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A New Method for Fredholm Integral Equations of 1D Backward Heat Conduction Problems

Chein-Shan Liu1

Department of Civil Engineering, National Taiwan University, Taipei, Taiwan. E-mail:

Computer Modeling in Engineering & Sciences 2009, 47(1), 1-22.


In this paper an analytical method for approximating the solution of backward heat conduction problem is presented. The Fourier series expansion technique is used to formulate a first-kind Fredholm integral equation for the temperature field u(x,t) at any time t < T, when the data are specified at a final time T. Then we consider a direct regularization, instead of the Tikhonov regularization, by adding the term αu(x,t) to obtain a second-kind Fredholm integral equation. The termwise separable property of kernel function allows us by transforming it to a two-point boundary value problem, and thus a closed-form solution is derived. The uniform convergence and error estimate of the regularized solution uα(x,t) are proved and a strategy to select the regularization parameter is provided. When numerical examples were tested, we find that the new method can retrieve the initial data very excellently, even the final data are seriously noised.


Cite This Article

Liu, C. (2009). A New Method for Fredholm Integral Equations of 1D Backward Heat Conduction Problems. CMES-Computer Modeling in Engineering & Sciences, 47(1), 1–22.

This work is licensed under a Creative Commons Attribution 4.0 International License , which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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