Solving an Inverse Heat Conduction Problem by a “Method of Lines”

1997 ◽  
Vol 119 (3) ◽  
pp. 406-412 ◽  
Author(s):  
L. Elde´n
Keyword(s):  
Heat Conduction ◽  
Heat Equation ◽  
Heat Conduction Problem ◽  
Time Derivative ◽  
Method Of Lines ◽  
Inverse Heat Conduction Problem ◽  
Heat Conducting ◽  
Inverse Heat Conduction ◽  
Use Of Time ◽  
Ill Posed

We consider a Cauchy problem for the heat equation in the quarter plane, where data are given at x = 1 and a solution is sought in the interval 0 < x < 1. This inverse heat conduction problem is a model of a situation where one wants to determine the surface temperature given measurements inside a heat-conducting body. The problem is ill-posed in the sense that the solution (if it exists) does not depend continuously on the data. In an earlier paper we showed that replacement of the time derivative by a difference stabilizes the problem. In this paper we investigate the use of time differencing combined with a “method of lines” for solving numerically the initial value problem in the space variable. We discuss the numerical stability of this procedure, and we show that, in most cases, a usual explicit (e.g., Runge-Kutta) method can be used efficiently and stably. Numerical examples are given. The approach of this paper is proposed as an alternative way of implementing space-marching methods for the sideways heat equation.

METHOD OF FUNDAMENTAL SOLUTION FOR AN INVERSE HEAT CONDUCTION PROBLEM WITH VARIABLE COEFFICIENTS

2013 ◽  
Vol 10 (02) ◽  
pp. 1341009 ◽  
Author(s):  
MING LI ◽  
XIANG-TUAN XIONG ◽  
YAN LI
Keyword(s):  
Heat Conduction ◽  
Heat Conduction Problem ◽  
Fundamental Solutions ◽  
Variable Coefficients ◽  
Inverse Heat Conduction Problem ◽  
Method Of Fundamental Solutions ◽  
Inverse Heat Conduction ◽  
Modified Method ◽  
On Line ◽  
Ill Posed

In this paper, we consider an inverse heat conduction problem with variable coefficient a(t). In many practical situations such as an on-line testing, we cannot know the initial condition for example because we have to estimate the problem for the heat process which was already started. Based on the method of fundamental solutions, we give a numerical scheme for solving the reconstruction problem. Since the governing equation contains variable coefficients, modified method of fundamental solutions was used to solve this kind of ill-posed problems. Some numerical examples are given for verifying the efficiency and accuracy of the presented method.

Numerical solution of two-dimensional radially symmetric inverse heat conduction problem

2015 ◽  
Vol 23 (2) ◽  
Author(s):  
Zhi Qian ◽  
Benny Y. C. Hon ◽  
Xiang Tuan Xiong
Keyword(s):  
Heat Conduction ◽  
Numerical Solution ◽  
Heat Conduction Problem ◽  
Inverse Heat Conduction Problem ◽  
Stability And Convergence ◽  
Two Dimensional ◽  
Inverse Heat Conduction ◽  
Convergence Error ◽  
Regularization Techniques ◽  
Ill Posed

AbstractWe investigate a two-dimensional radially symmetric inverse heat conduction problem, which is ill-posed in the sense that the solution does not depend continuously on input data. By generalizing the idea of kernel approximation, we devise a modified kernel in the frequency domain to reconstruct a numerical solution for the inverse heat conduction problem from the given noisy data. For the stability of the numerical approximation, we develop seven regularization techniques with some stability and convergence error estimates to reconstruct the unknown solution. Numerical experiments illustrate that the proposed numerical algorithm with regularization techniques provides a feasible and effective approximation to the solution of the inverse and ill-posed problem.

scholarly journals Optimal Tikhonov approximation for a sideways parabolic equation

2005 ◽  
Vol 2005 (8) ◽  
pp. 1221-1237 ◽  
Author(s):  
Chu-Li Fu ◽  
Hong-Fang Li ◽  
Xiang-Tuan Xiong ◽  
Peng Fu
Keyword(s):  
Heat Conduction ◽  
Parabolic Equation ◽  
Tikhonov Regularization ◽  
Regularization Method ◽  
Heat Conduction Problem ◽  
Inverse Heat Conduction Problem ◽  
Tikhonov Regularization Method ◽  
Inverse Heat Conduction ◽  
Convection Term ◽  
Ill Posed

We consider an inverse heat conduction problem with convection term which appears in some applied subjects. This problem is ill posed in the sense that the solution (if it exists) does not depend continuously on the data. A generalized Tikhonov regularization method for this problem is given, which realizes the best possible accuracy.

A NUMERICAL AND EXPERIMENTAL STUDY OF A ONE-DIMENSIONAL SPACE BOUNDARY INVERSE HEAT CONDUCTION PROBLEM

1999 ◽  
Vol 23 (2) ◽  
pp. 267-274
Author(s):  
T-F. Chen ◽  
S. Lin ◽  
J.C.-Y. Wang
Keyword(s):  
Heat Conduction ◽  
Heat Conduction Problem ◽  
Dimensional Space ◽  
Large Error ◽  
Inverse Heat Conduction Problem ◽  
Small Error ◽  
Inverse Heat Conduction ◽  
Ill Posed ◽  
Inverse Heat Conduction Problems ◽  
Space Boundary

Inverse Heat Conduction Problems (IHCPs) are not the same as direct heat conduction problems which are “well-posed”. The difficulty of the “ill-posed” IHCP is that a small error perturbation in the data will lead to a large error in the reconstructed solution. An inverse procedure for solving an IHCP should have the ability to handle the required information obtained from measurements containing noise. In the present work, a stable, accurate and reliable inversion solver is used with interior temperature measurements to predict surface temperatures at the both sides of a brick Numerical results in comparison with experimental data are presented.

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