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.

DECAY PROPERTY OF REGULARITY-LOSS TYPE FOR DISSIPATIVE TIMOSHENKO SYSTEM

2008 ◽  
Vol 18 (05) ◽  
pp. 647-667 ◽  
Author(s):  
KENTARO IDE ◽  
KAZUO HARAMOTO ◽  
SHUICHI KAWASHIMA
Keyword(s):  
Initial Data ◽  
Decay Property ◽  
General Situation ◽  
Decay Estimates ◽  
Timoshenko System ◽  
Linear Diffusion ◽  
Diffusion Wave ◽  
Estimates Of Solutions ◽  
Whole Space ◽  
The One

We study the decay property of the dissipative Timoshenko system in the one-dimensional whole space. We derive the L2decay estimates of solutions in a general situation and observe that this decay structure is of the regularity-loss type. Also, we give a refinement of these decay estimates for some special initial data. Moreover, under enough regularity assumption on the initial data, we show that the solution approaches the linear diffusion wave expressed in terms of the heat kernels as time tends to infinity. The proof is based on the detailed pointwise estimates of solutions in the Fourier space.

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.].

scholarly journals Simplified Grinding Temperature Model Study

2019 ◽  
Vol 6 (2) ◽  
pp. a1-a7
Author(s):  
N. V. Lishchenko ◽  
V. P. Larshin ◽  
H. Krachunov
Keyword(s):  
Differential Equation ◽  
Heat Conduction ◽  
Heat Source ◽  
Boundary Conditions ◽  
Grinding Temperature ◽  
Temperature Model ◽  
One Dimensional ◽  
Heating And Cooling ◽  
Equation Of Heat Conduction ◽  
The One

A study of a simplified mathematical model for determining the grinding temperature is performed. According to the obtained results, the equations of this model differ slightly from the corresponding more exact solution of the one-dimensional differential equation of heat conduction under the boundary conditions of the second kind. The model under study is represented by a system of two equations that describe the grinding temperature at the heating and cooling stages without the use of forced cooling. The scope of the studied model corresponds to the modern technological operations of grinding on CNC machines for conditions where the numerical value of the Peclet number is more than 4. This, in turn, corresponds to the Jaeger criterion for the so-called fast-moving heat source, for which the operation parameter of the workpiece velocity may be equivalently (in temperature) replaced by the action time of the heat source. This makes it possible to use a simpler solution of the one-dimensional differential equation of heat conduction at the boundary conditions of the second kind (one-dimensional analytical model) instead of a similar solution of the two-dimensional one with a slight deviation of the grinding temperature calculation result. It is established that the proposed simplified mathematical expression for determining the grinding temperature differs from the more accurate one-dimensional analytical solution by no more than 11 % and 15 % at the stages of heating and cooling, respectively. Comparison of the data on the grinding temperature change according to the conventional and developed equations has shown that these equations are close and have two points of coincidence: on the surface and at the depth of approximately threefold decrease in temperature. It is also established that the nature of the ratio between the scales of change of the Peclet number 0.09 and 9 and the grinding temperature depth 1 and 10 is of 100 to 10. Additionally, another unusual mechanism is revealed for both compared equations: a higher temperature at the surface is accompanied by a lower temperature at the depth. Keywords: grinding temperature, heating stage, cooling stage, dimensionless temperature, temperature model.

scholarly journals Fractional Heat Conduction Models and Thermal Diffusivity Determination

2015 ◽  
Vol 2015 ◽  
pp. 1-9 ◽  
Author(s):  
Monika Žecová ◽  
Ján Terpák
Keyword(s):  
Heat Conduction ◽  
Thermal Diffusivity ◽  
Numerical Methods ◽  
Fractional Order ◽  
Experimental Verification ◽  
Heat Conduction Equation ◽  
Historical Overview ◽  
One Dimensional ◽  
The One ◽  
Fractional Heat Conduction

The contribution deals with the fractional heat conduction models and their use for determining thermal diffusivity. A brief historical overview of the authors who have dealt with the heat conduction equation is described in the introduction of the paper. The one-dimensional heat conduction models with using integer- and fractional-order derivatives are listed. Analytical and numerical methods of solution of the heat conduction models with using integer- and fractional-order derivatives are described. Individual methods have been implemented in MATLAB and the examples of simulations are listed. The proposal and experimental verification of the methods for determining thermal diffusivity using half-order derivative of temperature by time are listed at the conclusion of the paper.

scholarly journals Existence and nonexistence of entire solutions to the logistic differential equation

2003 ◽  
Vol 2003 (17) ◽  
pp. 995-1003 ◽  
Author(s):  
Marius Ghergu ◽  
Vicentiu Radulescu
Keyword(s):  
Blow Up ◽  
Entire Solutions ◽  
One Dimensional ◽  
Nonexistence Result ◽  
Nonnegative Solutions ◽  
Existence And Nonexistence ◽  
Logistic Problem ◽  
Whole Space ◽  
The Mean ◽  
The One

We consider the one-dimensional logistic problem(rαA(|u′|)u′)′=rαp(r)f(u)on(0,∞),u(0)>0,u′(0)=0, whereαis a positive constant andAis a continuous function such that the mappingtA(|t|)is increasing on(0,∞). The framework includes the case wherefandpare continuous and positive on(0,∞),f(0)=0, andfis nondecreasing. Our first purpose is to establish a general nonexistence result for this problem. Then we consider the case of solutions that blow up at infinity and we prove several existence and nonexistence results depending on the growth ofpandA. As a consequence, we deduce that the mean curvature inequality problem on the whole space does not have nonnegative solutions, excepting the trivial one.

scholarly journals Internal Freezing of Snow Cover due to its Own Cold Heat Sources (Abstract)

1985 ◽  
Vol 6 ◽  
pp. 329-329
Author(s):  
Y. Yamada ◽  
T. Ikarashi
Keyword(s):  
Heat Conduction ◽  
Open System ◽  
The Other ◽  
Fundamental Aspect ◽  
Heat Sources ◽  
Field Observations ◽  
Snow Layer ◽  
One Dimensional ◽  
Wet Snow ◽  
The One

This report discusses the one-dimensional freezing of dry snow/ wet snow systems for the condition first examined by Stefan: the problem of heat conduction with phase change. There are two systems of internal freezing: one is a closed system of temperature rise in a dry snow layer sandwiched between upper and lower wet snow layers; the other an open system of freezing of a thin wet layer provoked mainly by an upper dry snow layer facing the atmosphere at its surface. The latter negatively concerns the release of some avalanches, because the weak layers of surface avalanches in districts where the melt-freeze metamorphism prevails (as in the Horuriku district of Japan) may be the thin wet granular snow layers.Numerical results are given for different conditions of internal freezing. A comparison with field observations reveals the fundamental aspect of this phenomenon and the possibility of avalanche release.

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