On time fractional pseudo-parabolic equations with nonlocal integral conditions

Nguyen Anh Tuan; Donal O'Regan; Dumitru Baleanu; Nguyen H. Tuan

doi:10.3934/eect.2020109

Article Contents

2022, Volume 11, Issue 1: 225-238. Doi: 10.3934/eect.2020109

This issue Previous Article Energy method for exponential stability of coupled one-dimensional hyperbolic PDE-ODE systems Next Article Existence results for fractional impulsive delay feedback control systems with Caputo fractional derivatives

On time fractional pseudo-parabolic equations with nonlocal integral conditions

1.
Division of Applied Mathematics, Thu Dau Mot University, Binh Duong Province, Vietnam
2.
School of Mathematics, Statistics and Applied Mathematics, National University of Ireland, Galway, Ireland
3.
Department of Mathematics, Faculty of Arts and Sciences, Cankaya University, 06530 Ankara, Turkey, Institute of Space Sciences, P.O.Box, MG-23, R 76900, Magurele-Bucharest, Romania

^* Corresponding author: Nguyen H. Tuan
^* Corresponding author: Nguyen H. Tuan

Received: July 2020

Revised: October 2020

Early access: December 2020

Published: February 2022

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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.

Keywords:

Fractional partial differential equation,
Caputo fractional,
well-posedness,
pseudo-parabolic equation,
nonlocal conditions,
nonlocal in time.

Mathematics Subject Classification: 26A33, 35B65, 35R11.

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References

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