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Study of certain partial differential equations of fractional order
(University August 20, 1955 –Skikda, 2023) Belhad, Tahar
In this thesis, we study some nonlinear elliptic problems involving rational operators. The difficulty lies in introducing non-classical functional spaces. In the first part, we studied the Dirichlet boundary value problem involving the rational operator p(x, y)−Laplacian. We base the outcome of existence under certain conditions on nonlinearity. Our goal is to apply Berkovits degree theorem to quasi-continuous operators of type (S+) generalized to real reflective Banach spaces. Through this theory, we prove the existence of weak, non-intuitive solutions to this problem. We then study another problem involving Ds, which is the distributive fractional Riesz gradient for 0 < s < 1, as a nonlocal operator generated by an Lèvy operation. Based on the invariance result for pseudotone operators, we have proven that there is at least one weak solution to such a problem. Moreover, we obtain the uniqueness of the solution to the problem under some considerations, and all of this from the theoretical side. From the numerical side, we have used a technique that is a combination of the Elzaki transformation method and the variational iteration method called the Elzaki variational iteration method (EVIM) on nonlinear fractional partial differential equations.