Reconstruction of Heat Sources for Parabolic Equations With Wentzell Boundary Conditions

YI Haihong; YANG Liu; TIAN Yu

doi:10.21656/1000-0887.450029

Volume 46 Issue 4

Apr. 2025

Turn off MathJax

Article Contents

Article Navigation > Applied Mathematics and Mechanics > 2025 > 46(4): 505-518

YI Haihong, YANG Liu, TIAN Yu. Reconstruction of Heat Sources for Parabolic Equations With Wentzell Boundary Conditions[J]. Applied Mathematics and Mechanics, 2025, 46(4): 505-518. doi: 10.21656/1000-0887.450029

Citation:

PDF( 433 KB)

Reconstruction of Heat Sources for Parabolic Equations With Wentzell Boundary Conditions

doi: 10.21656/1000-0887.450029

School of Mathematics and Physics, Lanzhou Jiaotong University, Lanzhou 730070, P.R.China

Funds:

11961042)

The National Science Foundation of China(61663018

Received Date: 2024-02-02
Rev Recd Date: 2024-11-17

Available Online: 2025-04-30

Abstract

Abstract

The inverse problem of reconstructing spatially related source terms in parabolic heat conduction equations was studied under the Wentzell boundary conditions and with the terminal temperature measurements. This study has important applications in determining the source terms in heat conduction engineering problems, and the difficulty lies in the handling of the Wentzell boundary conditions. Based on the divergence theorem, the boundary conditions were combined with parabolic equations. The extremum principle was proved differently under various boundary conditions. Due to the ill-posedness of the original problem, based on the framework of the optimal control theory, the original problem was optimized, and the existence and necessary conditions for the regularization solution were established. Furthermore, under the validness of the extremum principle, the uniqueness and stability of the regularization solution were proved.
- inverse source problem,
- Wentzell boundary condition,
- optimal control,
- divergence theorem

FullText(HTML)

References(17)

References

VOGT H, VOIGT J. Wentzell boundary conditions in the context of Dirichlet forms[J].Advances in Differential Equations,2003,8(7): 821-842.

[2]FAVINI A, GOLDSTEIN J A, ROMANELLI S, et al. The heat equation with nonlinear general Wentzell boundary condition[J].Advances in Differential Equations,2006,11(5): 481-510.

[3]GUIDETTI D. Parabolic problems with general Wentzell boundary conditions and diffusion on the boundary[J].Communications on Pure and Applied Analysis,2016,15(4): 1401-1417.

[4]VZQUEZ J L, VITILLARO E. Heat equation with dynamical boundary conditions of reactive-diffusive type[J].Journal of Differential Equations,2011,250(4): 2143-2161.

[5]ISMAILOV M I. Inverse source problem for heat equation with nonlocal wentzell boundary condition[J].Results in Mathematics,2018,73(2): 68.

[6]ISMAILOV M I, KANCA F, LESNIC D. Determination of a time-dependent heat source under nonlocal boundary and integral overdetermination conditions[J].Applied Mathematics and Computation,2011,218(8): 4138-4146.

[7]KERIMOV N B, ISMAILOV M I. Direct and inverse problems for the heat equation with a dynamic-type boundary condition[J].IMA Journal of Applied Mathematics,2015,80(5): 1519-1533.

[8]YIMAMU Y, DENG Z C, YANG L. An inverse volatility problem in a degenerate parabolic equation in a bounded domain[J].AIMS Mathematics,2022,7(10): 19237-19266.

[9]JOHANSSON T, LESNIC D. Determination of a spacewise dependent heat source[J].Journal of Computational and Applied Mathematics,2007,209(1): 66-80.

[10]YANG L, YU J N, LUO G W, et al. Reconstruction of a space and time dependent heat source from finite measurement data[J].International Journal of Heat and Mass Transfer,2012,55(23/24): 6573-6581.

[11]耿肖肖, 程浩. 一类球型区域上变系数反向热传导问题[J]. 应用数学和力学, 2021,42(7): 723-734. (GENG Xiaoxiao, CHENG Hao. The backward heat conduction problem with variable coefficients in a spherical domain[J].Applied Mathematics and Mechanics,2021,42(7): 723-734. (in Chinese))

[12]陈琛, 冯晓莉, 陈汉章. 一类随机微分方程的随机源反演方法和性质[J]. 应用数学和力学, 2023,44(7): 847-856. (CHEN Chen, FENG Xiaoli, CHEN Hanzhang. The random source inverse method and properties for a class of stochastic differential equations[J].Applied Mathematics and Mechanics,2023,44(7): 847-856. (in Chinese))

[13]刘翻丽, 解金鑫, 杨涛. 变系数导热方程的Robin系数反演问题[J]. 应用数学学报, 2021,44(4): 574-588. (LIU Fanli, XIE Jinxin, YANG Tao. Robin coefficient inversion problem of variable coefficient heat conduction equation[J].Acta Mathematicae Applicatae Sinica,2021,44(4): 574-588. (in Chinese))

[14]尹丽君, 温鑫亮. 具有Wentzel型边界条件的反源问题解的唯一性[J]. 兰州交通大学学报, 2020,39(4): 132-137. (YIN Lijun, WEN Xinliang. Uniqueness of the solution to the inverse source problem with a Wentzel boundary condition[J].Journal of Lanzhou Jiaotong University,2020,39(4): 132-137. (in Chinese))

[15]FAVINI A, GOLDSTEIN G R, GOLDSTEIN J A, et al. The heat equation with generalized Wentzell boundary condition[J].Journal of Evolution Equations,2002,2(1): 1-19.

[16]姜礼尚, 陈亚浙, 刘西垣, 等. 数学物理方程讲义[M]. 3版. 北京: 高等教育出版社, 2007. (JIANG Lishang, CHEN Yazhe, LIU Xiyuan, et al.Lectures on Mathematical and Physical Equations[M]. 3rd ed. Beijing: Higher Education Press, 2007. (in Chinese))

[17]伍卓群, 尹景学, 王春朋. 椭圆与抛物型方程引论[M]. 北京: 科学出版社, 2003. (WU Zhuoqun, YIN Jingxue, WANG Chunpeng.Introduction to Elliptic and Parabolic Equations[M]. Beijing: Science Press, 2003. (in Chinese))

Relative Articles

Supplements(0)

Cited By

Proportional views