Exact Solutions of Gas Dynamics Equations in Series in the Lagrangian Coordinate and Their Numerical Realization

Abstract

An approach to the determination of exact analytical solutions of the problems of gas flow in a tube with a piston governed by the equations of time-dependent, one-dimensional gas dynamics with plane waves is developed. The solutions are sought in the form of power series in a special Lagrangian coordinate, for example, the initial position of particles or a variable entropy. All time-dependent coefficients of the series are determined successively from recurrent relations via two given boundary conditions, namely, the law of piston motion and the temperature on the piston. The given quantities can be so chosen that the necessary initial data would be satisfied. To accurately calculate the terms of the series mathematical packages are used, whose functional includes symbolic transformations. The possibility of attaining the convergence of the solutions constructed is discussed. Examples of physical problems solved within the framework of the proposed approach are presented.