Re: solving differential equations

• To: mathgroup at smc.vnet.net
• Subject: [mg3544] Re: solving differential equations
• From: Robert Knapp <rknapp>
• Date: Thu, 21 Mar 1996 01:10:25 -0500
• Organization: Wolfram Research, Inc.
• Sender: owner-wri-mathgroup at wolfram.com

```Steven Wilkinson wrote:
>
> Mathgroup,
>
> I have a system of ODE I wish to solve numerically of the form X'[t] =
> F(X[t])/|F(X[t])| for a given function
>         F : plane -> plane
> and given intial conditions. NDSolve works well on this except where the
> system is ill-behaved, in particular, where F = 0. I can replace this
> system with another one that has equivalent solutions and is usually
> well-behaved where F = 0. However, I don't want to make the replacement
> unless I have to, and only near where F = 0. (I wish to construct a command
> that gives the solution for an inputted function F.)
>
> Is there any way possible to do this with NDSolve? In other words, as it
> steps along have it check the value of F. If its magnitude ever gets too
> close to 0, have it use the other system.
>
> If that is not possible, does anyone have a solver package that I could
> modify to do the above?
>
> Thanks,
> Steve Wilkinson
> Dept of Math & CS
> Northern Kentucky University
> wilkinson at nku.edu

The easiest thing to do is to redefine the function so that you change
to the equivalent solution only when F is near zero.   Below is a very
simple example of how you can do this.

In[1]:=
NDSolve[{x'[t] == t/Sin[t],x[0] == -1},x,{t,0,1}]
Infinity::indet:
Indeterminate expression 0. 1. ComplexInfinity
encountered.
NDSolve::ndnum:
Differential equation does not evaluate to a number
at t = 0..

Out[1]=
{{x -> InterpolatingFunction[{0., 0.}, <>]}}

In[2]:=
Series[t/Sin[t],{t,0,3}]

Out[2]=  (* find an approximation near the removeable singularity *)
2
t        4
1 + -- + O[t]
6

In[3]:=  (* define the function, so that the approximation is used
near the removable singularity.  100 \$MachineEpsilon was
an arbitrary choice, but I checked it with a plot to
see that it matched the approximation well *)

f[t_] := If[Abs[Sin[t]] < 100 \$MachineEpsilon,
1+t^2/6,t/Sin[t]]

In[4]:=
NDSolve[{x'[t] == f[t],x[0] == 1},x,{t,-1,1}]

Out[4]=
{{x -> InterpolatingFunction[{-1., 1.}, <>]}}

Note that you case is slightly different since your right hand side has
the function with singularity depending on the dependent variable.  You
can still use a similar definition and use f[x[t]] instead of just f[t].

Rob Knapp
Wolfram Research, Inc.

http://www.wri.com/~rknapp

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