scholarly journals PullbackD-Attractor of Coupled Rod Equations with Nonlinear Moving Heat Source

2014 ◽  
Vol 2014 ◽  
pp. 1-11
Author(s):  
Danxia Wang ◽  
Jianwen Zhang ◽  
Yinzhu Wang
Keyword(s):  
Heat Source ◽  
Boundary Conditions ◽  
Galerkin Method ◽  
Existence And Uniqueness ◽  
Nonlinear Operator ◽  
Initial Conditions ◽  
Global Solutions ◽  
Absorbing Set ◽  
Moving Heat Source ◽  
Condition C

We consider the pullbackD-attractor for the nonautonomous nonlinear equations of thermoelastic coupled rod with a nonlinear moving heat source. By Galerkin method, the existence and uniqueness of global solutions are proved under homogeneous boundary conditions and initial conditions. By prior estimates combined with some inequality skills, the existence of the pullbackD-absorbing set is obtained. By proving the properties of compactness about the nonlinear operatorg1(·),g2(·), and then proving the pullbackD-condition (C), the existence of the pullbackD-attractor of the equations previously mentioned is given.

scholarly journals Thermoelastic Behavior Inthin Hollow Cylinder using Internal Moving Heat Source

2019 ◽  
Vol 9 (2) ◽  
pp. 2578-2583
Keyword(s):  
Temperature Distribution ◽  
Heat Source ◽  
Boundary Conditions ◽  
Hollow Cylinder ◽  
Infinite Series ◽  
Thermal Stresses ◽  
Integral Transform ◽  
Elastic Problem ◽  
Moving Heat Source ◽  
Integral Transform Technique

A hollow cylinder having cylindrical hole at the center has been examined under the temperature variation condition. This composition deals with study of temperature distribution in thin hollow cylinder and corresponding stresses. The author has worked to carry out the transient thermo elastic problem for evaluation of temperature distribution, displacement and thermal stresses of a thin hollow cylinder. The known non homogeneous boundary conditions are applied to obtain the solution of this problem. The integral transform technique yields the solution to the problem. The analysis contains an infinite series. The variation of said parameters observed and analyzed by using necessary graphs

scholarly journals Nonexistence of Global Solutions to Time-Fractional Damped Wave Inequalities in Bounded Domains with a Singular Potential on the Boundary

2021 ◽  
Vol 5 (4) ◽  
pp. 258
Author(s):  
Areej Bin Sultan ◽  
Mohamed Jleli ◽  
Bessem Samet
Keyword(s):  
Boundary Conditions ◽  
Singular Potential ◽  
Initial Conditions ◽  
Sufficient Conditions ◽  
Global Solutions ◽  
Dirichlet Boundary Conditions ◽  
Test Functions ◽  
Test Function ◽  
Judicious Choice ◽  
Test Function Method

We first consider the damped wave inequality ∂2u∂t2−∂2u∂x2+∂u∂t≥xσ|u|p,t>0,x∈(0,L), where L>0, σ∈R, and p>1, under the Dirichlet boundary conditions (u(t,0),u(t,L))=(f(t),g(t)),t>0. We establish sufficient conditions depending on σ, p, the initial conditions, and the boundary conditions, under which the considered problem admits no global solution. Two cases of boundary conditions are investigated: g≡0 and g(t)=tγ, γ>−1. Next, we extend our study to the time-fractional analogue of the above problem, namely, the time-fractional damped wave inequality ∂αu∂tα−∂2u∂x2+∂βu∂tβ≥xσ|u|p,t>0,x∈(0,L), where α∈(1,2), β∈(0,1), and ∂τ∂tτ is the time-Caputo fractional derivative of order τ, τ∈{α,β}. Our approach is based on the test function method. Namely, a judicious choice of test functions is made, taking in consideration the boundedness of the domain and the boundary conditions. Comparing with previous existing results in the literature, our results hold without assuming that the initial values are large with respect to a certain norm.

scholarly journals Attractor of Beam Equation with Structural Damping under Nonlinear Boundary Conditions

2015 ◽  
Vol 2015 ◽  
pp. 1-10 ◽  
Author(s):  
Danxia Wang ◽  
Jianwen Zhang ◽  
Yinzhu Wang ◽  
Sufang Zhang
Keyword(s):  
Boundary Conditions ◽  
Nonlinear Boundary ◽  
Global Solutions ◽  
Nonlinear Boundary Conditions ◽  
Beam Equation ◽  
Rotational Inertia ◽  
Absorbing Set ◽  
Viscous Effect ◽  
Structural Damping ◽  
Longtime Behavior

Simultaneously, considering the viscous effect of material, damping of medium, and rotational inertia, we study a kind of more general Kirchhoff-type extensible beam equationutt-uxxtt+uxxxx-σ(∫0l‍(ux)2dx)uxx-ϕ(∫0l‍(ux)2dx)uxxt=q(x), in [0,L]×R+with the structural damping and the rotational inertia term. Little attention is paid to the longtime behavior of the beam equation under nonlinear boundary conditions. In this paper, under nonlinear boundary conditions, we prove not only the existence and uniqueness of global solutions by prior estimates combined with some inequality skills, but also the existence of a global attractor by the existence of an absorbing set and asymptotic compactness of corresponding solution semigroup. In addition, the same results also can be proved under the other nonlinear boundary conditions.

DYNAMICS OF THERMAL LOSSES BY CONVECTION AND RADIATION OF THE SPHERICAL HEAT ACCUMULATOR OF SOLAR PLANTS

2020 ◽  
Vol 4 (41) ◽  
pp. 57-62
Author(s):  
SHAVKAT KLYCHEV ◽  
◽  
BAKHRAMOV SAGDULLA ◽  
VALERIY KHARCHENKO ◽  
VLADIMIR PANCHENKO ◽  
...  
Keyword(s):  
Boundary Conditions ◽  
Heat Loss ◽  
Initial Conditions ◽  
Water System ◽  
Heat Losses ◽  
Research Purpose ◽  
Heat Accumulator ◽  
Water Heaters ◽  
Intrinsic Radiation ◽  
Over Time

There are needed energy (heat) accumulators to increase the efficiency of solar installations, including solar collectors (water heaters, air heaters, dryers). One of the tasks of designing heat accumulators is to ensure its minimal heat loss. The article considers the problem of determining the distribution of temperatures and heat losses by convection and radiation of the heat insulation-accumulating body (water) system for a ball heat accumulator under symmetric boundary conditions. The problem is solved numerically according to the program developed on the basis of the proposed «gap method». (Research purpose) The research purpose is in determining heat losses by convection and radiation of a two-layer ball heat accumulator with symmetric boundary conditions. (Materials and methods) Authors used the Fourier heat equation for spherical bodies. The article presents the determined boundary and initial conditions for bodies and their surfaces. (Results and discussion) The thickness of the insulation and the volume of the heat accumulator affect the dynamics and values of heat loss. The effect of increasing the thickness of the thermal insulation decreases with increasing its thickness, starting with a certain volume of the heat accumulator or with R > 0.3 meters, the heat losses change almost linearly over time. The dynamics of heat loss decreases with increasing shelf life, but the losses remain large. (Conclusions) Authors have developed a method and program for numerical calculation of heat loss and temperature over time in a spherical two-layer heat accumulator with symmetric boundary conditions, taking into account both falling and intrinsic radiation. The proposed method allows to unify the boundary conditions between contacting bodies.

scholarly journals Vibration of rotating circular cylindrical shells with distributed springs

2021 ◽  
Vol 37 ◽  
pp. 346-358
Author(s):  
Fuchun Yang ◽  
Xiaofeng Jiang ◽  
Fuxin Du
Keyword(s):  
Boundary Conditions ◽  
Galerkin Method ◽  
Shell Theory ◽  
Rotation Speed ◽  
Cylindrical Shells ◽  
Free Vibrations ◽  
Governing Equations ◽  
Natural Response ◽  
Circular Cylindrical Shells ◽  
The Galerkin Method

Abstract Free vibrations of rotating cylindrical shells with distributed springs were studied. Based on the Flügge shell theory, the governing equations of rotating cylindrical shells with distributed springs were derived under typical boundary conditions. Multicomponent modal functions were used to satisfy the distributed springs around the circumference. The natural responses were analyzed using the Galerkin method. The effects of parameters, rotation speed, stiffness, and ratios of thickness/radius and length/radius, on natural response were also examined.

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