Runge kutta method for solving PDE

[kthouz]

Hi!
If you don't see clearly this n terms please download the word file attached here.
Am a given a problem like f(x,y,y')= y''= x+y with y(0)=0, y'(0)=1 and h=0.1 and i want to solve it using Ringe Kutta.
As we know y_(n+1)= y_n + h(y'_n+(A_n + B_n + C_n)/3)
And y'_(n+1) = y'_n+(A_n + 2B_n + 2C_n + D_n)/3
x_(n+1)=x_0+(n+1)h
where
k=h/2
A_n=kf(x_n,y_n,y'_n)
$$\beta$$_n= k(y'_n+(1/2)A_n) and so for
B_n=kf(x_n +k,y_n + $$\beta$$_n,y'_n + A_n)
C_n= kf(x_n + k, y_n + $$\beta$$_n, y'_n + B_n)
$$\delta$$_n=h(y'_n + C_n)
D_n=kf(x_n +h, y_n + $$\delta$$_n,y'_n + 2 C_n)
My problem is this:
For that given function(above) we can see that the y'_n ,i.e the third term in the brackets, is useless in the funtion. So i all the calculations the third term will be neglected. First tell me if i am right or not!
I did like that but during the class, the lecturer took the third term into account and add it to both x + y with reasons that only y'_n is zero but the third term will not be always zero because it contains the sums of A_n which is not zero.

 

[gtoguy97]

So my question is that if the third term is zero or not. And in the Ring Kutta Method why do we need to use this third term?Yes, it is correct that you can neglect the third term in the function f(x,y,y'). This is because the y'_n term will always be zero, since it is the derivative of y at the current time step. The A_n, B_n, C_n and D_n terms are necessary components of the Runge-Kutta method, as they are used to calculate the numerical approximation of the derivatives in the function. As such, these terms must be included in the calculation of the solution.