Stochastic fractional evolution equations of order $1<\alpha<2$ with generalized operators

TL;DR

This paper studies stochastic fractional evolution equations with derivatives of order between 1 and 2, using generalized operators to establish unique solutions and applying the theory to fractional wave equations.

Contribution

It introduces a method for solving stochastic fractional evolution equations with generalized operators and proves the existence of unique solutions in a generalized stochastic process space.

Findings

Established existence and uniqueness of solutions for the equations.

Developed a solution approach using approximate problems and $L^2$-association.

Applied the theory to stochastic time and space fractional wave equations.

Abstract

We consider the Cauchy problem for stochastic fractional evolution equations with Caputo time fractional derivative of order $1<\alpha<2$ and space variable coefficients on an unbounded domain. The space derivatives that appear in the equations are of integer or fractional order such as the left and the right Liouville fractional derivative as well as the Riesz fractional derivative. To solve the problem we use generalized uniformly continuous solution operators. We obtain the unique solution within a certain Colombeau generalized stochastic process space. In our solving procedure, instead of the originate problem we solve a certain approximate problem, where operators of the original and the approximate problem are $L^2$-associated. Finally, application of the theory in solving stochastic time and time-space fractional wave equation is shown.