In this video I'll describe a method used
to solve ordinary differential equations and this is a better approximation than the Euler
Method which is described in another of our videos. The Runge-Kutta Method is a fourth
order approximation. The idea is we have initial conditions, namely at a certain location, a certain
value of x we know the value of y and then we have an ordinary differential equation
that relates how y changes as we change x and that's a function of x and y. So what
we're going to do in the Runge-Kutta Method is use a recurrence formula and say if I know
a value of y, then I can make a small step and the step is to say x_(n + 1) is equal
to x_n + h so we make a small step in h and calculate the new value of y that corresponds
to the x_(n + 1) and we have this term here that's a function of x_n and y_n and the size
of step. Smaller steps will give us more accurate calculations and of course this is something
that in a computer program can very easily be done to step through. Programs can be more
sophisticated than this but this gives you a good idea of how these programs work. So
what I'm going to do is write down the formula for T4 and then discuss what the terms are
in it. The T4 is 1/6 times the summation of the 6 terms essentially. k_1 is just the function
evaluated at x and y so at (x_n, y_n) for a given condition. k_2 is a function evaluated
but now x plus 1/2 the step size and y plus 1/2 the step size multiplied by the function,
k_1. k_3 is very similar except it's k_2 here and then k_4 is evaluated at the full step
size and k_3 is here. And so this gives us a better approximation than, say, the Euler
Method, which we just use the function at the previous point and the forth order approximation
means if we were to decrease the step size so instead if h, we used h over 2 as our step
size then we reduce the error by a factor of 2^5. We can get fairly accurate measurements
using this Runge-Kutta. So what I want to do next is apply this formula to a simple
differential equation and look at the first few steps to make it clear how we're doing
these calculations on a computer. So let's look at this example. I've written a simple
differential equation we could certainly solve this analytically but what we are interested
in is the technique. We're going to assume that at x_0 equals 1, y_0 equals 2. So
this is our initial condition. Then we use a step size to calculate y at a larger value
of x. So I've written down the recurrence formula and then I've substituted in the numbers.
y_0 is 2 the step size here is 0.1 and then k_1 plus 2k_2 plus 2k_3 plus k_4 and so we need
to calculate the values. k_1 is just the function evaluated at x_0 and y_0 which is then 3x_0^2
times y_0. So that's k1 and similarly we calculated the other values, k 2, 3, and 4. So I've written
the recurrence formula term now for k_2 so it's evaluated x_0 which is 1 plus the step size
0.1 divided by 2 and then again the step size over here 0.1, and then now the value of k_1
that we just calculated. So we have the value for k_2 and we do similar things to calculate
k_3 and k_4. So here are the values calculated for k_3 where we're using now k_2 that we calculated
in the previous step and we get the value of k_3, and to calculate k_4, now we use step
size h so this is 1.10 and we're using k_3 in this calculation. Once we have these four
values we can calculate y_1. So y_1 is y_0 plus the step size h over 6 k_1 plus 2 k_2 plus 2
k_3 plus k_4. So I'll substitute these values in. So substituting in the values we calculated
for k_1 , k_2, k_3, and k_4 we get y_1 that corresponds to an x_1 value of 1.10. Well this is much more accurate than the Euler Method calculation. Then
we continue the step through we get y_2 as y_1 plus hT_4 evaluated at x_1 and y_1, and the same
idea, we can calculate the function as we move through where k_1 in this calculation is a
function now of x_1 and y_1 the values we just calculated. This is then the technique that's
used in numerical programs like POLYMATH to solve ordinary differential equations and
we can certainly solve simultaneously a number of ordinary differential equations with this
technique.