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**On the numerical solution of the exterior elastodynamic problem by a boundary integral equation method.**
*(English)*
Zbl 1408.65093

Summary: A numerical method for the Dirichlet initial boundary value problem for the elastic equation in the exterior and unbounded region of a smooth, closed, simply connected two-dimensional domain, is proposed and investigated. This method is based on a combination of a Laguerre transformation with respect to the time variable and a boundary integral equation approach in the spatial variables. Using the Laguerre transformation in time reduces the time-dependent problem to a sequence of stationary boundary value problems, which are solved by a boundary layer approach resulting in a sequence of boundary integral equations of the first kind. The numerical discretization and solution are obtained by a trigonometrical quadrature method. Numerical results are included.

### MSC:

65N35 | Spectral, collocation and related methods for boundary value problems involving PDEs |

35L20 | Initial-boundary value problems for second-order hyperbolic equations |

42C10 | Fourier series in special orthogonal functions (Legendre polynomials, Walsh functions, etc.) |

45E05 | Integral equations with kernels of Cauchy type |

33C45 | Orthogonal polynomials and functions of hypergeometric type (Jacobi, Laguerre, Hermite, Askey scheme, etc.) |

65D32 | Numerical quadrature and cubature formulas |

74S15 | Boundary element methods applied to problems in solid mechanics |

65R20 | Numerical methods for integral equations |

35Q70 | PDEs in connection with mechanics of particles and systems of particles |

### Keywords:

elastic equation; initial boundary value problem; Laguerre transformation; fundamental sequence; single and double layer potentials; boundary integral equations of the first kind; trigonometrical quadrature method
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\textit{R. Chapko} and \textit{L. Mindrinos}, J. Integral Equations Appl. 30, No. 4, 521--542 (2018; Zbl 1408.65093)

### References:

[1] | H. Antes, Anwendungen der Methode der Randelemente in der Elastodynamik und der Fluiddynamik, Springer-Verlag, Berlin, 1988. · Zbl 0654.76002 |

[2] | R. Chapko, On the numerical solution of the Dirichlet initial boundary-value problem for the heat equation in the case of a torus, J. Eng. Math. 43 (2002), 45–87. · Zbl 1015.65057 |

[3] | ——–, On the combination of some semi-discretization methods and boundary integral equations for the numerical solution of initial boundary value problems, Proc. Appl. Math. Mech. 1 (2002), 424–425. · Zbl 1422.65230 |

[4] | R. Chapko and B.T. Johansson, Numerical solution of the Dirichlet initial boundary value problem for the heat equation in exterior \(3\)-dimensional domains using integral equations, J. Eng. Math. (2016), 1–17, DOI: 10.1007/s10665-016-9858-6. |

[5] | R. Chapko and R. Kress, Rothe’s method for the heat equation and boundary integral equations, J. Integral Equations Appl. 9 (1997), 47–69. · Zbl 0885.65101 |

[6] | ——–, On the numerical solution of initial boundary value problems by the Laguerre transformation and boundary integral equations, in Integral and integrodifferential equations: Theory, methods and applications, Math. Anal. Appl. 2 (2000), 55–69. · Zbl 0965.65121 |

[7] | M. Costabel, Time-dependent problems with the boundary integral equation method, in Encyclopedia of computational mechanics, E. Stein, R. Borst and T.J.R. Hughes, eds., Wiley, New York, 2003. |

[8] | V. Galazyuk and R. Chapko, The Chebyshev-Laguerre transformation and integral equations for exterior boundary value problems for the telegraph equation, Dokl. Akad. Nauk. 8 (1990), 11–14. |

[9] | R. Kress, Linear integral equations, Springer-Verlag, Berlin, 2014. · Zbl 1328.45001 |

[10] | V. Kupradze, Three-dimensional problems of the mathematical theory of elasticity and thermoelasticity, North-Holland Publishing Co., New York, 1979. |

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