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[Numerical] System of first order ordinary diff equations with given asymptotic

by ala
Tags: numerical diff
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ala
#1
Jul29-09, 03:44 AM
P: 23
I have system of first order ordinary diff equations, indipendent variable is x cordinate. I know asymptotic solution in left and right region (i.e. when x->-infinity or x->infinity, e.g. when abs(x)>1000), it's const plus exponentially falling function. I want to find numerical solution in middle region, witch will have given asymptotic in left and right region.

If I give initial value at left (i.e. at x=-1000) numerical solution blow up at right (I also have exponentially growing functions on right).

How to do this?
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#2
Jul29-09, 12:58 PM
P: 813
Okay, bellow is my attempt:

Let:

[tex]y(x)=c(1-w(x)exp(-kx))[/tex]

Then

[tex]
w(x)=(1-y/c)exp(kt)[/tex]
[tex]
\dot{w}=-\frac{\dot{y}}{c}exp(-kx)+k \ (1-y/c)exp(-kx)
[/tex]

Now time to make some substitutions
[tex]
\dot{w}=-\frac{f(y)}{c}exp(kt)+k \ (1-\left(c(1-w \ exp(-kx)) \right)/c)exp(kt)[/tex]
[tex]=-\frac{f(y)}{c}exp(kx)-w \ \left(1-\frac{k}{c}\right)exp(kx)+k \ w \ [/tex]
where [tex]y[/tex] is given above as:

[tex]y=c (1- w \ exp(-kx))[/tex]

and [tex]f(y)[/tex] is the original differential equation.

edit: The above only seems useful if [tex]\frac{1}{x}[/tex] is much bigger then [tex]k[/tex].
ala
#3
Jul30-09, 01:07 AM
P: 23
For that x, I have asymptotic solution. I want to find numerical solution in the middle, but don't know how. (I don't have 1 ODE, I have system of ODE)


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