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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:

\begin{displaymath}h(x)-{{c}\over{2}}\;\int_0^1{{xh(x)h(y)}\over{x+y}}\;{\rm d}y=1\end{displaymath}
where the unknown is the continuous function $\;h\colon[0,1]\longrightarrow{\rm i\kern -2.2pt r\hskip1pt}$.The easiest integration scheme
\begin{displaymath}\int_{a}^{b}f(y)\;{\rm d}y\cong f(a)(b-a)\end{displaymath}
conveys to a nonlinear system of equations described in what follows.

Let N be a positive integer number and $\;\;0<x_1<\ldots<x_j=j/n<\ldots<1\;\;$ a partition of the interval [0,1]. If Hk denotes the value of H(xk) then the evaluation of the considered integral equation at every xi produces the equations ($i\in\{0,\ldots,n\}$):

\begin{displaymath}h_i-{{x_ih_ic}\over{2}}\;\int_0^1{{h(y)}\over{x_i+y}}\;{\rm d}y=1\end{displaymath}
The approximation of every integral in the previous equation leads to the system of algebraic equations ($i\in\{0,\ldots,n\}$):
\begin{displaymath}h_i-h_i{{ic}\over{2n}}\;\sum_{j=0}^{n-1} {{h_j}\over{i+j}} =1\end{displaymath}
Since the case i=0 gives directly H0=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:

\begin{displaymath}\matrix{n=1&h_0=1,h_1={{2}\over{2-c}}\hfill\cr n=2&h_0=1,h_... ...2={{12(-24+c\mp2\sqrt{16-16c+c^{2}})}\over{-56c+48+3c^{2}}}\cr}\end{displaymath}
The system obtained when N=3 is easy to study since its Gröbner Basis with respect to $\gt _{\rm lex}$ provides the following description of the solution set (H0=1):
\begin{displaymath}\matrix{ h_1=&\!\!\!\!{{2(6-c)-\sqrt{36-36c+c^2}\pm\sqrt{5(... ...+(-60c^2+240c-2160)h_1+ 2880-480c}{-942c+720+20c^2}}\hfill\cr}\end{displaymath}
except when c is the real root of 20c2-942c+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 H1 and c has always four real roots greater than 1 for any value of $c\in(0,1]$.

 The case N=4 is more complicated. The Gröbner Basis of this system (considering c as a parameter) with respect to $\gt _{\rm lex}$ can be easily computed in Maple providing a new system with a structure ``á la Shape Lemma". The univariate polynomial to be solved is:

\begin{displaymath}\matrix{\scriptstyle c^4h_1^8+(24c^4-192c^3)h_1^7+(188c^4-1... ...8c+110886912)h_1^2+(21233664c-169869312)h_1+84934656\hfill\cr}\end{displaymath}
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 35c2-3184c+2240 in (0,1): in this case there are only four real solutions). The next picture shows the variation of H1 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 $c=0\rightarrow h\equiv 1$) this is the interesting solution.
$\qquad\qquad$
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 $n\geq 8$.

 

$\qquad\qquad$
The use of Gröbner Bases to deal with the case N=8 provides a polynomial of degree 256 in H1 with coefficients which are polynomials in c of degree 128.


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