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Abstract

Exterior Dirichlet and Robin initial boundary-value problems for the homogeneous wave equation with homogeneous initial conditions are considered in domains which are three-dimensional in spatial variables. Using a solution representation by the Kirchho formula these problems are reduced to time-domain boundary integral equations (TDBIEs) with unknown Cauchy data on a boundary surface. We have one nonstationary integral equation with an unknown normal derivative of the solution of the problem in the case of the Dirichlet problem and a system of two TDBIEs for both Cauchy data for the Robin problem. As a result of applying the Laguerre transform in the time variable to that TDBIEs and regrouping the corresponding Fourier-Laguerre coecients, we obtained innite sequences of boundary integral equations (BIEs), which depend only on the coordinates at boundary surfaces. All BIEs obtained from the Dirichlet problem are represented by the same elliptic boundary operator in the left-hand side of the equations and by recurrently dependent expressions in the right-hand sides. In the case of Robin problem, the structure of the resulting sequence is similar, but now we are dealing with a matrix operator, which is composed of four boundary operators and is elliptic in some Sobolev space. For the numerical solution of obtained BIEs, a fast boundary element method was developed as an implementation of the Galerkin method. The unknown traces and normal derivatives of the solutions are approximated by linear and piece-wise constant basis functions, respectively. To reduce the required storage and computational costs, an adaptive cross-approximation of the discretized boundary operators was implemented. A series of computational experiments on the numerical solution of the Dirichlet and Robin model problems were carried out. Their results demonstrate the high accuracy and estimated order of convergence of the proposed method.