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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2019-07-01
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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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language | English |
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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 |