Resolution of Fractional Differential Equation

Abstract

Motivated by the immense success of application of fractional equation in the branch of mathematical physics, the thesis studies three general models of fractional order partial differential equations using different definitions : Hilfer- Hadamard, Caputo and Caputo- Fabrizio fractional derivatives. In the first model, the study finds the critical exponents pc for which solutions cannot exist for all time in the subcritical case, whereas, in the critical and supercritical cases, global small data solutions exist. The discussion is based on the semi-group theory, fixed point theorem and the test function method. In the second model, the study shows that no solutions can exist for all time for certain values of p. Clearly, sufficient conditions for non-existence provide necessary conditions for existence of solutions. In many cases is difficult to find an analytical solution. For this reasons, the study uses a novel finite difference discretization scheme to solve numerically fractional-order’s partial differential equation involving a Caputo- Fabrizio fractional derivative supplemented with initial and boundary conditions (the third model).