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

  1. Jul 29, 2009 #1

    ala

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    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?
     
    ala, Jul 29, 2009
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  3. Jul 29, 2009 #2

    John Creighto

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    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].
     
    Last edited: Jul 29, 2009
    John Creighto, Jul 29, 2009
  4. Jul 30, 2009 #3

    ala

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    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)
     
    ala, Jul 30, 2009
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