User:Tohline/SSC/Stability/MoreGeneralApproach

More General Approach to the Parabolic Eigenvalue Problem

The material presented in this chapter is an extension of the chapter titled, Other Analytic Models and could also be considered to be a subsection of the associated chapter titled, Other Analytic Ramblings. More specifically, in the following "Introduction," we repeat a manipulation of the LAWE that was originally developed in the subsection of that chapter titled, "Consider Parabolic Case".

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Introduction

In the case of a parabolic density distribution, the LAWE becomes,

<math>~\frac{2x^2(5-3x^2)}{(1-x^2)(2-x^2)} \biggl[\alpha - \sigma^2 \biggl(5-3x^2\biggr)^{-1} + \frac{x \mathcal{G}_\sigma^'}{\mathcal{G}_\sigma} \biggr]</math>	<math>~=</math>	<math>~ \biggl(\frac{x^2 \mathcal{G}_\sigma^{' '}}{\mathcal{G}_\sigma}\biggr) +4 \cdot \frac{x \mathcal{G}_\sigma^'}{\mathcal{G}_\sigma} </math>
<math>~\Rightarrow ~~~~ \frac{2}{(1-x^2)(2-x^2)} \biggl[ \biggl( \alpha + \frac{x \mathcal{G}_\sigma^'}{\mathcal{G}_\sigma}\biggr)(5-3x^2) -\sigma^2 \biggr]</math>	<math>~=</math>	<math>~ \biggl(\frac{\mathcal{G}_\sigma^{' '}}{\mathcal{G}_\sigma}\biggr) +\frac{4}{x^2} \cdot \frac{x \mathcal{G}_\sigma^'}{\mathcal{G}_\sigma} </math>

Let's try,

<math>~\mathcal{G}_\sigma</math>

<math>~=</math>

<math>~(a_0 + a_2x^2)^n \cdot (b_0 + b_2x^2)^m \, ,</math>

which implies,

<math>~\mathcal{G}_\sigma^'</math>	<math>~=</math>	<math>~n(a_0 + a_2x^2)^{n-1}(2a_2x) \cdot (b_0 + b_2x^2)^m +m (a_0 + a_2x^2)^n \cdot (b_0 + b_2x^2)^{m-1}(2b_2x)</math>
<math>~\Rightarrow ~~~~ \frac{x \mathcal{G}_\sigma^'}{\mathcal{G}_\sigma}</math>	<math>~=</math>	<math>~n(a_0 + a_2x^2)^{-1}(2a_2x^2) +m (b_0 + b_2x^2)^{-1}(2b_2x^2) </math>
	<math>~=</math>	<math>~\frac{2x^2}{(a_0 + a_2x^2) (b_0 + b_2x^2)} \biggl[ n a_2 (b_0 + b_2x^2) +mb_2 (a_0 + a_2x^2) \biggr] </math>
	<math>~=</math>	<math>~\frac{2x^2}{(a_0 + a_2x^2) (b_0 + b_2x^2)} \biggl[ (n a_2 b_0 + mb_2 a_0) +(na_2 b_2+ mb_2 a_2)x^2\biggr] \, ,</math>

and,

<math>~\mathcal{G}_\sigma^{' '}</math>	<math>~=</math>	<math>~n m (a_0 + a_2x^2)^{n-1}(2a_2x) \cdot (b_0 + b_2x^2)^{m-1}(2b_2x) + n(a_0 + a_2x^2)^{n-1}(2a_2) \cdot (b_0 + b_2x^2)^m + n(n-1)(a_0 + a_2x^2)^{n-2}(2a_2x)^2 \cdot (b_0 + b_2x^2)^m </math>
		<math>~+m n(a_0 + a_2x^2)^{n-1}(2a_2x) \cdot (b_0 + b_2x^2)^{m-1}(2b_2x) +m (a_0 + a_2x^2)^n \cdot (b_0 + b_2x^2)^{m-1}(2b_2) +m(m-1) (a_0 + a_2x^2)^n \cdot (b_0 + b_2x^2)^{m-2}(2b_2x)^2</math>
<math>~\Rightarrow ~~~~ \frac{\mathcal{G}_\sigma^{' '}}{\mathcal{G}_\sigma}</math>	<math>~=</math>	<math>~8n m a_2b_2 x^2 (a_0 + a_2x^2)^{-1}\cdot (b_0 + b_2x^2)^{-1} + n2a_2 (a_0 + a_2x^2)^{-1} + n(n-1)4a_2^2 x^2 (a_0 + a_2x^2)^{-2} +m2b_2 (b_0 + b_2x^2)^{-1} +m(m-1)4 b_2^2 x^2 (b_0 + b_2x^2)^{-2}</math>
	<math>~=</math>	<math>~\frac{2n a_2(b_0 + b_2x^2) + 2m b_2 (a_0 + a_2x^2)}{ (a_0 + a_2x^2)(b_0 + b_2x^2)} + \biggl[ \frac{4n(n-1) a_2^2 }{ (a_0 + a_2x^2)^{2}} + \frac{8n m a_2b_2}{ (a_0 + a_2x^2)(b_0 + b_2x^2)}+ \frac{4m(m-1) b_2^2 }{(b_0 + b_2x^2)^{2}} \biggr]x^2 </math>

So, we have for the LAWE:

LHS	<math>~=</math>	<math>~ \frac{2}{(1-x^2)(2-x^2)} \biggl[ \biggl( \alpha + \frac{x \mathcal{G}_\sigma^'}{\mathcal{G}_\sigma}\biggr)(5-3x^2) -\sigma^2 \biggr]</math>
	<math>~=</math>	<math>~ \frac{2}{(1-x^2)(2-x^2)(a_0 + a_2x^2) (b_0 + b_2x^2)} \biggl\{ \biggl[ \alpha(a_0 + a_2x^2) (b_0 + b_2x^2) + 2x^2(n a_2 b_0 + mb_2 a_0) + 2x^4 (na_2 b_2+ mb_2 a_2) \biggr](5-3x^2) </math>
		<math>~ -\sigma^2 (a_0 + a_2x^2) (b_0 + b_2x^2) \biggr\} \, ;</math>
RHS	<math>~=</math>	<math>~ \biggl(\frac{\mathcal{G}_\sigma^{' '}}{\mathcal{G}_\sigma}\biggr) +\frac{4}{x^2} \cdot \frac{x \mathcal{G}_\sigma^'}{\mathcal{G}_\sigma} </math>
	<math>~=</math>	<math>~\frac{2n a_2(b_0 + b_2x^2) + 2m b_2 (a_0 + a_2x^2)}{ (a_0 + a_2x^2)(b_0 + b_2x^2)} + \biggl[ \frac{4n(n-1) a_2^2 }{ (a_0 + a_2x^2)^{2}} + \frac{8n m a_2b_2}{ (a_0 + a_2x^2)(b_0 + b_2x^2)}+ \frac{4m(m-1) b_2^2 }{(b_0 + b_2x^2)^{2}} \biggr]x^2 </math>
		<math>~ + \frac{8}{(a_0 + a_2x^2) (b_0 + b_2x^2)} \biggl[ (n a_2 b_0 + mb_2 a_0) +(na_2 b_2+ mb_2 a_2)x^2\biggr] </math>
	<math>~=</math>	<math>~\frac{1}{(a_0 + a_2x^2)(b_0 + b_2x^2)} \biggl\{ 2n a_2(b_0 + b_2x^2) + 2m b_2 (a_0 + a_2x^2) + 8(n a_2 b_0 + mb_2 a_0) + 8(na_2 b_2+ mb_2 a_2)x^2 </math>
		<math>~ + \biggl[8n m a_2b_2+ \frac{4n(n-1) a_2^2(b_0 + b_2x^2) }{ (a_0 + a_2x^2)} + \frac{4m(m-1) b_2^2(a_0 + a_2x^2) }{(b_0 + b_2x^2)} \biggr]x^2 \biggr\} \, . </math>

Putting these together gives,

<math>~ 0 </math>	<math>~=</math>	<math>~ \biggl[ \alpha(a_0 + a_2x^2) (b_0 + b_2x^2) + 2x^2(n a_2 b_0 + mb_2 a_0) + 2x^4 (na_2 b_2+ mb_2 a_2) \biggr](5-3x^2) -\sigma^2 (a_0 + a_2x^2) (b_0 + b_2x^2) </math>
		<math>~ - \biggl[ n a_2(b_0 + b_2x^2) + m b_2 (a_0 + a_2x^2) + 4(n a_2 b_0 + mb_2 a_0) + 4(na_2 b_2+ mb_2 a_2)x^2+ 4n m a_2b_2x^2 \biggr](1-x^2)(2-x^2) </math>
		<math>~ - \frac{(1-x^2)(2-x^2)}{ (a_0 + a_2x^2)(b_0 + b_2x^2)}\biggl[2n(n-1) a_2^2(b_0 + b_2x^2)^2 + 2m(m-1) b_2^2(a_0 + a_2x^2)^2 \biggr]x^2 \, . </math>

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