User:Tohline/SSC/Stability/MoreGeneralApproach
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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,



Let's try,



which implies,












and,












So, we have for the LAWE:
LHS 








RHS 














Putting these together gives,









Additional Setup
Benefitting from our earlier exploration of this problem, let's divide through by the product, , and introduce the new variable notations,
and
The LAWE becomes,









Multiplying through by the denominator of the last term(s) — that is, multiplying through by — will give us a polynomial with coefficient expressions for 6 terms expressed in terms of 5 unknowns .
Wouldn't a better strategy be to insert yet another quadratic factor — specifically, — which will introduce two additional unknowns but only add one more term into the polynomial expression? This would bring the total number of coefficient expressions to 7 while simultaneously raising the number of unknowns to 7. It will be tedious and messy, but worth the try.
Expanding from Two to Three Quadratic Terms
Here we rearrange terms in the "parabolic" LAWE to construct the governing ODE as,



Let's try,



or, in an effort to permit writing more compact expressions,



where,
and
This implies (after some whiteboard derivations),









Specific Values of Quadratic Coefficients
Now, if we assume that,
and
the "parabolic" LAWE becomes,



Then, plugging in the expressions for and its derivatives, we have,





































































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