Euler’s method question

  • Thread starter roam
  • Start date
  • #1
1,266
11

Homework Statement



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

http://img834.imageshack.us/img834/8366/eulere.jpg [Broken]


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 en 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.
 
Last edited by a moderator:

Answers and Replies

  • #2
rcgldr
Homework Helper
8,718
541
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.
 
Last edited:
  • #3
1,266
11
But I am wondering if I've even used the correct method for finding the number of steps?
 
  • #4
rcgldr
Homework Helper
8,718
541
I tried to use the following equation:
[itex]e_n \leq \frac{k}{n}[/itex]
But I am wondering if I've even used the correct method for finding the number of steps?
What was the source of the error equation you used, class notes, a textbook, ... ?
 
  • #5
1,266
11
What was the source of the error equation you used, class notes, a textbook, ... ?
It's from a textbook...

Edit: page 635 of the textbook called "Differential Equations" by Blanchard, Devany and Hall.
 
Last edited:
  • #6
rcgldr
Homework Helper
8,718
541
I am wondering if I've even used the correct method for finding the number of steps?
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).
 

Related Threads on Euler’s method question

  • Last Post
Replies
2
Views
2K
  • Last Post
Replies
1
Views
2K
  • Last Post
Replies
5
Views
1K
  • Last Post
Replies
2
Views
2K
  • Last Post
Replies
2
Views
2K
  • Last Post
Replies
2
Views
3K
  • Last Post
Replies
1
Views
1K
  • Last Post
Replies
6
Views
2K
  • Last Post
Replies
1
Views
1K
  • Last Post
Replies
4
Views
2K
Top