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Re: Re: How to NDSolve the differential equation

• To: mathgroup at smc.vnet.net
• Subject: [mg19656] Re: [mg19625] Re: How to NDSolve the differential equation
• From: "Carl K.Woll" <carlw at fermi.phys.washington.edu>
• Date: Tue, 7 Sep 1999 00:28:45 -0400
• Organization: Department of Physics
• References: <7qulo3$2r5@smc.vnet.net> <199909060820.EAA04788@smc.vnet.net.>
• Sender: owner-wri-mathgroup at wolfram.com

Kevin & Chee,

An alternate possibility in Kevin's second approach is to use an auxilliary
function which handles the special case x=0. separately, as follows:

In[16]:=
f[0.] := 1
f[x_?NumericQ] := Sin[x]/x

In[18]:=
NDSolve[{y''[x] + f[x]*y'[x] +
     2*y[x] == 0, y[0] == 0, y'[0] == 1}, y,
  {x, 0, 2}]

Out[18]=
{{y -> InterpolatingFunction[{{0., 2.}}, <>]}}

Carl Woll
Physics Dept
U of Washington

"Kevin J. McCann" wrote:

> Chee Lim,
>
> Look at your DE when x=0 with your IC.  The whole thing is zero.  When this
> happens NDSolve can't get any traction on the problem.  Maybe you could try
> something like this instead:
>
> NDSolve[{x y''[x] + Sin[x] y'[x] + 2 x y[x] == 0, y[0.001] == 0,
>     y'[0.001] == 1}, y,
> {x, 0.001, 2}]
>
> Alternatively, divide through by x and solve this DE:
>
> y''[x] + y'[x] Sin[x]/x + 2 y[x] == 0
>
> again you will have problems at x=0, this time because of the Sin[x]/x; so,
> why not expand it:
>
> s[x_] = Normal[Series[Sin[x]/x, {x, 0, 12}]]
>
> This gives about 9-place accuracy over the [0,2] range.  Then:
>
> q=y/.NDSolve[{y''[x] + y'[x] s[x]+ 2 y[x] == 0
> , y[0] == 0,
>     y'[0] == 1}, y,
> {x, 0, 2},WorkingPrecision->20][[1]]
>
> This gives a solution which satisfies the original DE to about 7 places.
>
> Plot[x q''[x] + q'[x]Sin[x] + 2x q[x], {x, 0, 2},
>   PlotRange -> {-0.000001, 0.000001},
>   PlotStyle -> RGBColor[1, 0, 0]];
>
> Kevin
>
> Chee Lim Cheung <cheelc at mbox2.singnet.com.sg> wrote in message
> news:7qulo3$2r5 at smc.vnet.net...
> > Dear Mathematica gurus & users,
> >
> > I encountered error messages with the tag Power::infy when I tried to do
> > the following:
> >
> > NDSolve[{x y''[x] + Sin[x] y'[x] + 2 x y[x] == 0,y[0]==0, y'[0]==1}, y,
> > {x,0,2}]
> >
> > Can anyone help me in getting Mathematica to produce an answer? I am using
> > Mathematica 4.0 & Mathematica 3.0.
> >
> > Thanks
> > Chee
> >

• References:
  • Re: How to NDSolve the differential equation
    • From: "Kevin J. McCann" <kevinmccann@Home.com>