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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:

$$h(x)-{{c}\over{2}}\;\int_0^1{{xh(x)h(y)}\over{x+y}}\;{\rm d}y=1$$ where the unknown is the continuous function $\;h\colon[0,1]\longrightarrow{\rm i\kern -2.2pt r\hskip1pt}$.The easiest integration scheme $$\int_{a}^{b}f(y)\;{\rm d}y\cong f(a)(b-a)$$ 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 H_k denotes the value of H(x_k) then the evaluation of the considered integral equation at every x_i produces the equations ($i\in\{0,\ldots,n\}$):

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

$$\matrix{n=1&h_0=1,h_1={{2}\over{2-c}}\hfill\cr n=2&h_0=1,h_... ...2={{12(-24+c\mp2\sqrt{16-16c+c^{2}})}\over{-56c+48+3c^{2}}}\cr}$$ 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): $$\matrix{ h_1=&\!\!\!\!{{2(6-c)-\sqrt{36-36c+c^2}\pm\sqrt{5(... ...+(-60c^2+240c-2160)h_1+ 2880-480c}{-942c+720+20c^2}}\hfill\cr}$$ 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:

$$\matrix{\scriptstyle c^4h_1^8+(24c^4-192c^3)h_1^7+(188c^4-1... ...8c+110886912)h_1^2+(21233664c-169869312)h_1+84934656\hfill\cr}$$ 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.