Solving Collocation Problems with Cubic Polynomial Approximations

[squenshl]

Homework Statement

Approximate y by a cubic polynomial on each of the sub-intervals [0,1/2] & [1/2,1] and then solve the collocation problem.
Approximate y by a cubic polynomial on each of the sub-intervals [0,1/4], [1/4,1/2], [1/2,3/4], [3/4,1] and then solve the collocation problem.

Homework Equations

The Attempt at a Solution

I have written the files below. Also:

function yp = f(x,y)
yp = 3/2*y.^2;

The fd_jac(x) file as in my other example.

I have an example of a function y = r(a)

function y = r(a)
x = [0.2 0.4 0.6 0.8]';
x0 = 0; xf = 1;
y0 = 4; yf = 1;

length(a)
y(1) = y0 - p(a,x0);
y(2:5) = ppp(a,x) - f(x,p(a,x));
y(6) = yf - p(a,xf);

y = y';

Please help.

 

[mighty2000]

Hi there,

Thank you for your forum post. It seems like you are working on solving a collocation problem using a cubic polynomial approximation. In order to solve this problem, you will need to follow a few steps:

1. Divide the interval [0,1] into smaller sub-intervals. In your first attempt, you have divided it into two sub-intervals [0,1/2] and [1/2,1]. In your second attempt, you have divided it into four sub-intervals [0,1/4], [1/4,1/2], [1/2,3/4], [3/4,1].

2. Approximate y by a cubic polynomial on each sub-interval. This means that for each sub-interval, you will have to find the coefficients of a cubic polynomial that best fits the data points.

3. Solve the collocation problem. This involves using the cubic polynomial approximation to solve a system of equations. The equations will be based on the boundary conditions and the continuity of the polynomial at the points where the sub-intervals meet.

I see that you have already written the files for the function and the fd_jac(x). Now, you will need to write a function for the cubic polynomial approximation. You can use the polyfit function in MATLAB to find the coefficients of the polynomial that best fits the data points.

Once you have the coefficients, you can use them to solve the system of equations in the collocation problem. This will give you the values of y at the points where the sub-intervals meet, which will then allow you to plot the cubic polynomial approximation for the entire interval [0,1].

I hope this helps. Good luck with your work!