The present paper is devoted to the development of the method for searching an approximate solution of non-stationary problems on a variable interval. This method was first proposed by L.I. Slepyan. Here, the field of its applicability has been expanded to systems of equations, including equations of both hyperbolic and parabolic type. We describe the procedure of such approach in detail on a number of classical partial differential equations and compare the obtained results with exact analytical solution. Also, we consider a dynamic non-coupled thermoelastic problem. The method of expansion on a variable interval allows us to estimate the material response to a thermal disturbance at a large distance from the source.

中文翻译：

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连续介质力学中非平稳问题的变区间展开法

本论文致力于开发在可变区间上搜索非平稳问题的近似解的方法。该方法最早由 LI Slepyan 提出。在这里，它的适用领域已经扩展到方程组，包括双曲线和抛物线类型的方程。我们在一些经典的偏微分方程上详细描述了这种方法的过程，并将得到的结果与精确解析解进行了比较。此外，我们还考虑了动态非耦合热弹性问题。可变区间的膨胀方法使我们能够估计材料对距离源很远的热扰动的响应。