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

Michel Cessenat; Philippe Genta

ESAIM: Mathematical Modelling and Numerical Analysis (2010)

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

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topCessenat, 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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