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 引用本文: 柏恩鹏, 熊向团. 一种新的正则化方法求解热传导方程的侧边值问题[J]. 应用数学和力学, 2021, 42(5): 541-550.
BAI Enpeng, XIONG Xiangtuan. A New Regularization Method for Solving Sideways Heat Equations[J]. Applied Mathematics and Mechanics, 2021, 42(5): 541-550. doi: 10.21656/1000-0887.410290
 Citation: BAI Enpeng, XIONG Xiangtuan. A New Regularization Method for Solving Sideways Heat Equations[J]. Applied Mathematics and Mechanics, 2021, 42(5): 541-550.

## 一种新的正则化方法求解热传导方程的侧边值问题

##### doi: 10.21656/1000-0887.410290

###### 作者简介:柏恩鹏(1995—)，男，硕士生(E-mail: baiepnwnu@163.com)；熊向团(1977—)，男，教授，博士生导师(通讯作者. E-mail: xiongxt@fudan.edu.com).
• 中图分类号: O241.1

## A New Regularization Method for Solving Sideways Heat Equations

Funds: The National Natural Science Foundation of China（11661072）
• 摘要: 考虑了四分之一平面内的热传导方程的侧边值问题，这类问题是严重不适定的.采用传统拟逆方法得到该问题的一个近似解，但发现它并不是一个正则化解.有趣的是，对解的分母项加以修正便可以得到侧边值问题的一个正则化解，进而提出了一种新的正则化方法，并分别给出先验和后验两种正则化参数选取规则下的Hlder型误差估计.数值实验验证了所提方法的可行性和有效性.
•  [1] XIONG X T, FU C L, LI H F. Central difference method of a non-standard inverse heat conduction problem for determining surface heat flux from interior observations[J]. Applied Mathematics and Computation,2005,173(2): 1265-1287. [2] TANTENHAHN U. Optimal stable approximations for the sideways heat equation[J]. Journal of Inverse and Ill-Posed Problems,2009,5(3): 287-307. [3] SEIDMAN T, ELDN L. An ‘optimal filtering’ method for the sideways heat equation[J]. Inverse Problems,1990,6(4): 681-696. [4] KU C Y, LIU C Y, XIAO J E, et al. A spacetime collocation Trefftz method for solving the inverse heat conduction problem[J]. Advances in Mechanical Engineering,2019,11(7): 1-11. [5] WANG J R. Uniform convergence of wavelet solution to the sideways heat equation[J]. Acta Mathematica Sinica(English Series),2010,26(10): 1981-1992. [6] 周焕林, 严俊, 余波, 等. 识别含热源瞬态热传导问题的热扩散系数[J]. 应用数学和力学, 2018,39(2): 160-169.(ZHOU Huanlin, YAN Jun, YU Bo, et al. Identify the thermal diffusivity of transient heat conduction problems with heat sources[J]. Applied Mathematics and Mechanics,2018,39(2): 160-169.(in Chinese)) [7] LIU J C, WEI T. A quasi-reversibility regularization method for an inverse heat conduction problem without initial data[J]. Applied Mathematics & Computation,2013,219(23): 10806-10821. [8] NGUYEN H T, LUU V C H. Two new regularization methods for solving sideways heat equation[J]. Journal of Inequalities and Applications,2015,2015: 65. DOI: 10.1186/s13660-015-0564-0. [9] ELDN L, BERNTSSON F, REGINSKA T. Wavelet and Fourier methods for solving the sideways heat equation[J]. SIAM Journal on Scientific Computing,2000,21(6): 2187-2205. [10] XIONG X T, FU C L, LI H F. Fourier regularization method of a sideways heat equation for determining surface heat flux[J]. Journal of Mathematical Analysis and Applications,2006,317(1): 331-348. [11] QIAN Z, FU C L, XIONG X T. A modified method for determining surface heat flux of IHCP[J]. Inverse Problems in Science,2007,15(3): 249-265. [12] ZHAO Z Y, MENG Z H. A modified Tikhonov regularization method for a backward heat equation[J]. Inverse Problems in Science and Engineering,2011,19(8): 1175-1182. [13] LIU J J, YAMAMOTO M. A backward problem for the time-fractional diffusion equation[J]. Applicable Analysis,2010,89(11): 1769-1788. [14] ELDN L. Approximations for a Cauchy problem for the heat equation[J]. Inverse Problems,1987,3(2): 263-273. [15] XIONG X T, XUE X M. Fractional Tikhonov method for an inverse time-fractional diffusion problem in 2-dimensional space[J]. Bulletin of the Malaysian Mathematical Sciences Society,2020,43(3): 25-38. [16] TAUTENHAHN U, SCHRTER T. On optimal regularization methods for the backward heat equation[J]. Journal for Analysis and Its Applications,1996,15(2): 475-493. [17] FENG X L, FU C L, CHENG H. A regularization method for solving the Cauchy problem for the Helmholtz equation[J]. Applied Mathematical Modelling,2011,〖STHZ〗 35(7): 3301-3315. [18] 薛雪敏, 熊向团. 时间分数阶反扩散问题的一种新的分数次Tikhonov方法[J]. 高校应用数学学报, 2018,33(4): 441-452.(XUE Xuemin, XIONG Xiangtuan. A new fractional Tikhonov method for a time-fractional inverse diffusion problem[J]. Applied Mathematics: a Journal of Chinese Universities,2018,33(4): 441-452.(in Chinese))
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##### 出版历程
• 收稿日期:  2020-09-24
• 修回日期:  2020-12-12
• 刊出日期:  2021-05-01

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