Build a program that utilises the "leapfrog" method of numerical integration

Question

unklerhaukus · Answer

The equation for the leapfrog method are given as:
$$x_{n+1} = x_n+v_{n+1/2}\delta t\v_{n+1/2} =v_{n-1/2}+a_n\delta t\v_{1/2} = v_0+\tfrac12a_0\delta t$$

unklerhaukus · Answer

The Non-linear equation is $$\frac{\mathrm d^2 x}{\mathrm dt^2}=-\omega_0^2x+\beta x^2$$

Initial conditions are $x_0 = A$,   $v_0=0$,   $a_0=-\omega_0^2x_0+\beta x_0^2$

$\omega_0=2\pi$,     $T=2\pi/\omega_0$

unklerhaukus · Answer

This is what i have at the moment, but it is clearly not quite right

unklerhaukus · Answer

attachment

e.mccormick · Answer

matlab?

unklerhaukus · Answer

yes MATLAB.

unklerhaukus · Answer

@dan815

dan815 · Answer

ok

dan815 · Answer

okay lemme get wht u are trying to do so we can code this fast

dan815 · Answer

can u explain leapfrop method with a picture

unklerhaukus · Answer

the green line should be more like this 

|dw:1417060575412:dw|

dan815 · Answer

k so what is leapfrop integration though

dan815 · Answer

is it like i just take a look at the slope, see where i end up with a linear estimation plot point there and continue

dan815 · Answer

ud have to do 2 approximations for first integration and then 2nd

dan815 · Answer

dy/dx numerical approxximation, then y

dan815 · Answer

right?