### Abstract

A boundary integral based method for the stable reconstruction of missing boundary data is presented for the governing hyperbolic equation of elastodynamics in annular planar domains. Cauchy data in the form of the solution and traction is reconstructed on the inner boundary curve from the similar data given on the outer boundary. The ill-posed data reconstruction problem is reformulated as a sequence of boundary integral equations using the Laguerre transform with respect to time and employing a single-layer approach for the stationary problem. Singularities of the involved kernels in the integrals are analyzed and made explicit, and standard quadrature rules are used for discretization. Tikhonov regularization is employed for the stable solution of the obtained linear system. Numerical results are included showing that the outlined approach can be turned into a practical working method for finding the missing data.

Original language | English |
---|---|

Article number | 112463 |

Journal | Journal of Computational and Applied Mathematics |

Volume | 367 |

Early online date | 17 Sep 2019 |

DOIs | |

Publication status | E-pub ahead of print - 17 Sep 2019 |

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### Bibliographical note

© 2019, Elsevier. Licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International http://creativecommons.org/licenses/by-nc-nd/4.0/### Keywords

- Boundary integral equations
- Cauchy problem
- Elastodynamics
- Laguerre transformation
- Nyström method
- Tikhonov regularization

### Cite this

*Journal of Computational and Applied Mathematics*,

*367*, [112463]. https://doi.org/10.1016/j.cam.2019.112463

}

*Journal of Computational and Applied Mathematics*, vol. 367, 112463. https://doi.org/10.1016/j.cam.2019.112463

**On a boundary integral solution of a lateral planar Cauchy problem in elastodynamics.** / Chapko, Roman; Johansson, B. Tomas; Mindrinos, Leonidas.

Research output: Contribution to journal › Article

TY - JOUR

T1 - On a boundary integral solution of a lateral planar Cauchy problem in elastodynamics

AU - Chapko, Roman

AU - Johansson, B. Tomas

AU - Mindrinos, Leonidas

N1 - © 2019, Elsevier. Licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International http://creativecommons.org/licenses/by-nc-nd/4.0/

PY - 2019/9/17

Y1 - 2019/9/17

N2 - A boundary integral based method for the stable reconstruction of missing boundary data is presented for the governing hyperbolic equation of elastodynamics in annular planar domains. Cauchy data in the form of the solution and traction is reconstructed on the inner boundary curve from the similar data given on the outer boundary. The ill-posed data reconstruction problem is reformulated as a sequence of boundary integral equations using the Laguerre transform with respect to time and employing a single-layer approach for the stationary problem. Singularities of the involved kernels in the integrals are analyzed and made explicit, and standard quadrature rules are used for discretization. Tikhonov regularization is employed for the stable solution of the obtained linear system. Numerical results are included showing that the outlined approach can be turned into a practical working method for finding the missing data.

AB - A boundary integral based method for the stable reconstruction of missing boundary data is presented for the governing hyperbolic equation of elastodynamics in annular planar domains. Cauchy data in the form of the solution and traction is reconstructed on the inner boundary curve from the similar data given on the outer boundary. The ill-posed data reconstruction problem is reformulated as a sequence of boundary integral equations using the Laguerre transform with respect to time and employing a single-layer approach for the stationary problem. Singularities of the involved kernels in the integrals are analyzed and made explicit, and standard quadrature rules are used for discretization. Tikhonov regularization is employed for the stable solution of the obtained linear system. Numerical results are included showing that the outlined approach can be turned into a practical working method for finding the missing data.

KW - Boundary integral equations

KW - Cauchy problem

KW - Elastodynamics

KW - Laguerre transformation

KW - Nyström method

KW - Tikhonov regularization

UR - https://linkinghub.elsevier.com/retrieve/pii/S0377042719304662

UR - http://www.scopus.com/inward/record.url?scp=85072290155&partnerID=8YFLogxK

U2 - 10.1016/j.cam.2019.112463

DO - 10.1016/j.cam.2019.112463

M3 - Article

VL - 367

JO - Journal of Computational and Applied Mathematics

JF - Journal of Computational and Applied Mathematics

SN - 0377-0427

M1 - 112463

ER -