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

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• 作者：Nguyen Huy Tuan^a ; ^{nguyenhuytuan@tdt.edu.vn" class="auth_mail" title="E-mail the corresponding author} ; Vo Anh Khoa^b ; ^{khoa.vo@gssi.infn.it" class="auth_mail" title="E-mail the corresponding author} ; ^{vakhoa.hcmus@gmail.com" class="auth_mail" title="E-mail the corresponding author} ; Mach Nguyet Minh^c ; ^{mach@math.uni-frankfurt.de" class="auth_mail" title="E-mail the corresponding author} ; Thanh Tran^d ; ^{thanh.tran@unsw.edu.au" class="auth_mail" title="E-mail the corresponding author}
• 关键词：35K05 ; 35K99 ; 47J06 ; 47H10
• 刊名：Journal of Computational and Applied Mathematics
• 出版年：2017
• 出版时间：1 January 2017
• 年：2017
• 卷：309
• 期：Complete
• 页码：56-78
• 全文大小：1212 K

文摘

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 Regińska and Regiński (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 L²$L^2$-norm of such regularized solutions. Besides, the regularization parameters are formulated properly. Finally, some illustrative examples are provided to corroborate our qualitative analysis.