Hi all, I'm hopeful someone can help you with this issue I am having in implementing a Maple procedure. I use to have a strong knowledge of the programme, but haven't used Maple in a couple of years and I appear to be more rusty than I recall, I keep hitting dead ends with multiple assignments, missing arguements and so forth, so I was hoping someone can help me smooth the edges!
The problem is as follows:
q1'' - f1(q1,q2) = - q1 / (q1^2 + q2^2) ^ (3/2)
q2'' = f2(q1,q2) = - q2 / (q1^2 + q2^2) ^ (3/2)
Initial conditions: q1(0) = 1-e, q2(0)=0, q1'(0) = 0, Q2'(0) ' sqrt( (1+e) / (1-e))
I have derived a system of 4 first order ODE's from this, these are as follows:
p1 = q1'
p2 = q2'
p1' = q1/((q1^2+q2^2)^(3/2))
p2' = - q1/((q1^2+q2^2)^(3/2))
With initial conditions:
p1(0) = 0
p2(0) = Sqrt((1 + e) / (1- e))
q1(0) = 1 - e
q2(0) = 0
I need to create a procedure that computes a solution that uses the explicit midpoint rule. I've managed to derive the following:
Gn = -(q1^2,n + q2^2,n)-(3/2)
G~n+1 = - (q1^2~,n + q2^2~,n)-(3/2)
q1,n+1 = q1,n + hp1~,n+1
q2, n+1 = q2,n + hp2~, n+1
p1,n+1 = p1,n + hgn~ + 1q1~,n+1
p2, n+1 = p2,n + hgn~ + q2~,n+1
q1~,n+1 = q1,n + h/2 p1,n
q2~, n+1 = q2,n + h/2 p2,n
p1~,n+1 = p1,n + h/2 gnq1,n
p2~, n+1 = p2,n + h/2 gnq2,n
Sorry it's incredibly longwinded, don't really expect much of a response! But I've tried implemented this procedure in maple, and I just can't seem to arrive at a suitable set of arrays.
Thanks for any help.