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=Find Analytic Solutions to an Eigenvalue Problem=

<font color="red">'''Note from J. E. Tohline to Students with Good Mathematical Skills'''</font>: 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.2C_Applied_Mathematicians|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'' such 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 any individual who is interested in pursuing an answer to the mathematical research problem that is being posed here.

==The Challenge==

Formally, this is an eigenvalue problem.

<table border="1" width="100%" cellpadding="10" align="center"><tr><td bgcolor="lightblue">
<table border="0" width="100%" cellpadding="5" align="left"><tr><td>

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 2<sup>nd</sup>-order, ordinary differential equation:

<div align="center">
<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><br />
</div>
<br />
where, <math>~\alpha</math> is a known constant. The desired functional solution is subject to the following two boundary conditions: <math>~\mathcal{G}_\sigma = 0</math> at <math>~x = 0</math>; and <math>~d\ln\mathcal{G}_\sigma/d\ln x = (\pi^2 \sigma^2/2 - \alpha)</math> at <math>~x = \pi</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>.

</td></tr></table>
</td></tr>
</table>

==Context==

The challenge posed above is one of a set of closely related eigenvalue problems that arise in the context of the study of the pulsating stars and the governing 2<sup>nd</sup>-order ODE is often referred to as the ''Linear Adiabatic Wave Equation'' (LAWE). In the most general context, the LAWE takes the form,

<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~\biggl[P \biggr]\frac{d^2\mathcal{G}_\sigma}{dx^2} + \biggl[\frac{4P}{x}
+ P^' \biggr]\frac{d\mathcal{G}_\sigma}{dx} + \biggl[ \sigma^2 \rho + \frac{\alpha P^'}{x} \biggr]\mathcal{G}_\sigma</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~0 \, ,</math>
</td>
</tr>
</table>
</div>

where, <math>~P</math> and <math>~\rho</math> are both functions of <math>~x</math> that have different prescriptions for each specified astrophysics problem — see the table of examples presented below — and primes denote differentiation with respect to <math>~x</math>. The symmetries associated with this broad set of eigenvalue problems can perhaps be better appreciated by rearranging terms in the LAWE to obtain,

<div id="LAWE">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~- \sigma^2 \mathcal{G}_\sigma </math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\frac{P}{\rho} \biggl[ \frac{4\mathcal{G}_\sigma^'}{x}+ \mathcal{G}_\sigma^{' '} \biggr]
+ \frac{P^'}{\rho} \biggl[ \frac{\alpha \mathcal{G}_\sigma}{x} + \mathcal{G}_\sigma^'\biggr] \, ,
</math>
</td>
</tr>
</table>
or, equivalently,

<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~- \sigma^2 \rho \mathcal{G}_\sigma </math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~ \frac{P}{x^4} \frac{d}{dx}\biggl( x^4 \mathcal{G}_\sigma^' \biggr)
+ \frac{P^'}{x^\alpha}\frac{d}{dx}\biggl(x^\alpha \mathcal{G}_\sigma\biggr) \, ,
</math>
</td>
</tr>
</table>
or, equivalently,

<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~\frac{d}{dx}\biggl(x^4 P \mathcal{G}_\sigma^'\biggr) + \biggl[ \biggl( \sigma^2 + \frac{\alpha P^'}{x\rho} \biggr) x^4 \rho
\biggr] \mathcal{G}_\sigma</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~ 0 \, .</math>
</td>
</tr>
</table>
</div>

<table border="1" cellpadding="5" align="center" width="90%">
<tr>
<th align="center" colspan="4"><font size="+1">Properties of Analytically Defined Astrophysical Structures</font></th>
</tr>
<tr>
<td align="center" width="10%">Model</td>
<td align="center"><math>~\rho(x)</math>
<td align="center"><math>~P(x)</math>
<td align="center"><math>~P^'(x)</math>
</tr>
<tr>
<td align="center">[[SSC/Stability/UniformDensity#The_Stability_of_Uniform-Density_Spheres|Uniform-density]]</td>
<td align="center"><math>~1</math>
<td align="center"><math>~1 - x^2</math>
<td align="center"><math>~-2x</math>
</tr>
<tr>
<td align="center">[[SSC/Structure/OtherAnalyticModels#Linear_Density_Distribution|Linear]]</td>
<td align="center"><math>~1-x</math>
<td align="center"><math>~(1-x)^2(1 + 2x - \tfrac{9}{5}x^2)</math>
<td align="center"><math>~-\tfrac{12}{5}x(1-x)(4-3x)</math>
</tr>
<tr>
<td align="center">[[SSC/Structure/OtherAnalyticModels#Parabolic_Density_Distribution|Parabolic]]</td>
<td align="center"><math>~1-x^2</math>
<td align="center"><math>~(1-x^2)^2(1 - \tfrac{1}{2} x^2)</math>
<td align="center"><math>~-x(1-x^2)(5-3x^2)</math>
</tr>
<tr>
<td align="center">[[SSC/Stability/Polytropes#n_.3D_1_Polytrope|<math>~n=1</math> Polytrope]]</td>
<td align="center"><math>~\frac{\sin }{ x}</math>
<td align="center"><math>~\biggl[\frac{\sin x}{x}\biggr]^2</math>
<td align="center"><math>~\frac{2}{x} \biggl[ \cos x - \frac{\sin x}{x} \biggr]
\frac{\sin x}{x}</math>
</tr>
</table>

Drawing the expressions for <math>~\rho(x)</math>, <math>~P(x)</math>, and <math>~P^'(x)</math> from the last row of this table and plugging them into this generic form of the LAWE leads to the ''specific'' statement of the astrophysically motivated eigenfunction problem presented above — inside the blue-framed box. As is discussed in the [[#Analogous_Problem_with_Known_Analytic_Solutions|subsection that follows]], an analogous eigenvalue problem whose analytic solution is ''known'' comes from plugging expressions presented in the first row of this table into the generic form of the LAWE.

==Analogous Problem with Known Analytic Solutions==

Here is an analogous problem whose analytic solution is known. Anyone interested in tackling the ''challenge'', provided above, should study — and even extend — the known set of solutions of this analogous problem. This exercise should provide at least partial preparation for addressing the above challenge.

===Statement of the Problem===

As above, the task here is to find one or more analytic expression(s) for the (eigen)function, <math>~\mathcal{F}_\sigma(x)</math> — and, simultaneously, the unknown value of the (eigen)frequency, <math>~\sigma</math> — that satisfies the following 2<sup>nd</sup>-order, ordinary differential equation:

<div align="center">
<math>
(1 - x^2) \frac{d^2 \mathcal{F}_\sigma}{dx^2} + \frac{4}{x}\biggl[1 - \frac{3}{2}x^2 \biggr] \frac{d\mathcal{F}_\sigma}{dx} + \biggl[3\sigma^2 - 2 \alpha \biggr] \mathcal{F}_\sigma = 0 .
</math><br />
</div>

===Try a Polynomial Expression for the Eigenfunction===
Let's ''guess'' that the proper eigenfunction is a polynomial expression in <math>~x</math>. Specifically, let's try a solution of the form,
<div align="center">
<math>\mathcal{F}_\sigma = a + bx + cx^2 + dx^3 + fx^4 + gx^5 + \cdots </math>
</div>
truncated at progressively higher- and higher-order terms.

====Lowest-order mode (Mode 0)====
Try,
<div align="center">
<math>\mathcal{F} = a \, ,</math>
</div>
in which case,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~\frac{d\mathcal{F}}{dx}</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~0 \, ,</math>
</td>
</tr>
</table>
</div>
and,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~\frac{d^2\mathcal{F}}{dx^2}</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~0 \, .</math>
</td>
</tr>
</table>
</div>
So, the governing 2<sup>nd</sup>-order ODE reduces to,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~(3\sigma^2 - 2 \alpha ) a</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~0 \, ,</math>
</td>
</tr>
</table>
</div>
which ''will'' be satisfied as long as, <math>~\sigma = (2\alpha/3)^{1/2} \, .</math> We conclude, therefore, that the eigenvector defining the lowest-order (the simplest) solution to the governing ODE has an eigenfunction given by,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~\mathcal{F}_0</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~a = \mathrm{constant} \, ,</math>
</td>
</tr>
</table>
</div>
with a corresponding eigenfrequency whose value is,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~\sigma_0</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\biggl(\frac{2\alpha}{3} \biggr)^{1/2} \, .</math>
</td>
</tr>
</table>
</div>

====Second Guess====
Try,
<div align="center">
<math>\mathcal{F} = a + bx \, ,</math>
</div>
in which case,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~\frac{d\mathcal{F}}{dx}</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~b \, ,</math>
</td>
</tr>
</table>
</div>
and,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~\frac{d^2\mathcal{F}}{dx^2}</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~0 \, .</math>
</td>
</tr>
</table>
</div>
Plugging this trial eigenfunction into the governing 2<sup>nd</sup>-order ODE gives,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~0</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\frac{4}{x}\biggl[1 - \frac{3}{2}x^2 \biggr] b + \biggl[3\sigma^2 - 2 \alpha \biggr] (a + bx)</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\frac{4b}{x} + (3\sigma^2 - 2 \alpha -6) bx + (3\sigma^2 - 2 \alpha ) a \, .</math>
</td>
</tr>
</table>
</div>
But this expression can be satisfied for all values of <math>~x</math> only if <math>~b = 0</math>, in which case the trial eigenfunction reduces to the earlier solution, <math>~\mathcal{F}_0</math>. We conclude, therefore, that our "second guess" does not generate a new solution to this eigenfunction problem.

====Third Guess====
Try,
<div align="center">
<math>\mathcal{F} = a + cx^2\, ,</math>
</div>
in which case,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~\frac{d\mathcal{F}}{dx}</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~2cx \, ,</math>
</td>
</tr>
</table>
</div>
and,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~\frac{d^2\mathcal{F}}{dx^2}</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~2c \, .</math>
</td>
</tr>
</table>
</div>
Plugging this trial eigenfunction into the governing 2<sup>nd</sup>-order ODE therefore gives,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~0</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~2c(1 - x^2) + 8c(1 - \frac{3}{2}x^2 ) + (3\sigma^2 - 2 \alpha ) (a + cx^2)</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~ \biggl[10c + (3\sigma^2 - 2 \alpha )a\biggr] + \biggl[ (3\sigma^2 - 2 \alpha ) -14 \biggr]cx^2 \, .</math>
</td>
</tr>
</table>
</div>
This relation will be satisfied for all values of <math>~x</math> if both expressions inside the square brackets are simultaneously zero, that is, if,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~10c + (3\sigma^2 - 2 \alpha )a</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~0 \, ,</math>
</td>
</tr>
</table>
</div>
and, simultaneously,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~(3\sigma^2 - 2 \alpha ) -14 </math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~0 \, .</math>
</td>
</tr>
</table>
</div>

* <font color="purple">Mode 1</font>:
: <math>x_1 = a + b\chi_0^2</math>, in which case,
<div align="center">
<math>
\frac{dx}{d\chi_0} = 2b\chi_0; ~~~~ \frac{d^2 x}{d\chi_0^2} = 2b;
</math>
</div>

<div align="center">
<math>
\frac{1}{(1 - \chi_0^2)} \biggl\{ 2b (1 - \chi_0^2) + 8b \biggl[1 - \frac{3}{2}\chi_0^2 \biggr] + A_1 \biggl(1 + \frac{b}{a}\chi_0^2 \biggr) \biggr\} = 0 ,
</math><br />
</div>
where,

<div align="center">
<math>
A_1 \equiv \frac{a}{\gamma_\mathrm{g}}\biggl[ \biggl( \frac{3}{2\pi G\rho_c} \biggr) \omega_1^2+ 2(4 - 3\gamma_\mathrm{g}) \biggr] .
</math>
</div>
Therefore,

<div align="center">
<math>
(A_1 + 10b) + \biggl[ \biggl(\frac{b}{a}\biggr) A_1 - 14b \biggr] \chi_0^2 = 0 ,
</math>
<br />
<br />
<math>
\Rightarrow ~~~~~ A_1 = - 10b ~~~~~\mathrm{and} ~~~~~ A_1 = 14a
</math>
<br />
<br />
<math>
\Rightarrow ~~~~~ \frac{b}{a} = -\frac{7}{5} ~~~~~\mathrm{and} ~~~~~ \frac{A_1}{a} = 14 = \frac{1}{\gamma_\mathrm{g}}\biggl[ \biggl( \frac{3}{2\pi G\rho_c} \biggr) \omega_1^2+ 2(4 - 3\gamma_\mathrm{g}) \biggr] .

</math>
</div>
Hence,
<div align="center">
<math>
\biggl( \frac{3}{2\pi G\rho_c} \biggr) \omega_1^2 = 20\gamma_\mathrm{g} -8
</math>
<br />
<br />
<math>
\Rightarrow ~~~~~ \omega_1^2 = \frac{2}{3}\biggl( 4\pi G\rho_c \biggr) (5\gamma_\mathrm{g} -2)
</math>
</div>
and, to within an arbitrary normalization factor,
<div align="center">
<math>
x_1 = 1 - \frac{7}{5}\chi_0^2 .
</math>
</div>

==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, [[SSC/Structure/Polytropes#n_.3D_1_Polytrope|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,
<div align="center">
<math>~\frac{\rho}{\rho_c} = \frac{\sin x}{x} \, ,</math>
</div>
where, <math>~\rho_c</math> is the density at the center of the star, and,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~x</math>
</td>
<td align="center">
<math>~\equiv</math>
</td>
<td align="left">
<math>~\pi\biggl(\frac{r}{R}\biggr) \, ,</math>
</td>
</tr>
</table>
</div>
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 2<sup>nd</sup>-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,
<div align="center">
<math>~\mathcal{G}_\sigma(x) \equiv \frac{\delta x}{x}</math>
</div>
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.



=Related Discussions=
* [[SSC/Perturbations#Spherically_Symmetric_Configurations_.28Stability_.E2.80.94_Part_II.29|Linear Stability Analysis]]
* [[SSC/Stability/UniformDensity#Uniform-Density_Configuration|Radial Pulsation Modes of Uniform-Density Spheres]]

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