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Third order differential equation numerical approximation

  1. Nov 13, 2008 #1
    1. The problem statement, all variables and given/known data
    There is a fluid flowing over a hot plate. We non-dimensionalized the problem from three partial diff eq's to two ode's. I am modeling I have two coupled differential equations that are a system of initial value problems. I am supposed to numerically integrate the two equations to come up with values... I'm then told to simplify this to a larger system of first order ode's. I'm not sure how to do this...


    2. Relevant equations
    I'm given F''' +1/2(F*F'')=0 and G''+Pr/2(F*G')=0 where Pr is the prandle number. F(0)=0 F'(0)=0 F'(infinity)=1 G(0)=1 G(infinity)=0


    3. The attempt at a solution

    I know I'm supposed to guess F'' and G' to get F' and G' to be what I want them to asymptote to. I just am not sure how to get to a place where I can use something like Runge-Kutta 4th order method...

    My attempt was U1=F1 U2=F' U1'=U2=F' U2'=U3=F'' U2''=F''' U2''+1/2(U1*U2')=0

    then

    V1=U1 V2=U2' V2'=U2'' and V2'+1/2(V1*V2)=0

    But I don't know if this is right...

    Any help is appreciated thanks,

    Matt
     
  2. jcsd
  3. Nov 14, 2008 #2

    HallsofIvy

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    Staff Emeritus
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    To use Runge-Kutta, you need a system of first order differential equations. You don't want U2"+ (1/2)(U1*U2')= 0, you want U3'+ (1/2)(U1*U3)= 0
    Your 3 equations are
    U1'= U2
    U2'= U3 and
    U3'+ (1/2)(U1*U3)= 0.

    Surely you don't mean this! What happened to G?
    Let V1= G, V2= V1'= G', and V3= V2'= G". Then
    V1'= V2
    V2'= V3 and
    V3'= Pr/(2*U2*V2)

     
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