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##
The Chandrasekhar *H*-Equation

`Extracted from `*A Collection of Nonlinear Problems* by J. J. More

`(Computational Solution of Nonlinear Systems of Equations, Lectures
in Applied Mathematics 26, American Mathematical Society, 723-762, 1990)
and ellaborated by L. Gonzalez-Vega`

The Chandrasekhar *H*-Equation was introduced by S. Chandrasekhar
in the context of radiative transfer problems. If *c* is a parameter
taking values in the interval [0,1], this equation is stated in the following
terms:

where the unknown is the continuous function .The
easiest integration scheme
conveys to a nonlinear system of equations described in what follows.
Let *N* be a positive integer number and
a partition of the interval [0,1]. If *H*_{k} denotes the
value of *H*(*x*_{k}) then the evaluation of the considered
integral equation at every *x*_{i} produces the equations
():

The approximation of every integral in the previous equation leads to the
system of algebraic equations ():
Since the case *i*=0 gives directly *H*_{0}=1, for every
*N* a system with *N* algebraic equations and *N* unknowns
is obtained.
Next the study of some of these systems is performed. The systems for
*N*=1 and *N*=2 are very easy to solve by hand:

The system obtained when *N*=3 is easy to study since its Gröbner
Basis with respect to
provides the following description of the solution set (*H*_{0}=1):
except when *c* is the real root of 20*c*^{2}-942*c*+720
in (0,1): in this case there are only two solutions. Previously it was
assured that the equation (of degree four) giving the relation between
*H*_{1} and *c* has always four real roots greater than
1 for any value of .
The case *N*=4 is more complicated. The Gröbner Basis
of this system (considering *c* as a parameter) with respect to
can be easily computed in *Maple* providing a new system with a structure
``á la Shape Lemma". The univariate polynomial to be solved is:

For every *c* in (0,1] this polynomial has eight real roots which
are greater than 1 and thus the considered system has eight real solutions
(except when *c* is the real root of 35*c*^{2}-3184*c*+2240
in (0,1): in this case there are only four real solutions). The next picture
shows the variation of *H*_{1} with respect to *c*, focusing
the attention on the smallest real root (which is always between 1 and
2) for *c* fixed since due to continuity reasons (for any *c*,
*H*(0)=1 and )
this is the interesting solution.
Finally the next picture shows a first approximation for the function *H*
for different values of *c*. These approximated values of *H*
can be used as starting points for using a numerical method to solve the
polynomial system of equations with .

The use of Gröbner Bases to deal with the case *N*=8 provides
a polynomial of degree 256 in *H*_{1} with coefficients which
are polynomials in *c* of degree 128.

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