On the well posedness of a mathematical model for a singular nonlinear fractional pseudo-hyperbolic system with nonlocal boundary conditions and frictional damping terms

Said Mesloub; Hassan Altayeb Gadain; Lotfi Kasmi; Said Mesloub; Hassan Altayeb Gadain; Lotfi Kasmi

doi:10.3934/math.2024146

AIMS Mathematics

2024, Volume 9, Issue 2: 2964-2992. doi: 10.3934/math.2024146

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On the well posedness of a mathematical model for a singular nonlinear fractional pseudo-hyperbolic system with nonlocal boundary conditions and frictional damping terms

1.
Department of Mathematics, King Saud University, P.O. Box 2455, Riyadh, Saudi Arabia
2.
Applied Mathematics Lab, University Kasdi Merbah, Ouargla, Algeria

Received: 25 August 2023 Revised: 19 November 2023 Accepted: 05 December 2023 Published: 02 January 2024
MSC : 35B45, 35D30, 35L70, 35R11

This paper is devoted to the study of the well-posedness of a singular nonlinear fractional pseudo-hyperbolic system with frictional damping terms. The fractional derivative is described in Caputo sense. The equations are supplemented by classical and nonlocal boundary conditions. Upon some a priori estimates and density arguments, we establish the existence and uniqueness of the strongly generalized solution for the associated linear fractional system in some Sobolev fractional spaces. On the basis of the obtained results for the linear fractional system, we apply an iterative process in order to establish the well-posedness of the nonlinear fractional system. This mathematical model of pseudo-hyperbolic systems arises mainly in the theory of longitudinal and lateral vibrations of elastic bars (beams), and in some special case it is propounded in unsteady helical flows between two infinite coaxial circular cylinders for some specific boundary conditions.
- pseudo-hyperbolic system,
- energy inequality,
- existence and uniqueness,
- iterative method,
- weak solution,
- Sobolev fractional space,
- nonlocal boundary condition
Citation: Said Mesloub, Hassan Altayeb Gadain, Lotfi Kasmi. On the well posedness of a mathematical model for a singular nonlinear fractional pseudo-hyperbolic system with nonlocal boundary conditions and frictional damping terms[J]. AIMS Mathematics, 2024, 9(2): 2964-2992. doi: 10.3934/math.2024146

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Abstract

This paper is devoted to the study of the well-posedness of a singular nonlinear fractional pseudo-hyperbolic system with frictional damping terms. The fractional derivative is described in Caputo sense. The equations are supplemented by classical and nonlocal boundary conditions. Upon some a priori estimates and density arguments, we establish the existence and uniqueness of the strongly generalized solution for the associated linear fractional system in some Sobolev fractional spaces. On the basis of the obtained results for the linear fractional system, we apply an iterative process in order to establish the well-posedness of the nonlinear fractional system. This mathematical model of pseudo-hyperbolic systems arises mainly in the theory of longitudinal and lateral vibrations of elastic bars (beams), and in some special case it is propounded in unsteady helical flows between two infinite coaxial circular cylinders for some specific boundary conditions.

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