Reaction-Advection-Diffusion Equations with Space Fractional Derivatives and Variable Coefficients on Infinite Domain

Abstract

In this paper we consider a space fractional reaction-advection-diffusion equation which is actually a semi-linear Cauchy problem with a spatial fractional derivative operator of order α, 0 < α < 2. We establish and prove assertions concerning the existence and uniqueness of solution within certain Colombeau space. The proofs are given in the case when the left Liouville fractional derivative is involved, but the results are also valid in the case of the right Liouville fractional derivative as well as for the Riesz fractional derivative. The solutions are obtained by using the theory of generalized uniformly continuous semigroups of operators. We have also proved that the non-regularized and corresponding regularized operators, as well as solutions of non-regularized and corresponding regularized equations are L 2 −associated.MSC 2010 : 26A33, 35R11, 46F30

On the unique solvability of a Cauchy problem with a fractional derivative

“…This equation has a special form of the problem from [16], however, we show that the solution of this equation can be presented explicitly, also this equation can be solved with more general assumptions about the function y(x). In [13], the authors establish and prove statements about the existence and uniqueness of the equation solution in some Colombo space.…”