TAILIEUCHUNG - Báo cáo sinh học: " Research Article A Spectral Regularization Method for a Cauchy Problem of the Modified Helmholtz Equation"

Tuyển tập các báo cáo nghiên cứu về sinh học được đăng trên tạp chí sinh học Journal of Biology đề tài: Research Article A Spectral Regularization Method for a Cauchy Problem of the Modified Helmholtz Equation | Hindawi Publishing Corporation Boundary Value Problems Volume 2010 Article ID 212056 13 pages doi 2010 212056 Research Article A Spectral Regularization Method for a Cauchy Problem of the Modified Helmholtz Equation Ailin Qian Jianfeng Mao and Lianghua Liu School of Mathematics and Statistics Xianning College Xianning Hubei 437100 China Correspondence should be addressed to Ailin Qian junren751113@ Received 15 December 2009 Accepted 9 May 2010 Academic Editor Salim Messaoudi Copyright 2010 Ailin Qian et al. This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited. We consider the Cauchy problem for a modified Helmholtz equation where the Cauchy data is given at x 1 and the solution is sought in the interval 0 x 1. A spectral method together with choice of regularization parameter is presented and error estimate is obtained. Combining the method of lines we formulate regularized solutions which are stably convergent to the exact solutions. 1. Introduction The Cauchy problem for Helmholtz equation arises from inverse scattering problems. Specific backgrounds can be seen in the existing literature we can refer to 1-6 and so forth. A number of numerical methods for stabilizing this problem are developed. Several boundary element methods combined with iterative conjugate gradient Tikhonov regularization and singular value decomposition methods are compared in 6 . Cauchy problem for elliptic equations is well known to be severely ill-posed 7 that is the solution does not depend continuously on the boundary data and small errors in the boundary data can amplify the numerical solution infinitely hence it is impossible to solve Cauchy problem of Helmholtz equation by using classical numerical methods and it requires special techniques for example regularization methods. Although theoretical concepts and .

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