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

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.