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.

图/表 3

参考文献 30

1	Beskos D E . Boundary element method in dynamic analysis: Part Ⅱ (1986-1996). ASME Appl Mech Rev, 1997, 50: 149- 197 doi: 10.1115/1.3101695
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 doi: 10.1006/jsvi.1998.1743
3	Fokas A S , Pelloni B . The dirichlet-to-neumann map for the elliptic Sine-Gordon equation. Nonlinearity, 2012, 25 (4): 1011- 1031 doi: 10.1088/0951-7715/25/4/1011
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 doi: 10.1007/BF01249332
5	Belgacem F B . Why is the Cauchy problem severely ill-posed?. Inverse Problems, 2007, 23 (2): 823 doi: 10.1088/0266-5611/23/2/020
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 doi: 10.1080/17415977.2013.850683
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 doi: 10.1088/0266-5611/25/5/055002
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 doi: 10.1016/j.amc.2010.09.019
11	Zhang H W , Zhang X J . Generalized-fractional Tikhonov-type method for the Cauchy problem of elliptic equation. Mathematics, 2020, 8 (1): 48 doi: 10.3390/math8010048
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 doi: 10.1016/j.amc.2015.03.115
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 doi: 10.1016/j.jmaa.2014.12.047
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 doi: 10.1016/j.jmaa.2016.08.045
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 doi: 10.1016/j.jmaa.2015.09.085
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 doi: 10.1080/17415977.2014.993983
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 doi: 10.1016/j.camwa.2016.11.001
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 doi: 10.1080/17415977.2016.1172222
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 doi: 10.1007/s11075-018-0573-4
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 doi: 10.1080/17415977.2018.1494168
23	Hochstenbach M E , Reichel L . Fractional Tikhonov regularization for linear discrete ill-posed problems. BIT Numer Math, 2011, 51: 197- 215 doi: 10.1007/s10543-011-0313-9
24	Eldén L , Simoncini V . A numerical solution of a Cauchy problem for an elliptic equation by krylov subspaces. Inverse Problems, 2009, 25: 065002 doi: 10.1088/0266-5611/25/6/065002
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 doi: 10.1080/00036811.2016.1276176
29	Qian Z , Feng X L . A fractional Tikhonov method for solving a Cauchy problem of Helmholtz equation. Applicable Analysis, 2017, 96: 1- 13 doi: 10.1080/00036811.2016.1246129
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 doi: 10.1007/s40840-015-0148-7

$q$	1.1	1.4	1.7	2	2.5	3	3.5
$\epsilon_{y=0.6}(u)$	0.0043	0.0069	0.0093	0.0122	0.0205	0.0296	0.0349
$\epsilon_{y=1}(u)$	0.0027	0.0059	0.0090	0.0122	0.0212	0.0302	0.0352

$\gamma$	0.5	0.8	1.1	1.5	1.8	2.5	3
$\epsilon_{y=0.6}(u)$	0.0052	0.0072	0.0095	0.0141	0.0192	0.0310	0.0353
$\epsilon_{y=1}(u)$	0.0034	0.0062	0.0090	0.0144	0.0200	0.0318	0.0358

$m$	1	2	3	4	5	6	7	8	9	10
$\epsilon_{y=0.6}(u)$	0.0380	0.0380	0.0353	0.0353	0.0353	0.0353	0.0353	0.0353	0.0353	0.0353
$\epsilon_{y=1}(u)$	0.0382	0.0382	0.0358	0.0358	0.0358	0.0358	0.0358	0.0358	0.0358	0.0358

[1]	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.
[2]	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.
[3]	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.
[4]	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.
[5]	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.
[6]	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.
[7]	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.
[8]	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.
[9]	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.
[10]	舒小保, 戴斌祥. 半线性中立型分数阶微分方程S -渐近ω周期解[J]. 数学物理学报, 2014, 34(1): 16-26.
[11]	赵振宇, 由雷. 求解热源识别问题的修正吉洪诺夫正则化方法[J]. 数学物理学报, 2014, 34(1): 186-192.
[12]	徐红梅, 许盈盈. 多维空间一般Benjamin-Bona-Mahony 方程解的逐点估计[J]. 数学物理学报, 2013, 33(6): 1112-1121.
[13]	安荣, 李开泰. Plate Contact问题的混合有限元逼近[J]. 数学物理学报, 2010, 30(3): 666-676.
[14]	康东升. 一种奇异临界椭圆方程的非平凡解[J]. 数学物理学报, 2006, 26(5): 716-720.
[15]	傅初黎, 邱春雨, 赵华. 求解一般抛物方程侧边值问题的Fourier正则化方法[J]. 数学物理学报, 2005, 25(6): 806-810.