Boundary Stabilization of a Thermoelastic Diffusion System of Type II
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Boundary Stabilization of a Thermoelastic Diffusion System of Type II Moncef Aouadi1 · Imed Mahfoudhi2,3 · Taoufik Moulahi4
Received: 3 May 2019 / Accepted: 28 December 2019 © Springer Nature B.V. 2020
Abstract In this paper we study the boundary stabilization of a one-dimensional thermoelastic diffusion problem of type II. The system of equations is a coupling of three hyperbolic equations. This poses some new mathematical and numerical difficulties. With the help of the semigroup theory of linear operators, we prove the well-posedness of the proposed problem. By using the frequency domain method combined with the multiplier technique, we prove the exponential stability of the solutions. Finally, we present a numerical scheme based on the Chebyshev spectral method and we give two numerical examples to validate the proposed model and to show its capability. Keywords Thermoelastic diffusion of type II · Well-posedness · Exponential decay · Numerical simulations Mathematics Subject Classification 35B40 · 65L07
1 Introduction Research conducted in the development of high technologies after the second world war, confirmed that the field of diffusion in solids cannot be ignored. Diffusion can be defined
B M. Aouadi
[email protected] I. Mahfoudhi [email protected] T. Moulahi [email protected]
1
Ecole Nationale d’Ingénieurs de Bizerte, Université de Carthage, BP66, Bizerte 7035, Tunisia
2
Peoples’ Friendship University of Russia (RUDN University), 6 Miklukho-Maklaya St, Moscow, 117198, Russian Federation
3
Faculté des Sciences de Monastir, Université de Monastir, Monastir 5000, Tunisia
4
Ecole Nationale d’Ingénieurs de Monastir, Université de Monastir, Monastir 5000, Tunisia
M. Aouadi et al.
as the random walk from a high concentration region to a low concentration region of a set of particles (such as an atom, ion, or molecule). This occurs in response to a concentration gradient expressed as the change in the concentration due to change in position. This phenomenon can be described by the classical Fick’s law. Today, the diffusion principle is increasingly applied in various fields, particularly in geophysics and industry. For instance, it can be used to measure the diffusion coefficients of various cations in minerals and improve the surface attributes of metals, such as wear, corrosion resistance, and hardness. Recently, in order to achieve more efficient extraction, the thermoelastic diffusion process has become a priority in the oil industry. In addition, the thermodiffusion process also contributes to studies in fields associated with the emergence of semiconductor devices and the advancement of microelectronics. Therefore, it is imperative to investigate the coupling effects among the temperature field, the diffusion field, and the strain field. Usually, the diffusion process is formulated using Fick’s Law. However, it does not consider the mutual exchange between the introduced substance and the substance or the influence of temperature on the interplay. The first critical evaluatio
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