Reconstruction of the electric field of the Helmholtz equation in three dimensions

JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS

卷:309

Article

Nguyen Huy Tuan¹  Vo Anh Khoa²  Mach Nguyet Minh³  Thanh Tran⁴ 

[1] Ton Duc Thang Univ, Fac Math & Stat, Appl Anal Res Grp, Ho Chi Minh City, Vietnam

[2] Gran Sasso Sci Inst, Math & Comp Sci Div, Laquila, Italy

[3] Goethe Univ Frankfurt, Dept Math, Frankfurt, Germany

[4] Univ New South Wales, Sch Math & Stat, Sydney, NSW, Australia

关键词: Cauchy problem;    Helmholtz equation;    Ill-posed problem;    Regularized solution;    Stability;    Error estimates;   

DOI  :  10.1016/j.cam.2016.05.021

来源: Elsevier

【 摘 要 】

In this paper, we rigorously investigate the truncation method for the Cauchy problem of Helmholtz equations which is widely used to model propagation phenomena in physical applications. The truncation method is a well-known approach to the regularization of several types of ill-posed problems, including the model postulated by Reginska and Reginski (2006). Under certain specific assumptions, we examine the ill-posedness of the non-homogeneous problem by exploring the representation of solutions based on Fourier mode. Then the so-called regularized solution is established with respect to a frequency bounded by an appropriate regularization parameter. Furthermore, we provide a short analysis of the nonlinear forcing term. The main results show the stability as well as the strong convergence confirmed by the error estimates in L-2-norm of such regularized solutions. Besides, the regularization parameters are formulated properly. Finally, some illustrative examples are provided to corroborate our qualitative analysis. (C) 2016 Elsevier B.V. All rights reserved.

【 授权许可】

Free