PatrickT

Dr. Patrick T

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15 years, 172 days

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I'd like to write a worksheet to check the validity (which I trust it must be, as it has been published in a reputable journal) and (above all) ease of implementability with Maple of Panayotounakos's solution of Abel ODEs.

A closed-form solution to the general Abel ODE should be an event celebrated with glee around the world: I'm a little puzzled by the absence of references to Panayotounakos's article, published in 2005 in Applied Mathematics Letters...

Here is a simple worksheet that follows the steps outlined in Panayotounakos, to the best of my understanding. The notation I have adopted below is mostly Panayotounakos's with some small changes. A summary is provided here: http://patrick.toche.free.fr/research/papers/abel.pdf

> restart: with(DEtools): with(plots): plotsetup(default):
> ode:= diff(y(x),x)*y(x)-y(x) = f(x); Abel ODE to be solved following Panoyotounakos's method
> f := x-> x;  Function on rhs of the Abel ODE to be specified. Simple cases are f(x)=0, f(x)=1, f(x)=x, etc
> A := 1:  Constant of integration to be specified. Will depend on (say) the given value of y(0)
> T := x-> ln(x+2*A):     change of variable from x to t
> X := t-> exp(t)-2*A:   reverse-change of variable from t to x
> F := t-> f(X(t)):      writing f as a function of t rather than x
> G := t-> ( ( t*sin(t) + cos(t) ) * Ci(t) + (cos(t))^(2)   ) * ( 4*t*Ci(t) + cos(t) ) / ( 2*(2*t*Ci(t))^(3) ) - 2*F(t); G(t);
> a := -4;
> b := t-> 3 + 4 * (G(t) + F(t)) * exp(-t);
> c := t-> -4 * (G(t) + 2*F(t)) * exp(-t);
> p := t-> b(t) - a^(2)/3;
> q := t-> c(t) - a*b(t)/3 + 2*(a/3)^(3);
> Cubic := r-> r^(3) + p*r + q;
> P := t-> p(t)/3; this normalization makes the cubic-root trigonometric formula cleaner
> Q := t-> q(t)/2; this normalization makes the cubic-root trigonometric formula cleaner
> R:= (P,Q,k)-> 2*(-P)^(1/2) * sin(( arcsin(Q/P^(2)*(-P)^(1/2)) +2*k*pi)/3); trigonometric formula for the cubic roots
> Y := (x,r) -> (1/2) * (x + 2*A) * (r + 1/3);  Panayotounakos formula for the solution y(x) in terms of the cubic-root-function r(x) defined above
> y := (x)-> Y(x,R(P(T(x)),Q(T(x)),0)): y(x); This is/should be the closed-form solution of y(x)
> py:= plot({y(x)}, x=-2..2, color=black): display({py});

 

P.S. I'm the PatrickT who has posted messages here in the past, for some reason my stars vanished after I updated my email...

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