Hodograph Method and Numerical Solution of the Two Hyperbolic Quasilinear Equations System. Part II. Zonal Electrophoresis Equations

TL;DR

This paper develops a method combining hodograph and numerical techniques to solve zonal electrophoresis equations, reducing PDE problems to ODEs and enabling multi-valued solutions, with applications to various initial data.

Contribution

The paper introduces a novel approach for solving hyperbolic quasilinear PDEs using the hodograph method and Riemann-Green functions, extending previous work to cases with sum-of-products Riemann-Green functions.

Findings

Effective reduction of PDEs to ODEs for electrophoresis equations

Ability to construct multi-valued solutions

Numerical results for different initial data types

Abstract

The paper presents the solutions for the zonal electrophoresis equations are obtained by analytical and numerical methods. The method proposed by the authors is used. This method allows to reduce the Cauchy problem for two hyperbolic quasilinear PDE's to the Cauchy problem for ODE's. In some respect, this method is analogous to the method of characteristics for two hyperbolic equations. The method is effectively applicable in all cases when the explicit expression for the Riemann-Green function of some linear second order PDE, resulting from the use of the hodograph method for the original equations, is known. One of the method advantages is the possibility of constructing a multi-valued solutions. Compared with the previous authors paper, in which, in particular, the shallow water equations are studied, here we investigate the case when the Riemann-Green function can be represent as the sum of the terms each of them is a product of two multipliers depended on different variables. The numerical results for zonal electrophoresis equations are presented. For computing the different initial data (periodic, wave packet, the Gaussian distribution) are used.