On Exact Analytical Solutions of Gas Dynamic Equations

摘要

The theory of construction of exact analytical solutions of the Cauchy problem using the power series depending on a special time variable whose form determines the particular class of motion is developed within one-dimensional time-dependent gas dynamics. Generally, the recurrent relations to the coefficients are finite and arranged so that there is no need to solve differential equations or integrate for calculation of the unknown functions and all the terms of series are determined successively from the initial conditions using only the algebraic operations and differentiation. This fact makes it possible also to find the terms of series exactly using any mathematical software package which admits of symbolic transformations. The necessary boundary conditions are discussed and the control techniques for the behavior of series are outlined. Some examples of the physical problems solved with the use of the method proposed are examined.