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

Author

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  • Hongwu Zhang

    (School of Mathematics and Information Science, North Minzu University, Yinchuan 750021, China
    These authors contributed equally to this work.)

  • Xiaoju Zhang

    (Development Center of Teachers’ Teaching, North Minzu University, Yinchuan 750021, China
    These authors contributed equally to this work.)

Abstract

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.

Suggested Citation

  • Hongwu Zhang & Xiaoju Zhang, 2019. "Generalized Tikhonov Method and Convergence Estimate for the Cauchy Problem of Modified Helmholtz Equation with Nonhomogeneous Dirichlet and Neumann Datum," Mathematics, MDPI, vol. 7(8), pages 1-19, July.
    • Handle: RePEc:gam:jmathe:v:7:y:2019:i:8:p:667-:d:251785
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    References listed on IDEAS

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    1. Qin, Hai-Hua & Wei, Ting, 2009. "Quasi-reversibility and truncation methods to solve a Cauchy problem for the modified Helmholtz equation," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 80(2), pages 352-366.
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