Résumé:
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).