[Numerical] System of first order ordinary diff equations with given asymptoticby ala Tags: numerical diff 

#1
Jul2909, 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? 



#2
Jul2909, 12:58 PM

P: 813

Okay, bellow is my attempt:
Let: [tex]y(x)=c(1w(x)exp(kx))[/tex] Then [tex] w(x)=(1y/c)exp(kt)[/tex] [tex] \dot{w}=\frac{\dot{y}}{c}exp(kx)+k \ (1y/c)exp(kx) [/tex] Now time to make some substitutions [tex] \dot{w}=\frac{f(y)}{c}exp(kt)+k \ (1\left(c(1w \ 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]. 



#3
Jul3009, 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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