User:Tohline/SSC/Stability/MoreGeneralApproach
More General Approach to the Parabolic Eigenvalue Problem
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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".
Material that appears after this point in our presentation is under development and therefore
may contain incorrect mathematical equations and/or physical misinterpretations.
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Introduction
In the case of a parabolic density distribution, the LAWE may be written in the form,
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<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>~ \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,
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<math>~\mathcal{G}_\sigma</math> |
<math>~=</math> |
<math>~(a_0 + a_2x^2)^n \cdot (b_0 + b_2x^2)^m \, ,</math> |
which implies,
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<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> |
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<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> |
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<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> |
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<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,
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<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> |
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<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> |
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<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> |
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<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:
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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> |
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<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> |
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<math>~ -\sigma^2 (a_0 + a_2x^2) (b_0 + b_2x^2) \biggr\} \, ;</math> |
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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> |
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<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> |
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<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> |
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<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> |
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<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,
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<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> |
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<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> |
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<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> |
Additional Setup
Benefitting from our earlier exploration of this problem, let's divide through by the product, <math>~(a_0 b_0)</math>, and introduce the new variable notations,
<math>~\lambda \equiv \frac{a_2}{a_0} \, ,</math> and <math>~\eta \equiv \frac{b_2}{b_0} \, .</math>
The LAWE becomes,
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<math>~ 0 </math> |
<math>~=</math> |
<math>~ \biggl[ \alpha(1 + \lambda x^2) (1 + \eta x^2) + 2x^2(n \lambda + m\eta ) + 2x^4 (n\lambda \eta + m\eta \lambda ) \biggr](5-3x^2) -\sigma^2 (1 + \lambda x^2) (1 + \eta x^2) </math> |
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<math>~ - \biggl[ n \lambda (1 + \eta x^2) + m \eta (1 + \lambda x^2) + 4(n \lambda + m\eta ) + 4(n\lambda \eta + m\eta \lambda )x^2+ 4n m \lambda \eta x^2 \biggr](1-x^2)(2-x^2) </math> |
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<math>~ - \frac{(1-x^2)(2-x^2)}{ (1 + \lambda x^2)(1 + \eta x^2)}\biggl[2n(n-1) \lambda^2(1 + \eta x^2)^2 + 2m(m-1) \eta ^2(1 + \lambda x^2)^2 \biggr]x^2 \, . </math> |
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© 2014 - 2021 by Joel E. Tohline |