User:Tohline/MathProjects/EigenvalueProblemN1
Find Analytic Solutions to an Eigenvalue Problem
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Note from J. E. Tohline to Students with Good Mathematical Skills: This is one of a set of well-defined research problems that are being posed, in the context of this online H_Book, as challenges to young, applied mathematicians. The astronomy community's understanding of the Structure, Stability, and Dynamics of stars and galaxies would be strengthened if we had, in hand, a closed-form analytic solution to the problem being posed here. (A solution can be obtained numerically with relative ease, but here the challenge is to find a closed-form analytic solution.) As is true with most meaningful scientific research projects, it is not at all clear whether this problem has a solution. In my judgment, however, it seems plausible that a closed-form solution can be discovered and such a solution would be of sufficient interest to the astronomical community that it would likely be publishable in a professional astronomy or physics journal. At the very least, this project offers an opportunity for a graduate student, an undergraduate, or even a talented high-school student (perhaps in connection with a mathematics science fair project?) to hone her/his research skills in applied mathematics. Also, I would be thrilled to include a solution to this problem — along with full credit to the solution's author — as a chapter in this online H_Book. Having retired from LSU, I am not in a position to financially support or formally advise students who are in pursuit of a higher-education degree. I would nevertheless be interested in sharing my expertise — and, perhaps, developing a collaborative relationship — with individuals who are interested in pursuing an answer to the mathematical research problem that is being posed here.
The Challenge
Formally, this is an eigenvalue problem. Find one or more analytic expression(s) for the (eigen)function, <math>~\mathcal{G}_\sigma(x)</math> — and, simultaneously, the unknown value of the (eigen)frequency, <math>~\sigma</math> — that satisfies the following 2nd-order, ordinary differential equation:
<math>
(x^2\sin x ) \frac{d^2\mathcal{G}_\sigma}{dx^2} + 2 \biggl[ x \sin x + x^2 \cos x \biggr] \frac{d\mathcal{G}_\sigma}{dx} +
\biggl[ \sigma^2 x^3 - 2\alpha ( \sin x - x\cos x ) \biggr] \mathcal{G}_\sigma = 0 \, ,
</math>
where, <math>~\alpha</math> is a known constant, subject to the following two boundary conditions: <math>~\mathcal{G}_\sigma = 0</math> and <math>~d\mathcal{G}_\sigma/dx = 0</math> at <math>~x = 0</math>. Note that, in the context of astrophysical discussions, the interval of <math>~x</math> that is of particular interest is <math>0 \le x \le \pi</math>.
Analogous Problem with Known Analytic Solutions
Astrophysical Context
A star that is composed entirely of a gas for which the pressure varies as the square of the gas density will have an equilibrium structure that is defined by, what astronomers refer to as, an <math>~n=1</math> polytrope. Inside such an equilibrium structure, the density of the gas will vary with radial position according to the expression,
<math>~\frac{\rho}{\rho_c} = \frac{\sin x}{x} \, ,</math>
where, <math>~\rho_c</math> is the density at the center of the star, and,
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<math>~x</math> |
<math>~\equiv</math> |
<math>~\pi\biggl(\frac{r}{R}\biggr) \, ,</math> |
where, <math>~R</math> is the radius of the equilibrium star. Notice that, according to this expression, the density will drop to zero when <math>~r = R</math>, in which case, <math>~x = \pi</math>. If a star of this type is nudged out of equilibrium — for example, squeezed slightly — in such a way that it maintains its spherical symmetry, the star will begin to undergo periodic, radial oscillations about its original equilibrium radius. The 2nd-order ODE whose solution is being sought in the above challenge is the equation that describes the behavior of these oscillations. In particular, the function,
<math>~\mathcal{G}_\sigma(x) \equiv \frac{\delta x}{x}</math>
describes the relative amplitude of the oscillation as a function of position, <math>~x</math>, within the star, and <math>~\sigma</math> gives the frequency of the oscillation.
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