一类半线性椭圆方程柯西问题的正则化方法

摘要/Abstract

摘要：

构造并利用一种广义分数Tikhonov正则化方法研究一类半线性椭圆方程柯西问题.基于所构造的正则化解满足一个非线性积分方程，首先证明正则化解的存在唯一性和稳定性; 继而在对精确解的先验假设下给出并证明正则化方法的收敛性; 最后设计一种迭代算法计算正则化解，并通过相应的计算结果验证了所提方法的稳定可行性.

关键词: 柯西问题, 半线性椭圆方程, 正则化方法, 收敛性估计, 数值模拟

Abstract:

In this paper, we construct and use a generalized fractional Tikhonov regularization method to study a Cauchy problem for semi-linear elliptic equation. Based on one nonlinear integral equation that the constructed regularization solution satisfies, we firstly prove the existence, uniqueness and stability for it. And then we give and prove the convergence for regularized method under an a-priori assumption on the exact solution. Ultimately, the regularized solution is calculated by designing an iteration algorithm, and we verify the stability and feasibility for proposed method by the corresponding computational results.

Key words: Cauchy problem, Semilinear elliptic equation, Regularization method, Convergence estimate, Numerical simulation

中图分类号:

O175.25

张宏武. 一类半线性椭圆方程柯西问题的正则化方法[J]. 数学物理学报, 2022, 42(1): 45-57.

Hongwu Zhang. One Regularization Method for a Cauchy Problem of Semilinear Elliptic Equation[J]. Acta mathematica scientia,Series A, 2022, 42(1): 45-57.

参考文献

1	Beskos D E . Boundary element method in dynamic analysis: Part Ⅱ (1986-1996). ASME Appl Mech Rev, 1997, 50: 149- 197
2	Chen J T , Wong F C . Dual formulation of multiple reciprocity method for the acoustic mode of a cavity with a thin partition. J Sound Vib, 1998, 217: 75- 95
3	Fokas A S , Pelloni B . The dirichlet-to-neumann map for the elliptic Sine-Gordon equation. Nonlinearity, 2012, 25 (4): 1011- 1031
4	Gutshabash E S , Lipovskii V D . Boundary value problem for the two-dimensional elliptic Sine-Gordon equation and its applications to the theory of the stationary josephson effect. J Math Sciences, 1994, 68: 197- 201
5	Belgacem F B . Why is the Cauchy problem severely ill-posed?. Inverse Problems, 2007, 23 (2): 823
6	Feng X L , Ning W T , Qian Z . A quasi-boundary-value method for a Cauchy problem of an elliptic equation in multiple dimensions. Inverse Problems in Science and Engineering, 2014, 22 (7): 1045- 1061
7	Hào D N , Duc D V , Lesnic D . A non-local boundary value problem method for the Cauchy problem for elliptic equations. Inverse Problems, 2009, 25: 055002
8	Lavrentiev M M, Romanov V G, Shishatski S P. Ill-Posed Problems of Mathematical Physics and Analysis. Providence, RI: American Mathematical Society, 1986
9	Tautenhahn U . Optimal stable solution of Cauchy problems of elliptic equations. Journal for Analysis and its Applications, 1996, 15 (4): 961- 984
10	Tuan N H , Trong D D , Quan P H . A note on a Cauchy problem for the Laplace equation: Regularization and error estimates. Applied Mathematics and Computation, 2010, 217: 2913- 2922
11	Zhang H W , Zhang X J . Generalized-fractional Tikhonov-type method for the Cauchy problem of elliptic equation. Mathematics, 2020, 8 (1): 48
12	Tuan N H , Thang L D , Khoa V A . A modified integral equation method of the nonlinear elliptic equation with globally and locally lipschitz source. Applied Mathematics and Computation, 2015, 265: 245- 265
13	Tuan N H , Thang L D , Khoa V A , Tran T . On an inverse boundary value problem of a nonlinear elliptic equation in three dimensions. Journal of Mathematical Analysis and Applications, 2015, 426: 1232- 1261
14	Zhang H W , Wei T . A Fourier truncated regularization method for a Cauchy problem of a semi-linear elliptic equation. Journal of Inverse and Ill-posed Problems, 2014, 22 (2): 143- 168
15	Tran Q V , Kirane M , Nguyen H T , Nguyen V T . Analysis and numerical simulation of the three dimensional Cauchy problem for quasi-linear elliptic equations. J Math Anal Appl, 2017, 446 (1): 470- 492
16	Tuan N H , Thang L D , Lesnic D . A new general filter regularization method for Cauchy problems for elliptic equations with a locally lipschitz nonlinear source. J Math Anal Appl, 2016, 434: 1376- 1393
17	Tuan N H , Thang L D , Trong D D , Khoa V A . Approximation of mild solutions of the linear and nonlinear elliptic equations. Inverse Problems in Science and Engineering, 2015, 23 (7): 1237- 1266
18	Khoa V A , Truong M T , Duy N H . A general kernel-based regularization method for coupled elliptic Sine-Gordon equations with Gevrey regularity. Computer Physics Communications, 2015, 183 (8): 1813- 1821
19	Khoa V A , Truong M T , Duy N H , Tuan N H . The Cauchy problem of coupled elliptic Sine-Gordon equations with noise: Analysis of a general kernel-based regularization and reliable tools of computing. Computers and Mathematics with Applications, 2017, 73 (1): 141- 162
20	Zhang H W , Wang R H . Modified boundary Tikhonov-type regularization method for the Cauchy problem of a semi-linear elliptic equation. Inverse Problems in Science and Engineering, 2016, 24 (7): 1249- 1265
21	Zhang H W , Zhang X J . Generalized Tikhonov-type regularization method for the Cauchy problem of a semi-linear elliptic equation. Numerical Algorithms, 2019, 81: 833- 851
22	Zhang H W , Zhang X J . Generalized Lavrentiev-type regularization method for the Cauchy problem of a semi-linear elliptic equation. Inverse Problems in Science and Engineering, 2019, 27 (8): 1120- 1144
23	Hochstenbach M E , Reichel L . Fractional Tikhonov regularization for linear discrete ill-posed problems. BIT Numer Math, 2011, 51: 197- 215
24	Eldén L , Simoncini V . A numerical solution of a Cauchy problem for an elliptic equation by krylov subspaces. Inverse Problems, 2009, 25: 065002
25	Engl H W , Hanke M , Neubauer A . Regularization of Inverse Problems. Dordrecht: Kluwer Academic Publishers, 1996
26	Kirsch A. An Introduction to the Mathematical Theory of Inverse Problems. New York: Springer-Verlag, 1996
27	Evans L C. Partial Diferential Equations. Providence, RI: Amer Math Soc, 1998
28	Doan V N , Nguyen H T , Khoa V A , Vo V A . A note on the derivation of filter regularization operators for nonlinear evolution equations. Applicable Analysis, 2018, 97 (1): 3- 12
29	Qian Z , Feng X L . A fractional Tikhonov method for solving a Cauchy problem of Helmholtz equation. Applicable Analysis, 2017, 96: 1- 13
30	Qin H H , Lu J M . A modified method for a Cauchy problem of the Helmholtz equation. Bull Malays Math Sci Soc, 2017, 40: 1493- 1522

[1]	李雅芝, 刘利利, 田妍妮. 考虑扩散的SVEITR肺结核传播模型的最优控制策略[J]. 数学物理学报, 2025, 45(3): 934-945.
[2]	吴鹏, 方诚. 具有异质空间扩散的梅毒模型的阈值动力学分析与仿真[J]. 数学物理学报, 2024, 44(5): 1352-1367.
[3]	张潇, 张宏武. 分数阶椭圆方程反边值问题的分数 Tikhonov 正则化方法[J]. 数学物理学报, 2024, 44(4): 978-993.
[4]	马亚妮, 袁海龙. 一类具有时滞的 Gierer-Meinhardt 活化抑制模型的分支分析[J]. 数学物理学报, 2023, 43(6): 1774-1788.
[5]	杨帆, 曹英, 李晓晓. 时空分数阶扩散波动方程的初值识别问题[J]. 数学物理学报, 2023, 43(2): 377-398.
[6]	何鑫海,刘梅,杨晗. 一类半线性时间分数阶扩散-波动方程解的整体存在唯一性[J]. 数学物理学报, 2022, 42(6): 1705-1718.
[7]	孟庆春,张磊. 一类三维逆时热传导问题的数值求解[J]. 数学物理学报, 2022, 42(1): 187-200.
[8]	储昌木,蒙璐. 一类带有变指数增长的半线性椭圆方程正解的存在性[J]. 数学物理学报, 2021, 41(6): 1779-1790.
[9]	杨帆,王乾朝,李晓晓. 识别Rayleigh-Stokes方程源项的分数阶Landweber迭代正则化方法[J]. 数学物理学报, 2021, 41(2): 427-450.
[10]	郑立飞,郭洁,吴美华,王小瑞,万阿英. 一类具有时滞的捕食者-猎物-共生者系统的研究[J]. 数学物理学报, 2018, 38(5): 1001-1013.
[11]	王丽. 超音速流越过弯曲坡面的反问题[J]. 数学物理学报, 2018, 38(4): 679-686.
[12]	谢瓯, 孟泽红, 赵振宇, 由雷. 求解拉普拉斯方程柯西问题的截断赫尔米特展开方法[J]. 数学物理学报, 2017, 37(3): 457-468.
[13]	邓继勤, 邓子明. 分数阶微分方程非局部柯西问题解的存在和唯一性[J]. 数学物理学报, 2016, 36(6): 1157-1164.
[14]	韩欢, 李星, 刘军, 倪成洲. 二维海底电缆铺设的微分方程模型与数值算法[J]. 数学物理学报, 2015, 35(4): 780-793.
[15]	姚宪忠, 穆春来, 张付臣. 带双参数的半线性椭圆方程正解存在与不存在性以及一个预值结论[J]. 数学物理学报, 2014, 34(5): 1144-1150.