Question

I need some help with numerical analysis:

I have 2 PDEs and I need to use the Runge-Kutta method of the 4th order. I'm not asking for the algorithm, I just need a bit of guidance for how to start. here are the PDEs:

(d^2r)/(dt^2) = -k/m*(r-L)-g*cos(phi)+r*[(dphi/dt)^2]

and

(d^2phi)/(dt^2) = g/r*sin(phi)-[2*(dphi/dt)*(dr/dt)]/r

while g,m,k,L are given

Answer 1

what makes these equations PDEs? they seem like ODEs to me...

Answer 2

@oldrin.bataku there are deriv wrt phi ANDr

Answer 3

for RK4 solving PDEs you typically have to discretize the domain yeah?

Answer 4

U(r,phi) while you use the derivative of phi and r in the same equation

Answer 5

no, @CutieSwag ...

Answer 6

I don't even see U in this system. are you sure this isn't just a typical coupled pair of ODEs?

Answer 7

I think I need to do an order reduction for the equations and then write the algorithm in a way that all 4 are connected, if that makes any sense

Answer 8

break it into a coupled system of four equations:$$\frac{d\phi}{dt}=u\\\frac{dr}{dt}=v\\\frac{d^2r}{dt^2}=\dots\text{using }u,v\dots\\\frac{d^2\phi}{dt^2}=\dots\text{using }u,v\dots$$or something to that effect

Answer 9

ok that might work, I need to think of it a bit, to see if it might help me to write the equations like this, but I also have to go to class. if you think of any thing else I'll be more than happy to listen

Answer 10

$$\frac{d\phi}{dt}=u\\\frac{dr}{dt}=v\\\frac{d^2\phi}{dt^2}=\frac1r\left(g\sin\phi-2uv\right)\\\frac{d^2r}{dt^2}=-\frac{k}m(r-L)-g\cos\phi+ru^2$$or something like that

Answer 11

then you evolve them each step independently

Answer 12

oh and I got what you were saying about the ODE , I think you are right. the derivative of all of them it t

Answer 13

so in that case treat \(\dfrac{du}{dt}=\dfrac{d^2\phi}{dt^2},\dfrac{dv}{dt}=\dfrac{d^2r}{dt^2}\)

Answer 14

as in singular derivative

Answer 15

right that's what I figured

Answer 16

ty :)

Answer 17

so now it's a coupled system of differential equations... sorry if I can't help much with the mechanical computation I usually throw it in mathematica

Answer 18

you'd be given the initial conditions:$$r(0)=r_0,\phi(0)=\phi_0,u(0)=\frac{d\phi}{dt}\Biggr|_{t=0},v(0)=\frac{dr}{dt}\Biggr|_{t=0}$$and you'd compute \(u(t),v(t),r(t),\phi(t)\) each step as a coupled system of 4 first-order ODEs

Answer 19

yeah that was the idea, and the computing is easy once you know how to put the equations into the method. thats why I ask only for a bit of guidance for how to start.

Answer 20

glad I could help :p

Answer 21

question: I use \[\frac{ dr }{ rt }=v\] in order to get a vector of the Vs or Rs? Vs, right??