On Time Fractional Pseudo-Parabolic Equations With Nonlocal Integral Conditions

Abstract

In this paper, we study the nonlocal problem for pseudo-parabolic equation with time and space fractional derivatives. The time derivative is of Caputo type and of order sigma, 0 < sigma < 1 and the space fractional derivative is of order alpha, beta > 0. In the first part, we obtain some results of the existence and uniqueness of our problem with suitably chosen alpha, beta. The technique uses a Sobolev embedding and is based on constructing a Mittag-Leffler operator. In the second part, we give the ill-posedness of our problem and give a regularized solution. An error estimate in L-p between the regularized solution and the sought solution is obtained.