Three algorithms for the numerical inversion of the Laplace transform are considered, for solving specific applied problems of wave dynamics. Comparing the algorithm based on the shifted Lagendre polynomials with the standart solutions shows that there exists an optimal number of terms in expansions. It is also established from the consideration of various algorithms, including modern ones, that the accuracy of all algorithms of numerical inversion decreases with increasing time t (with decreasing in the transformation parameter p in the complex plane). These two conclusions are a consequence of the incorrectness of the inversion Laplace transform problem. The application of the expansion method in the sine arcs to the solution of the initial-boundary value problem (IBV problem) of the study of the propagation of pulse pressure waves in blood vessels is presented. It is based on the equations of the cylindrical shell and blood pressure, and includes the matching conditions at the junction of the vessels, excitation of the pulse pressure wave, its propagation to the junction, reflected and transmitted waves. Application of the same method is presented for the problem of evolution of the free surface of water waves due to local bottom excitation sources that are repeated in time.
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