Sign in to follow this  

Two point boundary value problem NEW EXPLANATION ****

This topic is 4304 days old which is more than the 365 day threshold we allow for new replies. Please post a new topic.

If you intended to correct an error in the post then please contact us.

Recommended Posts

I am trying to use the secant method to obtain a numerical solution to a two point boundary problem. Here is the differential equation - x' = f(t, x) = x + 0.09 * 2 + cos(10 t) and here is the boundary condition - x(0) + x(1) - 3.0 = 0 I also have the algorithm for the secant method which looks like this xk+1 = x - (f(xk)(xk - xk-1))/(f(xk) - f(xk-1)) Whereby k denotes the current iteration k-1 denotes the previous and k+1 denotes the next(I think). Also, f is the differential equation above (I think). Could somebody confirm that this is correct so far? Then I am asked, using 0.7 and 1.0 as initial guesses for the value of x(0) to find an approximation for x(0) (for x(t)) such that the boundary condition is satisfied to within a tolerance of 10-4. Could somebody explain what this tolerance of 10-4 means? Then I am instructed to used a fixed stepsize of 0.025 for t between 0 and 1. So am I to assume that I do the following: 1. Work out x(1) using the boundary condition and an intial guess (lets say 0.7). 2. Use x(0) and x(1) as x(k-1) and x(k) in the secant algorithm to calculate x(k+1). 3. Increment t by 0.025 and carry out step 2 again etc. Does this make sense to anyone? I can't figure what I am supposed to be doing here. Here is the actual assignment incase I was unclear anywhere. ************************************************ Construct a computer program that uses both the secant method and the Runge-Kutta method (that you developed in assignment #3) to obtain a numerical solution to the two-point boundary-value problem: x' = f(t,x) = x + 0.09 x 2 + cos(10 t) differential equation x(0) + x(1) - 3.0 = 0 boundary condition Starting with the initial guesses 0.7 and 1.0 for the (unknown) initial value, x(0), obtain an approximation to x(0) {for the final solution, x(t)} such that the boundary condition is satisfied to within a tolerance of 10-4 . Use a fixed stepsize of 0.025 (i.e., take 40 steps each time you integrate the differential equation from t=0 to t=1). Write your program so that the output shows the values of x(0), x(1), and x(0)+x(1)-3 (the error in satisfying the boundary condition) at the end of each iteration of the secant method. After the last iteration of the secant method, re-integrate from t=0 to t=1 and print out the solution for x(t) over the range [0,1]. ************************************************ Thanks for any help, I am completely dumbfounded by this one. Mark [Edited by - mrmrcoleman on February 28, 2006 10:02:52 AM]

Share this post


Link to post
Share on other sites

This topic is 4304 days old which is more than the 365 day threshold we allow for new replies. Please post a new topic.

If you intended to correct an error in the post then please contact us.

Guest
This topic is now closed to further replies.
Sign in to follow this