Solve Euler Integration for dx/dt = (x-at)/(x/a+t)

[Jimbo]

Hi

Im a bit stuck on the method for Euler Integration. I have the following first order differential equation:

dx/dt = (x-at) / (x / a+t)

where constant a = 1.0V/s, and initial condition x = 1.0V at t=0s

I have a time step of 0.02 and I need to calculate the output voltage at a time t = 0.2s

So I have the Euler Integration formula: x(t + ^t) = x(t) + ^t.f(x,t)
and I've started putting the values into an Excel spreadsheet, with columns as follows:

t | x(t) | f(x,t) | ^x = ^t.f(x,t)

I have values for all the first row for these columns but an example in a textbook has an additional column called x(exact) which appears to calculate an error value? I am not sure how to calculate this value, and am also unsure about how to move on with the iteration? What values should I move down for further rows?

I have searched the web but can't find any sites I can make sense of - can anybody help me? I hope my question makes sense...

Thanks

Jimbo :)

 

[quantumdude]

Hi Jimbo, and welcome to PF.

Originally posted by Jimbo
So I have the Euler Integration formula: x(t + ^t) = x(t) + ^t.f(x,t)
and I've started putting the values into an Excel spreadsheet, with columns as follows:

This bit I don't understand. Could you try to clear that up? If you are trying to do superscripts, do this:

x[ sup ]2[ /sup ]

with no spaces in the brackets. The result will be:

x²

Thanks,

 

[Jimbo]

Thanks

Hi

Thanks for your reply

Sorry about the unclear formula - the ^ were meant to be deltas to signify the small change in t

x(t + delta(t)) = x(t) + delta(t).f(x,t)

Hope that makes more sense now

Jimbo :)

 

[quantumdude]

Originally posted by Jimbo
x(t + delta(t)) = x(t) + delta(t).f(x,t)

OK, that helps. If you want to make it look prettier, check the Announcement at the top of the forum Howto: Making Math Symbols.

& Delta ;

without the spaces gives you:

Δ

Oooohhhh! Pretty!

Hope that makes more sense now

We're getting there, but I don't know what f(x,t) is. [?]

edit: typo

 

[Integral]

Here is what I have on Euler's Method,

Given
y' = f(x,y)
y(a)= y₀

For a solution over the interval [a,b]
Choose a step h=(b-a)/N. Set x_n= a + nh for n= 0,1,2...N

y_n+1 = y_n+ hf(x_n,y_n)

Error = h²y''(ξ)/2

Where x_n < &xi; < x_n+ h

I am haveing trouble intrepeting your notation, perhaps you can apply my notation to your problem.

Edit: Typo+ some content

 

[Integral]

I just reread your initial post, you did say EXACT and not error term. The ONLY way to generate the exact solution is to solve the DE. This solution is what Eulers, or any other numerical method, is approximating.

 

[gnome]

My textbook has an "exact" column too. That's not part of the Euler's method computation. It's there just for purposes of the example, to show you the amount of the error in the Euler approximation.

As Integral said, that value was obtained by solving the DE (i.e. by integration), then evaluating the solution for various values of t so that the error could be determined for each iteration.

 

[Jimbo]

Thanks

Hi

Thanks very much everyone for your help

I have it sorted now. gnome was correct about the exact column not being part of the calculation.

To move onto the next iteration I needed to add the value &Delta;(t).f(x,t) to x(t) to get a new value for x(t)

Thanks again

Jimbo :)