Numerical Analysis: Solving y0 = Ly; y(0) = 1 w/ Methods I & II

In summary, the conversation discusses a problem involving solving a differential equation numerically using two different methods. The solution to the equation is verified as y(t) = e^(Lt) and the two methods are described as Method I and Method II. The conversation then goes on to discuss computing the truncation error for both methods, obtaining expressions for y_(n+1) as a function of y0, and computing the limit as n goes to infinity for y_(n). The question of whether the methods converge is also raised. Finally, the conversation includes a request for help in solving the problem and determining which method performs better.
  • #1
proplaya201
14
0

Homework Statement



Consider the equation
y0 = Ly; y(0) = 1:
**L = lamda**
Verify that the solution to this equation is y(t) = e^(Lt). We want to solve this equation numerically to obtain an approximation to y(1). Consider the two following methods to approximate
the solution to this equation:
Method I: y_(n+1) = y_(n) + hf(y_(n+1); t_(n+1))
Method II:y_(n+1) = y_(n) +(h/2)*(f(y_(n); t_(n)) + f(y_(n+1); t_(n+1)))

(a) Compute the truncation error for both methods. Which one is more accurate?
(b) i was able to do this one!
(c) Obtain an expression for y_(n+1) as a function of y0 for methods I and II.
(d) Using the expressions obtained in the previous part, compute lim (as n goes to inf) y_(n)
(Remember: h = 1/n).
(e) In view of the results in the previous part, do the methods converge?
(f) Assume that h = :1, and L = -1000. Plot y0, y1, y2, y3, and y4 for both
methods. Plot the function y(t) = e^(Lt). In view of these plots, which method performs
better?

i am just soo lost when it comes to solving this problem. i have idea how to approach it. any help with this will be greatly appreciated!
 
Physics news on Phys.org
  • #2
proplaya201 said:

Homework Statement



Consider the equation
y0 = Ly; y(0) = 1:
**L = lamda**
Surely this isn't what you meant. Did you mean y'= Ly?

Verify that the solution to this equation is y(t) = e^(Lt). We want to solve this equation numerically to obtain an approximation to y(1). Consider the two following methods to approximate
the solution to this equation:
Method I: y_(n+1) = y_(n) + hf(y_(n+1); t_(n+1))
Method II:y_(n+1) = y_(n) +(h/2)*(f(y_(n); t_(n)) + f(y_(n+1); t_(n+1)))

(a) Compute the truncation error for both methods. Which one is more accurate?
(b) i was able to do this one!
(c) Obtain an expression for y_(n+1) as a function of y0 for methods I and II.
(d) Using the expressions obtained in the previous part, compute lim (as n goes to inf) y_(n)
(Remember: h = 1/n).
(e) In view of the results in the previous part, do the methods converge?
(f) Assume that h = :1, and L = -1000. Plot y0, y1, y2, y3, and y4 for both
methods. Plot the function y(t) = e^(Lt). In view of these plots, which method performs
better?

i am just soo lost when it comes to solving this problem. i have idea how to approach it. any help with this will be greatly appreciated!
 
  • #3
yes that is what i meant, sorry, working on this for a long time... kinda burnt out
 

What is Numerical Analysis?

Numerical Analysis is a branch of mathematics that focuses on developing efficient algorithms for solving mathematical problems that are too complex to be solved by hand or with traditional methods.

What is the main goal of Numerical Analysis?

The main goal of Numerical Analysis is to find numerical solutions to mathematical problems, especially those that cannot be solved analytically. It involves using computers and algorithms to approximate solutions and obtain accurate results.

What is the difference between Methods I and II in solving y0 = Ly; y(0) = 1?

Method I, also known as the Euler method, is a basic numerical method that uses a first-order approximation to solve initial value problems. Method II, also known as the Runge-Kutta method, is a more accurate and efficient method that uses higher-order approximations to solve initial value problems.

How do these methods work to solve y0 = Ly; y(0) = 1?

Both methods involve breaking down the problem into smaller steps and using iterative calculations to approximate the solution. Method I uses a linear approximation while Method II uses a combination of linear and quadratic approximations to achieve a more accurate result.

What are the advantages of using numerical methods to solve mathematical problems?

Numerical methods allow us to solve complex mathematical problems that would otherwise be impossible to solve by hand. They also provide a way to check the accuracy of solutions obtained using traditional methods. Additionally, numerical methods can be easily automated and can handle a wide range of problems, making them a valuable tool for scientists and engineers.

Similar threads

  • Calculus and Beyond Homework Help
Replies
2
Views
657
Replies
7
Views
1K
  • Calculus and Beyond Homework Help
Replies
1
Views
1K
  • Calculus and Beyond Homework Help
Replies
6
Views
5K
  • Quantum Physics
Replies
1
Views
525
Replies
6
Views
1K
  • Calculus
Replies
1
Views
956
  • Calculus and Beyond Homework Help
Replies
2
Views
1K
  • Calculus and Beyond Homework Help
Replies
2
Views
1K
  • Calculus and Beyond Homework Help
Replies
6
Views
2K
Back
Top