Third order differential equation numerical approximation

[blue2004STi]

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

[HallsofIvy]

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
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.

then

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

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

Any help is appreciated thanks,

Matt