Study of certain partial differential equations of fractional order
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Date
2023
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University August 20, 1955 –Skikda
Abstract
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.