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.


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
Full Text: DOI EuDML


[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 · doi:10.1007/978-1-4612-5338-9
[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 1998, 41: 1105-1131. 10.1002/(SICI)1097-0207(19980330)41:6<1105::AID-NME327>3.0.CO;2-0 · Zbl 0911.76035 · doi:10.1002/(SICI)1097-0207(19980330)41:6<1105::AID-NME327>3.0.CO;2-0
[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 2003,192(5-6):709-722. 10.1016/S0045-7825(02)00592-3 · Zbl 1022.78012 · doi:10.1016/S0045-7825(02)00592-3
[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 2003,31(3-4):367-377. · Zbl 1047.65097
[5] Regińska T, Regiński K: Approximate solution of a Cauchy problem for the Helmholtz equation.Inverse Problems 2006,22(3):975-989. 10.1088/0266-5611/22/3/015 · Zbl 1099.35160 · doi:10.1088/0266-5611/22/3/015
[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 2004,60(11):1933-1947. 10.1002/nme.1031 · Zbl 1062.78015 · doi:10.1002/nme.1031
[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 2007,26(2):285-307. · 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 2009,59(10):2625-2640. 10.1016/j.apnum.2009.05.014 · Zbl 1169.65333 · doi:10.1016/j.apnum.2009.05.014
[10] Qin HH, Wei T: Modified regularization method for the Cauchy problem of the Helmholtz equation.Applied Mathematical Modelling 2009,33(5):2334-2348. 10.1016/j.apm.2008.07.005 · Zbl 1185.65203 · doi:10.1016/j.apm.2008.07.005
[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 2010,233(8):1969-1979. 10.1016/j.cam.2009.09.031 · Zbl 1185.65171 · doi:10.1016/j.cam.2009.09.031
[12] Eldén, L.; Berntsson, F., Spectral and wavelet methods for solving an inverse heat conduction problem, 3-10 (1998), Oxford, UK · doi:10.1016/B978-008043319-6/50004-2
[13] Berntsson F: A spectral method for solving the sideways heat equation.Inverse Problems 1999,15(4):891-906. 10.1088/0266-5611/15/4/305 · Zbl 0934.35201 · doi:10.1088/0266-5611/15/4/305
[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 2009,79(5):1668-1678. 10.1016/j.matcom.2008.08.009 · Zbl 1162.65050 · doi:10.1016/j.matcom.2008.08.009
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