- #1
iamalexalright
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Homework Statement
In brief, I am working on a stellar model.
We started with these four equations:
[tex]\delta q/\delta x = px^{2}/t[/tex]
[tex]\delta p /\delta x = -pq/tx^{2}[/tex]
[tex]\delta t /\deltax = -Cp^{1.75} / x^{2}t^{8325}[/tex]
[tex]\delta t /\delta x = -2q/5x^{2}[/tex]
In the model you switch to the 4th equation instead of the 3rd when the ratio of the 3rd to the 4th is less than 1.
Because the variables p and t vary over orders of magnitude we switch to logarithmetic variables.
[tex]g_{1}=log(p)[/tex]
[tex]g_{2}=log(q)[/tex]
[tex]g_{3}=log(t)[/tex]
[tex]y=log(x)[/tex]
The 4 equations transform to:
[tex]log[-\delta g_{1}/\delta y] = g_{2} - g_{3} - y[/tex]
[tex]log[\delta g_{2}/\delta y] = g_{1} + 3y - g_{2} - g_{3}[/tex]
[tex]log[-\delta g_{3}/\delta y] = log(C) + 1.75g_{1} - y - 9.25g_{3}[/tex]
[tex]log[-\delta g_{3}/\delta y] = log(2/5) + g_{2} - g_{3} - y[/tex]
log(C) = -5.51674
The initial conditions are:
x = 1, q = 1, t = 0, p = 0
Since we encounter problems we use:
[tex]p^{1.75} = 1.75t^{8.25}/8.25C[/tex]
[tex]t = 1.75/8.25(1/x - 1)[/tex]
to find these initial values.
We then work on way inward (from x = 1 to x = 0)
2. The attempt at a solution
I worked out
g1 = -8.37512874
g2 = 1
g3 = -2.36361205
y = -0.00877392431
to be initial values.
Now I am having troubles doing the actual numerical integration...for instance if I try to find g1 at a point x-h(where h I have .02) i do this:
[tex]g_{1 at x-h} = g_{1 at x} + (1/6)(k_{1} + 2k_{2} + 2k_{3} + k_{4})}[/tex]
Now [tex]k_{1}[/tex] is simply the derivative of [tex]g_{1}[/tex] at x.
For [tex]k_{2}[/tex] I use [tex]x + .5hk_{1}[/tex] instead of x in the derivative equation(and the same method for k3 and k4).
Is this correct? Also I don't know what to do about that log outside(meaning, do I raise 10 by the right-hand side to get the actual derivative and not the log of it?).
Thanks in advance