Geodesic
Interaction of weak discontinuities and the hodograph method as applied to electric field fractionation of a two-component mixture
Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki, Tome 56 (2016) no. 8, pp. 1455-1469

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The hodograph method is used to construct a solution describing the interaction of weak discontinuities (rarefaction waves) for the problem of mass transfer by an electric field (zonal electrophoresis). Mathematically, the problem is reduced to the study of a system of two first-order quasilinear hyperbolic partial differential equations with data on characteristics (Goursat problem). The solution is constructed analytically in the form of implicit relations. An efficient numerical algorithm is described that reduces the system of quasilinear partial differential equations to ordinary differential equations. For the zonal electrophoresis equations, the Riemann problem with initial discontinuities specified at two different spatial points is completely solved. 
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     author = {M. S. Elaeva and M. Yu. Zhukov and E. V. Shiryaeva},
     title = {Interaction of weak discontinuities and the hodograph method as applied to electric field fractionation of a two-component mixture},
     journal = {\v{Z}urnal vy\v{c}islitelʹnoj matematiki i matemati\v{c}eskoj fiziki},
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     number = {8},
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     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZVMMF_2016_56_8_a7/}
}
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M. S. Elaeva; M. Yu. Zhukov; E. V. Shiryaeva. Interaction of weak discontinuities and the hodograph method as applied to electric field fractionation of a two-component mixture. Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki, Tome 56 (2016) no. 8, pp. 1455-1469. http://geodesic.mathdoc.fr/item/ZVMMF_2016_56_8_a7/