Timoshenko system with fractional operator in the memory and spatial fractional thermal effect
- PDF / 3,390,745 Bytes
- 29 Pages / 439.37 x 666.142 pts Page_size
- 13 Downloads / 192 Views
Timoshenko system with fractional operator in the memory and spatial fractional thermal effect Hanni Dridi1 · Abdelhak Djebabla1 Received: 5 November 2019 / Accepted: 22 April 2020 © Springer-Verlag Italia S.r.l., part of Springer Nature 2020
Abstract We consider the differential system
𝜌1 𝜑tt − 𝜅(𝜑xx + 𝜓x ) = 0, 𝜌2 𝜓tt − b𝜓xx + 𝜅(𝜑x + 𝜓) − 𝜌3 𝜃t +
1 𝛽 ∫0
∫0
∞
h(s)A𝜎 𝜓(t − s)ds + 𝛿𝜃x = 0,
∞
g(s)A𝜎 𝜃(t − s)ds + 𝛿𝜓tx = 0,
describing a Timoshenko system with fractional operator in the memory and spatial fractional thermal effect of Gurtin–Pipkin type, which depends on a parameter 𝜎 ∈ [0, 1]. Under some assumptions on the kernels, the paper proved the global existence of a weak solution. In addition the utilisation of the semigroup method in fractional Hilbert space shows some results about the system stability which is related to the number of stability 𝜉g and the parameter of the fractional order 𝜎. Keywords Timoshenko system · Fractional thermal effect · Asymptotic behavior · Contraction semigroup · Stability number · Coupled systems Mathematics Subject Classification 35B40 · 45K05 · 74D05 · 74D03
1 Introduction Timoshenko’s theory [30] is an improvement of the Euler–Bernoulli theory [6]. Indeed, these systems modeling beams under several vibration. Recently, the system stability of Timoshenko is one of important posed question . Widely speaking, there is different types of damping that have been used to dampen undesirable vibrations, as portrayed by several authors, among * Hanni Dridi [email protected] Abdelhak Djebabla [email protected] 1
Laboratory of Applied Mathematics, University of Badji Mokhtar, P.O. Box 12, 23000 Annaba, Algeria
13
Vol.:(0123456789)
H. Dridi, A. Djebabla
them Kim and Renardy [18], Raposo et al. [25], Messaoudi and Mustafa [20], and Tian and Zhang [29]. Subsequently, many results of stability wether (exponential or polynomial) have prevailed in the literature. Consequently, when the term damping occurs in the system, it will give the wave speeds a crucial role in determining the behavior of the solutions at infinite time. This paper will study the following problem
𝜌1 𝜑tt − 𝜅(𝜑xx + 𝜓x ) = 0, 𝜌2 𝜓tt − b𝜓xx + 𝜅(𝜑x + 𝜓) −
𝜌3 𝜃t +
1 𝛽 ∫0
∫0
(1.1)
∞
h(s)A𝜎 𝜓(t − s)ds + 𝛿𝜃x = 0,
(1.2)
∞
g(s)A𝜎 𝜃(t − s)ds + 𝛿𝜓tx = 0,
(1.3)
Here, the unknows variables 𝜑, 𝜓, 𝜃 ∶ (x, t) ∈ (0, L) × ℝ+ ↦ ℝ are the transverse displacement, the rotation angle and the relative temperature, as stated respectevely. Furthemore, the operator A represents the derivatives (−𝜕xx ) and 𝜎 is a parameter in the interval [0, 1]. At last, 𝜅, b, 𝛿, 𝛽 represent the positive coefficients in addition to 𝜌i for i = {1, 2, 3}. The aforementioned system (1.1)–(1.3) is complemented with the Dirichlet boundary conditions for 𝜑 and 𝜃
𝜑(0, t) = 𝜑(L, t) = 𝜃(0, t) = 𝜃(L, t) = 0,
(1.4)
In addition to Neumann boundary condition for 𝜓
𝜓x (0, t) = 𝜓x (L, t) = 0.
(1.5)
The system describes a model for elastic beams vibrations. It is the coupling of the shear force and the bending moment acting on the syste
Data Loading...
