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Re: NDSolve::ndsz question

• To: mathgroup at smc.vnet.net
• Subject: [mg63674] Re: NDSolve::ndsz question
• From: BGHEkaya at gmail.com
• Date: Mon, 9 Jan 2006 04:49:27 -0500 (EST)
• References: <dpnqoa$6rh$1@smc.vnet.net>
• Sender: owner-wri-mathgroup at wolfram.com

Flip, The  easiest way to extend the integration using NDSolve is to
work with exact arithmetic. Thus you can readily solve for x=400 with
high accuracy

κ = 400000000;
β = 23/10000;
ϵ = 24/1000;

sol = NDSolve[{y'[x] == -κ
    x^(-1/2)\[ExponentialE]^(-ϵ x)(
            y[x]^2 - (β x^2BesselK[2, x])^2), y[1/10] ==
             β BesselK[2, 1/10]/(1/10)},
              y, {x, 1/10, 2000}, WorkingPrecision -> 30,  MaxSteps ->
Infinity]

For large x the solution to the ODE is

y[x]~-(Sqrt[ϵ]/(Sqrt[ϵ]*C[1] + Sqrt[Pi]*κ*
     Erf[Sqrt[x]*Sqrt[ϵ]]))

and thus as x->Infinity, y[x] tends to a constant value given by

-(Sqrt[ϵ]/(Sqrt[ϵ]*C[1] + Sqrt[Pi]*κ))

We can use the numerical value of y[400] to evaluate the constant of
integration C[1]

I found this value to be

5.824503762*^-10

By increasing the accuracy of NDSolve you can improve on this value.

Hope this helps,

Cheers,

Brian