基于邊界加擾動(dòng)的Helmhotz方程柯西問(wèn)題的正則化

劉利平

基于邊界加擾動(dòng)的Helmhotz方程柯西問(wèn)題的正則化

劉利平

（甘肅政法學(xué)院計(jì)算機(jī)科學(xué)學(xué)院，甘肅，蘭州 730070）

Helmhotz方程的柯西問(wèn)題是一類典型的反問(wèn)題而且是不適定的，也就是說(shuō)其解不連續(xù)依賴于柯西數(shù)據(jù)，即小的擾動(dòng)都會(huì)導(dǎo)致解的爆破。文章給出了邊界加擾動(dòng)的正則化方法，恢復(fù)了解對(duì)數(shù)據(jù)的連續(xù)依賴性，并給出了收斂性估計(jì)。最后用數(shù)值例子說(shuō)明我們的方法是有效可行的。

Helmhotz方程；柯西問(wèn)題；不適定問(wèn)題；邊界加擾動(dòng)的正則化方法；誤差估計(jì)

Helmhotz型方程最初來(lái)源于物理中聲、波的傳播和散射，建筑物的振動(dòng)以及電磁場(chǎng)等等[1-3]。它是一種典型的不適定問(wèn)題[4]，即問(wèn)題的解如果存在，將不連續(xù)依賴于定解數(shù)據(jù)[5-6]。給數(shù)值計(jì)算帶來(lái)了極大的困難[7]。由于這一問(wèn)題在理論和實(shí)際中有著重要的價(jià)值，人們對(duì)它的正問(wèn)題（Dirichlet，Neumann 或者混合邊值問(wèn)題）在上世紀(jì)已經(jīng)做過(guò)廣泛的研究，許多數(shù)值方法被用來(lái)求解Helmhotz 方程[8-10]。

對(duì)方程(1.1)我們利用Fourier變換可以求得變換后的問(wèn)題在頻域中的解為：

利用Fourier逆變換可以得到該問(wèn)題的精確解為:

下面來(lái)看頻域中的解。

1 問(wèn)題的邊界加擾動(dòng)正則化方法

基于這一思想，我們用邊界加擾動(dòng)的方法來(lái)修改原問(wèn)題得

利用Fourier 變換得

2 收斂性估計(jì)

證明:由Parserval等式以及(2.1)和(3.1)的兩個(gè)假設(shè)，可以得到

證明:類似于定理1的證明，得到

當(dāng)

當(dāng)

3 數(shù)值算例

圖1 精確解與近似解的誤差

[1] Beskos D E. Boundary element methods in dynamic analysis[J].Applied Mechanics Reviews, 1997,50(3):149- 197.

[2] Chen J T, Wong F C. Dual formulation of multiple reciprocity methodfor the acoustic mode of a cavity with a thin partition[J]. Journal of Sound and Vibration, 1998, 217(1) :75-95.

[3] Hrycak T, Isakov V. Increased stability in the continuation of solutions to the Helmholtz equation[J]. Inverse Problems, 2004, 20 (3):697-712.

[4] Hadamard J. Lectures on Cauchy’s problem in linear partial differential equations[M]. New York:Dover Publications, 1953.

[5] 劉繼軍.不適定問(wèn)題的正則化方法及應(yīng)用[M]. 北京:科學(xué)出版社, 2005.

[6] Isakov V. Inverse problems for partial differential equations[M]. vol. 127 of Applied Mathematical Sciences. New York :Springer-Verlag, 1998.

[7] Tikhonov A., Arsenin V. Solutions of ill-posed problems[M]. New York:V. H. Winston& Sons, Washington, D.C.: John Wiley & Sons, 1977.

[8] Jin B, Zheng Y. A meshless method for some inverse problems associated with the Helmholtz equation[J]. Comput. Methods Appl. Mech. Engrg., 2006,195 (19-22 ):2270-2288.

[9] Reginska T, Reginski K. Approximate solution of a Cauchy problem forthe Helmholtz equation[J]. Inverse Problems, 2006, 22(3):975-989.

[10] Marin L, Elliott L, Heggs P J, et al. Conjugate gradient-boundary element solution to the Cauchy problem for Helmholtz-type equations[J]. Comput. Mech., 2003,31(3-4):367-377.

Based on the boundary modification regularization method for the Cauchy problem of Helmholtz equations

LIU Li-ping

(School of Computer Science, Gansu Institute of Politics Science and Law, Gansu, Lanzhou 730070, China)

The Cauchy problems for the Helmholtz equations are considered. The problem is ill-posed in the sense that the solution (if exists) does not depend continuously on the given data. In order to obtain a stability approximation solution of the problem, it is necessary to employ some regularized techniques. Furthermore, we use the boundary modification regularized method to solve the Cauchy problems for Helmholtz equations and give the convergence estimates. Finally, the numerical examples show that the proposed numerical method works effectively.

Helmhotz equation; Cauchy problem; ill-posed problem; the boundary modification regularization method; error estimate

O175.8

A

10.3969/j.issn.1674-8085.2012.04.005

1674-8085(2012)04-0021-04

2012-04-22；

2012-05-20

劉利平(1984-)，女，湖南株洲人，助教，碩士，主要從事偏微分方程研究(E-mail: liulipinglele@163.com).