On time fractional pseudo-parabolic equations with nonlocal integral conditions

The main objective of the paper is to study the non-local problem for a pseudo-parabolic equation with fractional time and space. The derivative of time is understood in the sense of the time derivative of the Caputo fraction of the order alpha, 0 < alpha < 1. The first result is an investigation of the existence and uniformity of the solution; the formula for mild solution and the regularity properties will be given. The proofs are based on a number of sophisticated techniques using the Sobolev embedding and also on the construction of the Mittag-Lefler operator. In the second part, we investigate the convergence of the mild solution for non-local problem to the solution of the local problem when two non-local parameters reach 0. Finally, we present some numerical examples to illustrate the proposed method.

Automated summary

The main objective of this paper is to study the non-local problem for a pseudo-parabolic equation with fractional time and space. In the first part, the existence and uniformity of the solution are investigated, and the formula for the mild solution and its regularity properties are provided. In the second part, the convergence of the mild solution for the non-local problem to the solution of the local problem is examined when two non-local parameters approach 0. Finally, numerical examples are presented to illustrate the proposed method.