×

A spectral regularization method for a Cauchy problem of the modified Helmholtz equation. (English) Zbl 1198.65114

Summary: 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 the choice of the regularization parameter is presented and an error estimate is obtained. Combining the method of lines, we formulate regularized solutions which are stably convergent to the exact solutions.

MSC:

65L05 Numerical methods for initial value problems involving ordinary differential equations
34A30 Linear ordinary differential equations and systems
65L60 Finite element, Rayleigh-Ritz, Galerkin and collocation methods for ordinary differential equations
65L20 Stability and convergence of numerical methods for ordinary differential equations
65L70 Error bounds for numerical methods for ordinary differential equations
PDF BibTeX XML Cite
Full Text: DOI EuDML

References:

[1] Kirsch A: An Introduction to the Mathematical Theory of Inverse Problems, Applied Mathematical Sciences. Volume 120. Springer, New York, NY, USA; 1996:x+282. · Zbl 0865.35004
[2] Harari, I; Barbone, PE; Slavutin, M; Shalom, R, Boundary infinite elements for the Helmholtz equation in exterior domains, International Journal for Numerical Methods in Engineering, 41, 1105-1131, (1998) · Zbl 0911.76035
[3] Marin, L; Elliott, L; Heggs, PJ; Ingham, DB; Lesnic, D; Wen, X, An alternating iterative algorithm for the Cauchy problem associated to the Helmholtz equation, Computer Methods in Applied Mechanics and Engineering, 192, 709-722, (2003) · Zbl 1022.78012
[4] Marin, L; Elliott, L; Heggs, PJ; Ingham, DB; Lesnic, D; Wen, X, Conjugate gradient-boundary element solution to the Cauchy problem for Helmholtz-type equations, Computational Mechanics, 31, 367-377, (2003) · Zbl 1047.65097
[5] Regińska, T; Regiński, K, Approximate solution of a Cauchy problem for the Helmholtz equation, Inverse Problems, 22, 975-989, (2006) · Zbl 1099.35160
[6] Marin, L; Elliott, L; Heggs, PJ; Ingham, DB; Lesnic, D; Wen, X, Comparison of regularization methods for solving the Cauchy problem associated with the Helmholtz equation, International Journal for Numerical Methods in Engineering, 60, 1933-1947, (2004) · Zbl 1062.78015
[7] Hadamard J: Lectures on the Cauchy Problem in Linear Partial Differential Equations. Yale University Press, New Haven, Conn, USA; 1923. · JFM 49.0725.04
[8] Xiong, XT; Fu, CL, Two approximate methods of a Cauchy problem for the Helmholtz equation, Computational & Applied Mathematics, 26, 285-307, (2007) · Zbl 1182.35237
[9] Fu, CL; Feng, XL; Qian, Z, The Fourier regularization for solving the Cauchy problem for the Helmholtz equation, Applied Numerical Mathematics, 59, 2625-2640, (2009) · Zbl 1169.65333
[10] Qin, HH; Wei, T, Modified regularization method for the Cauchy problem of the Helmholtz equation, Applied Mathematical Modelling, 33, 2334-2348, (2009) · Zbl 1185.65203
[11] Qian, AL; Xiong, XT; Wu, YJ, On a quasi-reversibility regularization method for a Cauchy problem of the Helmholtz equation, Journal of Computational and Applied Mathematics, 233, 1969-1979, (2010) · Zbl 1185.65171
[12] Eldén, L; Berntsson, F, Spectral and wavelet methods for solving an inverse heat conduction problem, 3-10, (1998), Oxford, UK
[13] Berntsson, F, A spectral method for solving the sideways heat equation, Inverse Problems, 15, 891-906, (1999) · Zbl 0934.35201
[14] Xiong, XT; Fu, CL; Cheng, J, Spectral regularization methods for solving a sideways parabolic equation within the framework of regularization theory, Mathematics and Computers in Simulation, 79, 1668-1678, (2009) · Zbl 1162.65050
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.