Interactions between discontinuities for binary mixture separation problem and hodograph method

TL;DR

This paper presents an analytical method using the hodograph transformation to solve the interaction of discontinuities in first-order PDEs, with applications to mixture separation problems like electrophoresis.

Contribution

It introduces a novel approach transforming PDEs into ODEs on level lines, enabling explicit solutions for complex discontinuity interactions in first-order PDE systems.

Findings

Complete solution to the Cauchy problem with piecewise discontinuities.

Method allows solving Goursat problems with initial data on characteristics.

Applicable to systems with known Riemann-Green functions.

Abstract

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.