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.