求解多维非特征柯西问题的修正核方法
A Modified Kernel Method for Solving a Non-Characteristic Cauchy Problem in Multiple Dimensions
DOI: 10.12677/PM.2023.137208, PDF,   
作者: 韩晶晶：西北师范大学数学与统计学院，甘肃 兰州
关键词: 非特征柯西问题；修正核方法；误差估计；Non-Characteristic Cauchy Problem； Modified Kernel Method； Error Estimation
摘要: 讨论了多维非特征柯西问题，这是一个严重不适定的问题，即解(如果存在的话)不连续地依赖于数据。在简单地分析了频率空间中柯西问题的不适定性之后，提出了一种修正核方法来解决这个问题，从而产生了一个适定的问题，并在正则化参数的先验选择规则下，给出并证明了正则化解和精确解之间的误差估计。
Abstract: We study a non-characteristic Cauchy problem in multiple dimensions. This is a severely ill-posed problem, i.e., the solution (if it exists) does not depend continuously on the data. After simply ana-lyzing the ill-posedness of the Cauchy problem in the frequency space, we propose to solve this problem by modifying the kernel, which generates a well-posed problem. Error estimates between the exact solution and the regularized solution are given. At last, we employ some numerical ex-amples to illustrate the behavior of the proposed methods.
文章引用：韩晶晶. 求解多维非特征柯西问题的修正核方法[J]. 理论数学, 2023, 13(7): 2024-2031. https://doi.org/10.12677/PM.2023.137208

参考文献

[1] Knabner, P. and Vessella, S. (1988) The Optimal Stability Estimate for Some Ill-Posed Cauchy Problems for a Parabolic Equation. Mathematical Methods in the Applied Sciences, 10, 575-583. [Google Scholar] [CrossRef] 
[2] Meyer, Y. (1992) Wavelets and Operators. Cambridge University Press, Cambridge.
[3] Elden, L. (1987) Approximations for a Cauchy Problem for the Heat Equation. Inverse Prob-lems, 3, 263-273. [Google Scholar] [CrossRef] 
[4] Zhi, Q. (2009) Regularization Methods for a Cauchy Problem for a Parabolic Equation in Multiple Dimensions. Journal of Inverse and Ill-Posed Problems, 17, 891-911. [Google Scholar] [CrossRef] 
[5] Knosowski, Y., Lieres, E. and Schneider, A. (1999) Regularization of a Non-Characteristic Cauchy Problem for a Parabolic Equation in Multiple Dimensions. Inverse Problems, 15, 731-743. [Google Scholar] [CrossRef] 
[6] Qian, Z. and Fu, C.L. (2007) Regularization Strategies for a Two-Dimensional Inverse Heat Conduction Problem. Inverse Problem, 23, 1053-1068. [Google Scholar] [CrossRef] 
[7] Qian, Z., Fu, C.L. and Xiong, X.T. (2007) A Modified Method for Determining the Surface Heat Flux of IHCP. Inverse Problems in Science and Engineering, 15, 249-265. [Google Scholar] [CrossRef] 
[8] Hao, D.N., Schneider, A. and Reinhardt, H.-J. (1995) Regular-ization of a Non-Characteristic Cauchy Problem for a Parabolic Equation. Inverse Problems, 11, 1247-1263. [Google Scholar] [CrossRef] 

为你推荐