# Some special solutions of self similar type in MHD, obtained by a separation method of variables

• Volume: 33, Issue: 5, page 939-963
• ISSN: 0764-583X

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## Abstract

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We use a method based on a separation of variables for solving a first order partial differential equations system, using a very simple modelling of MHD. The method consists in introducing three unknown variables Φ1, Φ2, Φ3 in addition to the time variable t and then in searching a solution which is separated with respect to Φ1 and t only. This is allowed by a very simple relation, called a “metric separation equation”, which governs the type of solutions with respect to time. The families of solutions for the system of equations thus obtained, correspond to a radial evolution of the fluid. Solving the MHD equations is then reduced to find the transverse component H∑ of the magnetic field on the unit sphere Σ by solving a non linear partial equation on Σ. Thus, we generalize ideas of Courant-Friedrichs [7] and of Sedov [11], on dimensional analysis and self-similar solutions.

## How to cite

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Cessenat, Michel, and Genta, Philippe. "Some special solutions of self similar type in MHD, obtained by a separation method of variables." ESAIM: Mathematical Modelling and Numerical Analysis 33.5 (2010): 939-963. <http://eudml.org/doc/197450>.

@article{Cessenat2010,
abstract = { We use a method based on a separation of variables for solving a first order partial differential equations system, using a very simple modelling of MHD. The method consists in introducing three unknown variables Φ1, Φ2, Φ3 in addition to the time variable t and then in searching a solution which is separated with respect to Φ1 and t only. This is allowed by a very simple relation, called a “metric separation equation”, which governs the type of solutions with respect to time. The families of solutions for the system of equations thus obtained, correspond to a radial evolution of the fluid. Solving the MHD equations is then reduced to find the transverse component H∑ of the magnetic field on the unit sphere Σ by solving a non linear partial equation on Σ. Thus, we generalize ideas of Courant-Friedrichs [7] and of Sedov [11], on dimensional analysis and self-similar solutions. },
author = {Cessenat, Michel, Genta, Philippe},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis},
keywords = {Magnetohydrodynamic (MHD); separation of variables; selfsimilar solutions; dimensional analysis.; transverse component of magnetic field; MHD equations; self-similar solutions; dimensional analysis; metric separation equation; unit sphere; spherical harmonics; polarizations},
language = {eng},
month = {3},
number = {5},
pages = {939-963},
publisher = {EDP Sciences},
title = {Some special solutions of self similar type in MHD, obtained by a separation method of variables},
url = {http://eudml.org/doc/197450},
volume = {33},
year = {2010},
}

TY - JOUR
AU - Cessenat, Michel
AU - Genta, Philippe
TI - Some special solutions of self similar type in MHD, obtained by a separation method of variables
JO - ESAIM: Mathematical Modelling and Numerical Analysis
DA - 2010/3//
PB - EDP Sciences
VL - 33
IS - 5
SP - 939
EP - 963
AB - We use a method based on a separation of variables for solving a first order partial differential equations system, using a very simple modelling of MHD. The method consists in introducing three unknown variables Φ1, Φ2, Φ3 in addition to the time variable t and then in searching a solution which is separated with respect to Φ1 and t only. This is allowed by a very simple relation, called a “metric separation equation”, which governs the type of solutions with respect to time. The families of solutions for the system of equations thus obtained, correspond to a radial evolution of the fluid. Solving the MHD equations is then reduced to find the transverse component H∑ of the magnetic field on the unit sphere Σ by solving a non linear partial equation on Σ. Thus, we generalize ideas of Courant-Friedrichs [7] and of Sedov [11], on dimensional analysis and self-similar solutions.
LA - eng
KW - Magnetohydrodynamic (MHD); separation of variables; selfsimilar solutions; dimensional analysis.; transverse component of magnetic field; MHD equations; self-similar solutions; dimensional analysis; metric separation equation; unit sphere; spherical harmonics; polarizations
UR - http://eudml.org/doc/197450
ER -

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