| Topic: |
Science > Physics |
| User: |
"OsherD" |
| Date: |
02 Nov 2005 01:52:44 PM |
| Object: |
Riccati-izable and Painleve-izable Equations |
From Osher Doctorow
COPYRIGHT NOTICE
Riccati-izable and Painleve-izable Equations
Copyright By Owner Osher Doctorow Ph.D.
First Published 2005
I use the expression "-izable" to mean: "convertable to". So
Ricati-izable in my usage means "convertable to Riccati Differential
equation." The Painleve differential equations are closely related to
the Riccati Differential equation(s).
One of the best and most comprehensive (if not the most comprehensive)
treatment of such equations and others is P. L. Sachdev (Dept. Math.
Indian Institute of Science, India) A Compendium on Nonlinear Ordinary
Differential Equations, Wiley: N.Y. 1997.
The most interesting such equations, in my opinion, belong to the
following types:
A. y" + a(y' )^2 + g(x, y)y' + h(x, y) = 0
B. y" + k(y' )^2/y + g(x, y)y' + h(x, y) = 0
C. y" + f(y)(y' )^2 + g(x, y)y' + h(x, y) = 0
D. y" + f(x, y)(y' )^2 + g(x, y)y' + h(x, y) = 0
where various terms can be 0 or constant which among other things
results in considerable simplifications, or particular functions can
enormously simplify the forms indicated.
Osher Doctorow
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| User: "OsherD" |
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| Title: Re: Riccati-izable and Painleve-izable Equations |
02 Nov 2005 02:13:21 PM |
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From Osher Doctorow
Some really simplified equations of the above types which even simplify
to direct linearity mostly are:
1) y" + [y^2 + (y' )^2]y = 0
2) y" - y(y' )^2 = 0
3) y" - [a/(ay + b)](y' )^2 = 0
4) y" - (2/3)[1/y + 1/(y - 1)](y' )^2 = 0
5) y" + 2(y' )^2 tan(y) = 0
Another interesting type has e (meaning very small peturbation epsilon)
as a coefficient of y", and is still one of the main types listed last
time by multiplying through by e or 1/e or something similar.
One surprise is the equation:
6) y" - (y' )^2 - [2y^2 + a(x)](y' ) - y^4 - 2y^3 - b1(x)y^2 - b2(xy)y
- b3(x) = 0, a(x) and bj(x) holomorphic in a region
where the 1, 2, 3 in bj(x) are subscripts. Suitable transformations
and a particular choice of the bj(x) for example as 2 + a(x) and 1 +
a(x) respectively for j = 1, 2 yield the Riccati equation:
7) y' = -y^2 - y + constant
Osher Doctorow
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| User: "OsherD" |
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| Title: Re: Riccati-izable and Painleve-izable Equations |
02 Nov 2005 02:30:08 PM |
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From Osher Doctorow
Does energy have a tendency to expand/contract and to interact with
force? From indications that equations of the 4 types of this thread
often reduce to Riccati equations and the presence of y"
("acceleration" of y) and (y' )^2 ("kinetic" velocity or speed
squared), this is at least a possible Conjecture in the scenarios where
these equations hold. Maybe even my old WLF threads in sci.physics of
form E = w^a L^b F^c might benefit from such a Conjecture.
Osher Doctorow
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