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Second ODE, initial conditions are zeros at infinity

  1. Jul 5, 2012 #1
    second ODE, initial conditions are zeros at infinity!!

    I want to know the temperature profile of phase transition layer in the interstellar medium.
    For stationary solution, the dimensionless differential equation I ended up with is

    [tex]\frac{d^2T}{dx^2} = \frac{f(T)}{T^2} - \frac{1}{T} [/tex]
    where [itex]f(T)[/itex] is some complicated but well-behaved function.
    Boundary conditions are
    [tex]T(x=-\infty) = 1 ,[/tex]
    [tex]\frac{dT}{dx}(x=-\infty) = 0,[/tex]

    However, [itex]f(T=1) = 1[/itex], one obtains [itex]\frac{d^2T}{dx^2}(x=-\infty) = 0[/itex]

    How do I solve this numerically? where should I start the integration? and what should be the initial condition?
    Do I need to Taylor expand the differential equation?

    Thank you for your attention.
     
  2. jcsd
  3. Jul 5, 2012 #2

    MathematicalPhysicist

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    Gold Member

    Re: second ODE, initial conditions are zeros at infinity!!

    One approach is to multiply by dT/dx this eqaution to get:

    [itex]1/2 d/dx (dT/dx)^2 = (f(T)/T^2-1/T)dT/dx = (f(T)-T)/T^2 dT/dx[/itex]

    Now perhaps use an iterating scheme of recursion eqautions, and integrate from minus infinity to some arbitrary x.

    Hope I helped somehow.
     
  4. Jul 5, 2012 #3
    Re: second ODE, initial conditions are zeros at infinity!!

    hmm..can you elaborate on the iterating scheme of recursion eq. or share some links I can refer to?
     
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