A New Regularization Method for Solving Sideways Heat Equations

BAI Enpeng; XIONG Xiangtuan

doi:10.21656/1000-0887.410290

Volume 42 Issue 5

May 2021

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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. 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. doi: 10.21656/1000-0887.410290

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A New Regularization Method for Solving Sideways Heat Equations

doi: 10.21656/1000-0887.410290

College of Mathematics and Statistics, Northwest Normal University,Lanzhou 730070, P.R.China

Funds: The National Natural Science Foundation of China（11661072）

Received Date: 2020-09-24
Rev Recd Date: 2020-12-12
Publish Date: 2021-05-01

Abstract

Abstract

The seriously ill-posed sideways heat equations were considered in the quarter plane. The classical quasi-reversibility method was applied to acquire an approximate but non-regularized solution to the problem. Interestingly, a regularization solution to the sideways heat equation was obtained through modification of the denominator of the solution. Then, a new regularization method was proposed, and the Hlder-type error estimates under a priori and a posteriori parameter choice rules were proved, respectively. Numerical experiments show the feasibility and effectiveness of the proposed method.
- sideways heat equation,
- ill-posed problem,
- regularization method,
- regularization parameter,
- error estimation

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References(18)

References

[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))