Euler’s method question

roam

1. The problem statement, all variables and given/known data

I need some help with the last part of the following problem:




3. The attempt at a solution

My approximation to the solution to the IVP at t=-0.8 using 1 step of the Euler's method was:

x(-0.8)=0.8

Whereas the approximation with 1 step of 4th order Runge-Kutta method was:

x(-0.8)=0.8214

And since the exact solution is

[itex]x(-0.8) = e^{-0.8 +1} -2 \times (-0.8) -2 = 0.8214027582[/itex]

the error in Euler's method would be

[itex]|0.8214027582-0.8| =0.0214027582[/itex]

And the error for Runge-Kutta is

[itex]|0.8214027582-0.8214| =2.7582 \times 10^{-6}[/itex]

I'm stuck here. So how many steps does Euler's method take to produce an answer with an error no larger than 2.7582 x 10^-6 (the error of Runge-Kutta)?

I tried to use the following equation:

[itex]e_n \leq \frac{k}{n}[/itex]

Where k is a constant and n is the number of steps and e_n is the error. I then tried to solve for the constant bu substituting in the values from Euler's method:

[itex]0.021402758 = \frac{k}{1} \ \implies k =0.021402758[/itex]

Then substituting in the new error

[itex]2.7582 \times 10^{-6}=\frac{0.021402758}{n} \ \implies n = 7760[/itex]

But doesn't 7760 steps seem too much? Where did I go wrong? I appreciate it if anyone could help me with this problem.

rcgldr

7760 seems to be OK. You could confirm this using a program or a spreadsheet with Δt = (0.2 / 7760) to see if it corresponds with your answer.

roam

But I am wondering if I've even used the correct method for finding the number of steps?

rcgldr

What was the source of the error equation you used, class notes, a textbook, ... ?

roam

It's from a textbook...

Edit: page 635 of the textbook called "Differential Equations" by Blanchard, Devany and Hall.

rcgldr

It appears that you have. As mentioned before if you want to check this, you could confirm this using a program or a spreadsheet using Euler method with Δt = (0.2 / 7760) to see if it corresponds with your answer (for the spread sheet you would need to use 7761 rows, the initial state and 7760 steps).