Two regularization methods for identification of the heat source depending only on spatial variable for the heat equation

2009 ◽  
Vol 17 (8) ◽  
Author(s):  
Fan Yang ◽  
Chu-Li Fu
Keyword(s):  
Heat Source ◽  
Heat Equation ◽  
Spatial Variable ◽  
Regularization Methods

scholarly journals On the approximate numerical solutions of fractional heat equation with heat source and heat loss

2021 ◽  
pp. 321-321
Author(s):  
Hami Gundogdu ◽  
Ömer Gozukizil
Keyword(s):  
Heat Source ◽  
Heat Equation ◽  
Heat Loss ◽  
Numerical Solutions ◽  
Adomian Decomposition Method ◽  
Approximate Solutions ◽  
Absolute Error ◽  
Maximum Absolute Error ◽  
Fractional Heat Equation ◽  
Adomian Decomposition

In this paper, we are interested in obtaining an approximate numerical solution of the fractional heat equation where the fractional derivative is in Caputo sense. We also consider the heat equation with a heat source and heat loss. The fractional Laplace-Adomian decomposition method is applied to gain the approximate numerical solutions of these equations. We give the graphical representations of the solutions depending on the order of fractional derivatives. Maximum absolute error between the exact solutions and approximate solutions depending on the fractional-order are given. For the last thing, we draw a comparison between our results and found ones in the literature.

scholarly journals Heat diffusion in a channel under white noise modeling of turbulence

2021 ◽  
Vol 4 (4) ◽  
pp. 1-21
Author(s):  
Franco Flandoli ◽  
◽  
Eliseo Luongo
Keyword(s):  
Velocity Field ◽  
Heat Source ◽  
Heat Equation ◽  
White Noise ◽  
Diffusion Coefficients ◽  
Weak Sense ◽  
Passive Scalar ◽  
Heat Diffusion ◽  
Turbulent Fluid ◽  
Deterministic Solution

<abstract><p>A passive scalar equation for the heat diffusion and transport in an infinite channel is studied. The velocity field is white noise in time, modelling phenomenologically a turbulent fluid. Under the driving effect of a heat source, the phenomenon of eddy dissipation is investigated: the solution is close, in a weak sense, to the stationary deterministic solution of the heat equation with augmented diffusion coefficients.</p></abstract>

scholarly journals Сorrection сalculation of the heat source power during plasma-powder surfacing of the necks of stepped shafts

2019 ◽  
Vol 298 ◽  
pp. 00123
Author(s):  
G.R. Latypova ◽  
N.N. Karpenko ◽  
R.A. Latypov
Keyword(s):  
Mathematical Model ◽  
Heat Source ◽  
Heat Equation ◽  
Penetration Depth ◽  
Calculation Method ◽  
Temperature Fields ◽  
Lambert Function ◽  
Practical Application ◽  
Differential Heat ◽  
Source Power

A mathematical model is proposed for calculating temperature fields during arc surfacing of limited cylinders, eliminating the need for dimensionless parameters, when solving the differential heat equation, which facilitates the practical application of the developed calculation method. It is proposed that the penetration depth be calculated using the Lambert function. The obtained dependences make it possible to quantify the required correction of the heat source power in the process of surfacing the necks of stepped shafts.

scholarly journals Some similarity temperature profiles for the microwave heating of a half-space

1992 ◽  
Vol 33 (3) ◽  
pp. 290-320 ◽  
Author(s):  
James M. Hill ◽  
Adrian H. Pincombe
Keyword(s):  
Magnetic Properties ◽  
Heat Source ◽  
Heat Equation ◽  
Microwave Heating ◽  
Exponential Decay ◽  
Transient Temperature ◽  
Exponential Dependence ◽  
Temperature Profiles ◽  
Electrical And Magnetic Properties ◽  
Increasing Temperature

AbstractThere is presently considerable interest in the utilisation of microwave heating in areas such as cooking, sterilising, melting, smelting, sintering and drying. In general, such problems involve Maxwell's equations coupled with the heat equation, for which all thermal, electrical and magnetic properties of the material are nonlinear. The heat source arising from microwaves is proportional to the square of the modulus of the electric field intensity, and is known to increase with increasing temperature. In an attempt to find a simple model of microwave heating, we examine here simple transient temperature profiles corresponding to a heat source with spatial exponential decay but increasing with temperature, for which we assume either a power-law dependence or an exponential dependence. The spatial exponential decay is known to apply exactly when the electrical and magnetic properties of the material are assumed constant. A number of transient temperature profiles for this model are examined which arise from the invariance of the governing heat equation under simple one-parameter transformation groups. Some closed analytical expressions are obtained, but in general the resulting ordinary differential equations need to be solved numerically, and extensive numerical results are presented. For both models, these results indicate the appearance of moving fronts.

scholarly journals Boundary Stabilization of Heat Equation with Multi-Point Heat Source

Mathematics ◽  
2021 ◽  
Vol 9 (8) ◽  
pp. 834
Author(s):  
Qing-Qing Hu ◽  
Feng-Fei Jin ◽  
Bao-Qiang Yan
Keyword(s):  
Heat Source ◽  
Heat Equation ◽  
Exponential Stability ◽  
Output Feedback ◽  
Closed Loop ◽  
Feedback Controller ◽  
Loop System ◽  
Boundary Stabilization ◽  
Closed Loop System ◽  
Point Heat Source

In this paper, we consider boundary stabilization problem of heat equation with multi-point heat source. Firstly, a state feedback controller is designed mainly by backstepping approach. Under the designed state controller, the exponential stability of closed-loop system is guaranteed. Then, an observer-based output feedback controller is proposed. We prove the exponential stability of resulting closed-loop system using operator semigroup theory. Finally, the designed state and output feedback controllers are effective via some numerical simulations.

scholarly journals Simultaneous Identification of Thermal Conductivity and Heat Source in the Heat Equation

2021 ◽  
pp. 1968-1978
Author(s):  
M. J. Huntul ◽  
M.S. Hussein
Keyword(s):  
Thermal Conductivity ◽  
Inverse Problem ◽  
Heat Source ◽  
Heat Equation ◽  
Mathematical Formulation ◽  
Approximate Solutions ◽  
Ill Posed ◽  
The One ◽  
Nicolson Scheme ◽  
Noisy Measurements

This paper presents a numerical solution to the inverse problem consisting of recovering time-dependent thermal conductivity and  heat source coefficients  in the one-dimensional  parabolic heat equation.   This  mathematical  formulation  ensures that the inverse problem  has a unique  solution.   However, the problem  is still  ill-posed since small errors  in the input data lead to a drastic  amount  of errors in the output coefficients.  The  finite  difference method  with  the Crank-Nicolson  scheme is adopted  as a direct  solver of the problem in a fixed domain.   The inverse problem is solved subjected to both exact and noisy measurements  by using the MATLAB  optimization  toolbox  routine  lsqnonlin , which is also applied to minimize the nonlinear  Tikhonov  regularization functional.  The thermal conductivity and heat source coefficients are reconstructed using heat flux measurements. The root mean squares error is used to assess the accuracy of the approximate solutions of the problem. A couple of  numerical  examples are presented to verify the accuracy and stability of the solutions.

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