On time fractional pseudo-parabolic equations with nonlocal integral conditions

Abstract

<p style='text-indent:20px;'>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 <inline-formula><tex-math id="M1">\begin{document}$ \sigma,\; \; 0<\sigma<1 $\end{document}</tex-math></inline-formula> and the space fractional derivative is of order <inline-formula><tex-math id="M2">\begin{document}$ \alpha,\beta >0 $\end{document}</tex-math></inline-formula>. In the first part, we obtain some results of the existence and uniqueness of our problem with suitably chosen <inline-formula><tex-math id="M3">\begin{document}$ \alpha, \beta $\end{document}</tex-math></inline-formula>. The technique uses a Sobolev embedding and is based on constructing a Mittag-Leffler operator. In the second part, we give the ill-posedness of our problem and give a regularized solution. An error estimate in <inline-formula><tex-math id="M4">\begin{document}$ L^p $\end{document}</tex-math></inline-formula> between the regularized solution and the sought solution is obtained.</p>