The asymptotic behavior of the Bresse–Cattaneo system

2016 ◽  
Vol 18 (04) ◽  
pp. 1550045 ◽  
Author(s):  
Belkacem Said-Houari ◽  
Taklit Hamadouche
Keyword(s):  
Heat Conduction ◽  
Differential Equations ◽  
Decay Rate ◽  
Stability Number ◽  
Timoshenko System ◽  
Decay Properties ◽  
Bresse System ◽  
Whole Space ◽  
The Stability ◽  
The One

In this paper, we investigate the decay properties of the Bresse–Cattaneo system in the whole space. We show that the coupling of the Bresse system with the heat conduction of the Cattaneo theory leads to a loss of regularity of the solution and we prove that the decay rate of the solution is very slow. In fact, we show that the [Formula: see text]-norm of the solution decays with the rate of [Formula: see text]. The behavior of solutions depends on a certain number [Formula: see text] (which is the same stability number for the Timoshenko–Cattaneo system [Damping by heat conduction in the Timoshenko system: Fourier and Cattaneo are the same, J. Differential Equations 255(4) (2013) 611–632; The stability number of the Timoshenko system with second sound, J. Differential Equations 253(9) (2012) 2715–2733]) which is a function of the parameters of the system. In addition, we show that we obtain the same decay rate as the one of the solution for the Bresse–Fourier model [The Bresse system in thermoelasticity, to appear in Math. Methods Appl. Sci.].

DECAY PROPERTY FOR THE TIMOSHENKO SYSTEM WITH FOURIER'S TYPE HEAT CONDUCTION

2014 ◽  
Vol 11 (01) ◽  
pp. 135-157 ◽  
Author(s):  
NAOFUMI MORI ◽  
SHUICHI KAWASHIMA
Keyword(s):  
Heat Conduction ◽  
Energy Method ◽  
Decay Estimates ◽  
Fourier Space ◽  
Pointwise Estimates ◽  
Timoshenko System ◽  
One Dimensional ◽  
Estimates Of Solutions ◽  
Whole Space ◽  
The One

We study the Timoshenko system with Fourier's type heat conduction in the one-dimensional (whole) space. We observe that the dissipative structure of the system is of the regularity-loss type, which is somewhat different from that of the dissipative Timoshenko system studied earlier by Ide–Haramoto–Kawashima. Moreover, we establish optimal L2decay estimates for general solutions. The proof is based on detailed pointwise estimates of solutions in the Fourier space. Also, we introuce here a refinement of the energy method employed by Ide–Haramoto–Kawashima for the dissipative Timoshenko system, which leads us to an improvement on their energy method.

scholarly journals Researches in physical astronomy

1837 ◽  
Vol 3 ◽  
pp. 128-129
Keyword(s):  
Differential Equations ◽  
The Other ◽  
Disturbing Function ◽  
Direct Integration ◽  
Planetary Theory ◽  
The Moon ◽  
Disturbing Force ◽  
Other Hand ◽  
The Stability ◽  
The One

The present paper contains some further developments of the theory of the moon, which are given at length, in order to save the trouble of the calculator, and to avoid the danger of mistake. The author remarks, that while it seems desirable, on the one hand, to introduce into the science of physical astronomy a greater degree of uniformity, by bringing to perfection a theory of the moon founded on the integration of the equations employed in the planetary theory, it is also no less important, on the other hand, to complete, in the latter, the method hitherto applied solely to the periodic inequalities. Hi­therto those terms in the disturbing function which give rise to the secular inequalities, have been detached, and the stability of the system has been inferred by means of the integration of certain equations, which are linear when the higher powers of the eccentri­cities are neglected and from considerations founded on the varia­tion of the elliptic constants. But the author thinks that the stability of the system may be inferred also from the expressions which result at once from the direct integration of the differential equations. The theory, he states, may be extended, without any analytical difficulty, to any power of the disturbing force, or of the eccentricities, ad­mitting the convergence of the series; nor does it seem to be limited by the circumstance of the planet’s moving in the same direction.

Decay property of the Timoshenko–Cattaneo system

2016 ◽  
Vol 14 (03) ◽  
pp. 393-413 ◽  
Author(s):  
Naofumi Mori ◽  
Shuichi Kawashima
Keyword(s):  
Heat Conduction ◽  
Decay Estimate ◽  
Low Frequency ◽  
Pointwise Estimate ◽  
Frequency Region ◽  
Decay Property ◽  
General Situation ◽  
High Frequency Region ◽  
Timoshenko System ◽  
Whole Space

We study the Timoshenko system with Cattaneo’s type heat conduction in the one-dimensional whole space. We investigate the dissipative structure of the system and derive the optimal [Formula: see text] decay estimate of the solution in a general situation. Our decay estimate is based on the detailed pointwise estimate of the solution in the Fourier space. We observe that the decay property of our Timoshenko–Cattaneo system is of the regularity-loss type. This decay property is a little different from that of the dissipative Timoshenko system (see [K. Ide, K. Haramoto and S. Kawashima, Decay property of regularity-loss type for dissipative Timoshenko system, Math. Models Methods Appl. Sci. 18 (2008) 647–667]) in the low frequency region. However, in the high frequency region, it is just the same as that of the Timoshenko–Fourier system (see [N. Mori and S. Kawashima, Decay property for the Timoshenko system with Fourier’s type heat conduction, J. Hyperbolic Differential Equations 11 (2014) 135–157]) or the dissipative Timoshenko system (see [K. Ide, K. Haramoto and S. Kawashima, Decay property of regularity-loss type for dissipative Timoshenko system, Math. Models Methods Appl. Sci. 18 (2008) 647–667]), although the stability number is different. Finally, we study the decay property of the Timoshenko system with the thermal effect of memory-type by reducing it to the Timoshenko–Cattaneo system.

scholarly journals Uniform decay rates of a Bresse thermoelastic system in the whole space

2021 ◽  
Author(s):  
Mounir Afilal ◽  
Baowei Feng ◽  
Abdelaziz Soufyane
Keyword(s):  
Energy Method ◽  
Decay Rates ◽  
Lyapunov Functionals ◽  
Fourier Space ◽  
Uniform Decay ◽  
Decay Properties ◽  
Bresse System ◽  
Whole Space

In this paper, we investigate the decay properties of the thermoelastic Bresse system in the whole space. We consider many cases depending on the parameters of the model and we establish new decay rates. We need to mention here that, in some cases we don’t have the regularity-loss phenomena as in the previous works in the literature. To prove our results, we use the energy method in the Fourier space to build a very delicate Lyapunov functionals that give the desired results.

Difference Schemes with Nonlocal Boundary Conditions

2001 ◽  
Vol 1 (1) ◽  
pp. 62-71 ◽  
Author(s):  
Alexei V. Goolin ◽  
Nikolai I. Ionkin ◽  
Valentina A. Morozova
Keyword(s):  
Heat Conduction ◽  
Boundary Conditions ◽  
Heat Conduction Equation ◽  
Nonlocal Boundary ◽  
Nonlocal Boundary Conditions ◽  
Difference Schemes ◽  
Conduction Equation ◽  
The Stability ◽  
The One ◽  
Necessary And Sufficient

AbstractThe paper deals with the stability, with respect to initial data, of difference schemes that approximate the heat-conduction equation with constant coefficients and nonlocal boundary conditions. Some difference schemes are considered for the one-dimensional heat-conduction equation, the energy norm is constructed, and the necessary and sufficient stability conditions in this norm are established for explicit and weighted difference schemes.

scholarly journals Direct Two-Point Block One-Step Method for Solving General Second-Order Ordinary Differential Equations

2012 ◽  
Vol 2012 ◽  
pp. 1-16 ◽  
Author(s):  
Zanariah Abdul Majid ◽  
Nur Zahidah Mokhtar ◽  
Mohamed Suleiman
Keyword(s):  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Multistep Method ◽  
Second Order ◽  
Variable Step Size ◽  
Step Size ◽  
Step Method ◽  
One Step ◽  
The Stability ◽  
The One

A direct two-point block one-step method for solving general second-order ordinary differential equations (ODEs) directly is presented in this paper. The one-step block method will solve the second-order ODEs without reducing to first-order equations. The direct solutions of the general second-order ODEs will be calculated at two points simultaneously using variable step size. The method is formulated using the linear multistep method, but the new method possesses the desirable feature of the one-step method. The implementation is based on the predictor and corrector formulas in thePE(CE)mmode. The stability and precision of this method will also be analyzed and deliberated. Numerical results are given to show the efficiency of the proposed method and will be compared with the existing method.

scholarly journals Lack of Exponential Decay for a Laminated Beam with Structural Damping and Second Sound

2017 ◽  
Author(s):  
Wenjun Liu ◽  
Xiangyu Kong ◽  
Gang Li
Keyword(s):  
Heat Conduction ◽  
Exponential Decay ◽  
Polynomial Decay ◽  
Second Sound ◽  
Structural Damping ◽  
Stability Number ◽  
Laminated Beam ◽  
One Dimensional ◽  
The Stability ◽  
Math Phys

In previous work (Z. Angew. Math. Phys. 68(2), 2017), Apalara considered a one dimensional thermoelastic laminated beam under Cattaneo’s law of heat conduction and proved the exponential and polynomial decay results depend on the stability number χT . In this paper, we continue to study the same system and show that the solution of the concerned system lacks of exponential decay result in the case χT ≠ 0 which solves the open problem proposed by Apalara (Z. Angew. Math. Phys. 68(2), 2017).

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