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An LGDAE Method to Solve Nonlinear Cauchy Problem Without Initial Temperature

Chein-Shan Liu 1
Department of Civil Engineering, National Taiwan University, Taipei 106-17, Taiwan. Email: liucs@ntu.edu.tw

Computer Modeling in Engineering & Sciences 2014, 99(5), 371-391. https://doi.org/10.3970/cmes.2014.099.371

Abstract

We recover an unknown initial temperature for a nonlinear heat conduction equation ut(x,t) = uxx(x,t) + H(x,t,u,ux), under the Cauchy boundary conditions specified on the left-boundary. The method in the present paper transforms the Cauchy problem into an inverse heat source problem to find F(x) in Tt(x,t) = Txx(x,t) + H + F(x). By using the GL(N,R) Lie-group differential algebraic equations (LGDAE) algorithm to integrate the numerical method of lines discretized equations from sideways heat equation, we can fast recover the initial temperature and two boundary conditions on the right-boundary. The accuracy and efficiency are confirmed by comparing the exact solutions with the recovered results, where a large noisy disturbance is imposed on the Cauchy data.

Keywords

Cauchy problem, Missing initial condition, Lie-group differential algebraic equations (LGDAE) method, Sideways heat equation.

Cite This Article

Liu, C. (2014). An LGDAE Method to Solve Nonlinear Cauchy Problem Without Initial Temperature. CMES-Computer Modeling in Engineering & Sciences, 99(5), 371–391.



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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