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Re: Solving differential equations in the complex plane

  • To: mathgroup at smc.vnet.net
  • Subject: [mg103772] Re: [mg103768] Solving differential equations in the complex plane
  • From: Dan Dubin <ddubin at ucsd.edu>
  • Date: Mon, 5 Oct 2009 13:16:34 -0400 (EDT)
  • References: <200910051139.HAA28779@smc.vnet.net>

>Hi !
>How can I solve an ordinary differential equation of order n in the
>complex plane following a prescribed contour ?
>I can of course write my own Runge-Kutta package but is there a quickest
>way to do that (maybe NDSolve but how to define the contour ??) ?
>
>Example : NDSolve[{y'[x] == Exp[y[x]], y[1] == 1}, y, {x, 1, 3}]
>fails because of a singularity in x=1+1/e.
>However integrating the ODE following a path which avoids the
>singularity should be possible eventually leading to a multivalued function.
>
>Thanks for a hint.


Let x[s] be a complex path parametrized by a real variable s, for instance

x[s_] = 2 + Exp[-I s];

Then Re[x] runs from 1 to 3 as s runs from Pi to 0. This is the range 
of x you wanted. You can choose other paths if this one is not 
convenient.

Now transform the ODE in x to an ODE in s using the chain rule, and 
solve this ODE in s:

NDSolve[{y'[s]/x'[s]==Exp[y[s]],y[Pi]==1},y,{s,0,Pi}]

yields the function y[s] along the chosen contour.

-- 
---------------
| Professor Dan Dubin
| Dept of Physics , Mayer Hall Rm 3531,
| UC San Diego La Jolla CA 92093-0319
| phone (858) - 534-4174 fax: (858)-534-0173
| ddubin at ucsd.edu


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