二元混合分离问题中不连续性相互作用及 hodograph 方法
Interactions between discontinuities for binary mixture separation problem and hodograph method
AI总结：
本文研究了二元混合分离问题中不连续性相互作用，利用 hodograph 方法求解首 order PDE 的 Cauchy 问题，并提出通过将 PDE 转换为 ODE 来恢复显式解的方法。
摘要：
对于具有分段不连续初始数据的首 order PDE 的 Cauchy 问题，完全求解。通过 hodograph 方法研究初始不连续性破裂后不连续性之间的相互作用。解以分析隐式形式构造。为恢复解的显式形式，将 PDE 转换为某些在隐式解的等位线（等时线）上的 ODE。特别是，该方法允许求解具有在特征线上初始数据的 Goursat 问题。本文描述了区电泳（混合分离方法）的具体问题。然而，所提出的方法可用于解决任何两个首 order 青线性 PDE 系统，其中 hodograph 变换后出现的二阶线性 PDE 具有显式 Riemann-Green 函数。
英文摘要：
The Cauchy problem for first-order PDE with the initial data which have a piecewise discontinuities localized in different spatial points is completely solved. The interactions between discontinuities arising after breakup of initial discontinuities are studied with the help of the hodograph method. The solution is constructed in analytical implicit form. To recovery the explicit form of solution we propose the transformation of the PDEs into some ODEs on the level lines (isochrones) of implicit solution. In particular, this method allows us to solve the Goursat problem with initial data on characteristics. The paper describes a specific problem for zone electrophoresis (method of the mixture separation). However, the method proposed allows to solve any system of two first-order quasilinear PDEs for which the second order linear PDE, arising after the hodograph transformation, has the Riemann-Green function in explicit form.
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