# Runge-Kutta method for ut = f(x,y)?

Hi,

I am trying to solve something similar to ut = f(x,y), and ut = f(t,x,y) using RK4. I asked a few friends, and nobody knew for sure how to go about it. I've also looked online, without much success. Can anyone give me a hint on this one?

In reality, it is probably preferable to derive a scheme from the first principle, but if I could avoid this, that would be nice.

Thanks.

## Answers and Replies

Integral
Staff Emeritus
Science Advisor
Gold Member
Sorry, I don't understand your notation.

R-K algorithms are for solving differential equations, is that what you have?

Sorry, how about something like the following?

$\frac{du}{dt} = x^2 + sin(y)$

Isn't this solved by using simultaneous RK4's?

I think it's supposed to be u'(t) = f(x,y) and u'(t) = f(t,x,y)? I don't understand the question clearly

^Thanks, I think that is probably what I was looking for!

One bonus question, while we're at it. Is there still value to using the RK method for the differential equation I listed in my previous post? That is, one with f(x,y), rather than f(t,x,y)?

Redbelly98
Staff Emeritus
Science Advisor
Homework Helper
One bonus question, while we're at it. Is there still value to using the RK method for the differential equation I listed in my previous post? That is, one with f(x,y), rather than f(t,x,y)?
If u, x, & y are all functions of t, then I'm pretty sure you need 3 differential equations to solve it. So additionally you need expressions for dx/dt and dy/dt. You can't solve a single differential equation with all those variables, by RK4 or any other means -- unless there is some relation between u, x, & y that you have omitted, or x and y are specified functions of t.

And if x and y are being treated as dependent variables, so u is supposed to be a function of x, y, & t, then RK4 won't work there either. In that case you need to go to a multivariable method, such as using difference equations or perhaps finite element methods.