On an initial boundary value problem for fractional pseudo-parabolic equation with conformable derivative

Huy Tuan Nguyen; Nguyen Van Tien; Chao Yang; Huy Tuan Nguyen; Nguyen Van Tien; Chao Yang

doi:10.3934/mbe.2022524

Mathematical Biosciences and Engineering

2022, Volume 19, Issue 11: 11232-11259. doi: 10.3934/mbe.2022524

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On an initial boundary value problem for fractional pseudo-parabolic equation with conformable derivative

1.
Division of Applied Mathematics, Science and Technology Advanced Institute, Van Lang University, Ho Chi Minh City, Vietnam
2.
Faculty of Applied Technology, School of Engineering and Technology, Van Lang University, Ho Chi Minh City, Vietnam
3.
Department of Mathematics and Computer Science, University of Science, Ho Chi Minh City, Vietnam
4.
Vietnam National University, Ho Chi Minh City, Vietnam
5.
Faculty of Math, FPT University HCM, Saigon Hi-Tech Park, Thu Duc City, Ho Chi Minh City, Vietnam
6.
College of Mathematical Sciences, Harbin Engineering University, Harbin 150001, China
7.
Faculty of Applied Mathematics, AGH University of Science and Technology, 30-059 Kraków, Poland

Received: 01 June 2022 Revised: 22 July 2022 Accepted: 26 July 2022 Published: 05 August 2022

In this paper, we study the initial boundary value problem of the pseudo-parabolic equation with a conformable derivative. We focus on investigating the existence of the global solution and examining the derivative's regularity. In addition, we contributed two interesting results. Firstly, we proved the convergence of the mild solution of the pseudo-parabolic equation to the solution of the parabolic equation. Secondly, we examine the convergence of solution when the order of the derivative of the fractional operator approaches $ 1^- $. Our main techniques used in this paper are Banach fixed point theorem and Sobolev embedding. We also apply different techniques to evaluate the convergence of generalized integrals encountered.
- conformable derivative,
- pseudo-parabolic equation,
- well-posedness,
- regularity estimates
Citation: Huy Tuan Nguyen, Nguyen Van Tien, Chao Yang. On an initial boundary value problem for fractional pseudo-parabolic equation with conformable derivative[J]. Mathematical Biosciences and Engineering, 2022, 19(11): 11232-11259. doi: 10.3934/mbe.2022524

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Abstract

In this paper, we study the initial boundary value problem of the pseudo-parabolic equation with a conformable derivative. We focus on investigating the existence of the global solution and examining the derivative's regularity. In addition, we contributed two interesting results. Firstly, we proved the convergence of the mild solution of the pseudo-parabolic equation to the solution of the parabolic equation. Secondly, we examine the convergence of solution when the order of the derivative of the fractional operator approaches $ 1^- $. Our main techniques used in this paper are Banach fixed point theorem and Sobolev embedding. We also apply different techniques to evaluate the convergence of generalized integrals encountered.

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