一类非线性反向热传导问题的Fourier正则化方法

数学物理学报 ›› 2017, Vol. 37 ›› Issue (1): 62-71.

一类非线性反向热传导问题的Fourier正则化方法

杨帆¹, 傅初黎², 李晓晓¹, 任玉鹏¹

1. 兰州理工大学理学院兰州 730050;
2. 兰州大学数学与统计学院兰州 730000

收稿日期:2016-05-19 修回日期:2016-10-17 出版日期:2017-02-26 发布日期:2017-02-26
通讯作者: 杨帆 E-mail:yfggd114@163.com
基金资助:
国家自然科学基金（11561045）和兰州理工大学博士基金

Fourier Truncation Regularization for a Class of Nonlinear Backward Heat Equation

Yang Fan¹, Fu Chuli², Li Xiaoxiao¹, Ren Yupeng¹

1. School of Science, Lanzhou University of Technology, Lanzhou 730050;
2. School of Mathematics and Statistics, Lanzhou University, Lanzhou 730000

Received:2016-05-19 Revised:2016-10-17 Online:2017-02-26 Published:2017-02-26
Supported by:
Supported by the NSFC (11561045) and the Doctor Fund of Lan Zhou University of Technology

摘要/Abstract

摘要：

考虑了非线性抛物方程反向热传导问题，这类问题是不适定的，即问题的解不连续依赖于测量数据.利用Fourier截断正则化方法恢复其不适定性，得到问题的一个正则近似解，并且给出正则解和精确解之间具有Hölder型的误差估计.

关键词: 反向热传导方程, 非线性不适定问题, Fourier截断方法, 误差估计

Abstract:

In this paper, we consider the nonlinear backward heat equation, which is severely ill-posed in the sense that the solution does not depend continuously on the data. A Fourier regularization method is proposed to solve this problem. For the regularization solution, the Hölder type stability estimate between the regularization solution and the exact solution is given.

Key words: Backward heat equation, Nonlinear ill-posed problem, Fourier truncation method, Error estimate

中图分类号:

O175.25

杨帆, 傅初黎, 李晓晓, 任玉鹏. 一类非线性反向热传导问题的Fourier正则化方法[J]. 数学物理学报, 2017, 37(1): 62-71.

Yang Fan, Fu Chuli, Li Xiaoxiao, Ren Yupeng. Fourier Truncation Regularization for a Class of Nonlinear Backward Heat Equation[J]. Acta mathematica scientia,Series A, 2017, 37(1): 62-71.

参考文献

[1] Carasso A. Error bounds in the final value problem for the heat equation. SIAM J Math Anal, 1976, 7:195-199
[2] Lavrentév M M, Romanov V G, Shishat'skii S P. Ill-posed Problems of Mathematical Physics and Analysis. Providence RI:American Mathematical Society, 1986
[3] Isakov V. Inverse Problems for Partial Differential Equations. New York:Springer-Verlag, 1998:93-98
[4] Lattés R, Lions J L. Methode de Quasi-Réversibilité et Applications. Paris:Dunod, 1967
[5] Showalter R E. The final value problem for evolution equations. J Math Anal Appl, 1974, 47:563-572
[6] Ames K A, Gordon W C, Epperson J F, Oppenhermer S F. A comparison of regularizations for an ill-posed problem. Math Comput, 1998, 67:1451-1471
[7] Miller K. Stabilized quasi-reversibilite and other nearly-best-possible methods for non-well-posed problems//Symposium on Non-Well-Posed Problems and Logarithmic Convexity. Berlin, Heidelberg:Springer, 1973:161-176
[8] Tautenhahn U, Schröter T. On optimal regularization methods for the backward heat equation. Z Anal Anwend, 1996, 15:475-493
[9] Seidman T I. Optimal filtering for the backward heat equation. SIAM J Numer Anal, 1996, 33:162-170
[10] Mera N S, Elliott L, Ingham D B, Lesnic D. An iterative boundary element method for solving the onedimensional backward heat conduction problem. Internat J Heat Mass Transfer, 2001, 44:1937-1946
[11] Jourhmane M, Mera N S. An iterative algorithm for the backward heat conduction problem based on variable relaxation factors. Inverse Probl Sci Eng, 2002, 10:293-308
[12] Liu C S. Group preserving scheme for backward heat conduction problems. Internat J Heat Mass Transfer, 2004, 47:2567-2576
[13] Kirkup S M, Wadsworth M. Solution of inverse diffusion problems by operator-splitting methods. Appl Math Model, 2002, 24:1003-1018
[14] Fu C L, Xiong X T, Qian Z. Fourier regularization for a backward heat equation. J Math Anal Appl, 2007, 33:472-480
[15] Qian Z, Fu C L, Shi R. A modified method for a backward heat conduction problem. Appl Math Comput, 2007, 185:564-573
[16] Trong D D, Quan P H, Khanh T V, Tuan N H. A nonlinear case of the 1-D backward heat problem:Regularization and error estimate. Z Anal Anwend, 2007, 26:231-245
[17] Eldén L, Berntsson F, Regińska T. Wavelet and Fourier method for solving the sideways heat equation. SIAM J Sci Comput, 2000, 21:2187-2205
[18] Fu C L, Xiong X T, Fu P. Fourier regularization method for solving the surface heat flux from interior observations. Math Comput Model, 2005, 42:489-498
[19] Qian Z, Fu C L, Xiong X T, Wei T. Fourier truncation method for high order numerical derivatives. Appl Math Comput, 2006, 181:940-948
[20] Fu C L, Feng X L, Qian Z. The Fourier regularization for solving the Cauchy problem for the Helmholtz equation. Appl Numer Math, 2009, 59:2625-2640
[21] Dou F F, Fu C L, Yang F L. Optimal error bound and Fourier regularization for identifying an unknown source in the heat equation. J Comput Appl Mat, 2009, 230:728-737
[22] Yang F, Fu C L. Two regularization methods to identify time-dependent heat source through an internal measurement of temperature. Math Comput Model, 2011, 53:793-804
[23] Quan P H, Trong D D. A nonlinearly backward heat problem:Regularization and error estimate. Appl Anal, 2006, 85:641-657
[24] Evans L C. Partial Diferential Equation. Providence, RI:American Mathematical Society, 1998