Generalized Tikhonov Method and Convergence Estimate for the Cauchy Problem of Modified Helmholtz Equation with Nonhomogeneous Dirichlet and Neumann Datum

We investigate a Cauchy problem of the modified Helmholtz equation with nonhomogeneous Dirichlet and Neumann datum, this problem is ill-posed and some regularization techniques are required to stabilize numerical computation. We established the result of conditional stability under an a priori assum...

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Main Authors: Hongwu Zhang, Xiaoju Zhang
Format: Article
Language:English
Published: MDPI AG 2019-07-01
Series:Mathematics
Subjects:
Online Access:https://www.mdpi.com/2227-7390/7/8/667
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author Hongwu Zhang
Xiaoju Zhang
author_facet Hongwu Zhang
Xiaoju Zhang
author_sort Hongwu Zhang
collection DOAJ
description We investigate a Cauchy problem of the modified Helmholtz equation with nonhomogeneous Dirichlet and Neumann datum, this problem is ill-posed and some regularization techniques are required to stabilize numerical computation. We established the result of conditional stability under an a priori assumption for an exact solution. A generalized Tikhonov method is proposed to solve this problem, we select the regularization parameter by a priori and a posteriori rules and derive the convergence results of sharp type for this method. The corresponding numerical experiments are implemented to verify that our regularization method is practicable and satisfied.
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spelling doaj.art-5411c9da4b654e589631268c2c89ac182022-12-22T02:58:03ZengMDPI AGMathematics2227-73902019-07-017866710.3390/math7080667math7080667Generalized Tikhonov Method and Convergence Estimate for the Cauchy Problem of Modified Helmholtz Equation with Nonhomogeneous Dirichlet and Neumann DatumHongwu Zhang0Xiaoju Zhang1School of Mathematics and Information Science, North Minzu University, Yinchuan 750021, ChinaDevelopment Center of Teachers’ Teaching, North Minzu University, Yinchuan 750021, ChinaWe investigate a Cauchy problem of the modified Helmholtz equation with nonhomogeneous Dirichlet and Neumann datum, this problem is ill-posed and some regularization techniques are required to stabilize numerical computation. We established the result of conditional stability under an a priori assumption for an exact solution. A generalized Tikhonov method is proposed to solve this problem, we select the regularization parameter by a priori and a posteriori rules and derive the convergence results of sharp type for this method. The corresponding numerical experiments are implemented to verify that our regularization method is practicable and satisfied.https://www.mdpi.com/2227-7390/7/8/667ill-posed problemCauchy problemmodified Helmholtz equationgeneralized Tikhonov regularization methodconvergence estimate
spellingShingle Hongwu Zhang
Xiaoju Zhang
Generalized Tikhonov Method and Convergence Estimate for the Cauchy Problem of Modified Helmholtz Equation with Nonhomogeneous Dirichlet and Neumann Datum
Mathematics
ill-posed problem
Cauchy problem
modified Helmholtz equation
generalized Tikhonov regularization method
convergence estimate
title Generalized Tikhonov Method and Convergence Estimate for the Cauchy Problem of Modified Helmholtz Equation with Nonhomogeneous Dirichlet and Neumann Datum
title_full Generalized Tikhonov Method and Convergence Estimate for the Cauchy Problem of Modified Helmholtz Equation with Nonhomogeneous Dirichlet and Neumann Datum
title_fullStr Generalized Tikhonov Method and Convergence Estimate for the Cauchy Problem of Modified Helmholtz Equation with Nonhomogeneous Dirichlet and Neumann Datum
title_full_unstemmed Generalized Tikhonov Method and Convergence Estimate for the Cauchy Problem of Modified Helmholtz Equation with Nonhomogeneous Dirichlet and Neumann Datum
title_short Generalized Tikhonov Method and Convergence Estimate for the Cauchy Problem of Modified Helmholtz Equation with Nonhomogeneous Dirichlet and Neumann Datum
title_sort generalized tikhonov method and convergence estimate for the cauchy problem of modified helmholtz equation with nonhomogeneous dirichlet and neumann datum
topic ill-posed problem
Cauchy problem
modified Helmholtz equation
generalized Tikhonov regularization method
convergence estimate
url https://www.mdpi.com/2227-7390/7/8/667
work_keys_str_mv AT hongwuzhang generalizedtikhonovmethodandconvergenceestimateforthecauchyproblemofmodifiedhelmholtzequationwithnonhomogeneousdirichletandneumanndatum
AT xiaojuzhang generalizedtikhonovmethodandconvergenceestimateforthecauchyproblemofmodifiedhelmholtzequationwithnonhomogeneousdirichletandneumanndatum