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.

中文翻译：

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拉格朗日坐标系气体动力学方程的精确解及其数值实现

摘要：开发了一种确定具有活塞的管中气体流动问题的精确解析解的方法，该问题由具有平面波的时间相关的一维气体动力学方程控制。以特殊拉格朗日坐标中的幂级数形式寻求解，例如粒子的初始位置或可变熵。该系列的所有瞬态系数通过两个给定的边界条件，即活塞运动定律和活塞上的温度，从循环关系中依次确定。可以选择给定的数量，以满足必要的初始数据。为了准确计算系列的项，使用了数学包，其功能包括符号变换。讨论了实现所构建解收敛的可能性。提出了在建议方法的框架内解决的物理问题的例子。