Science:Math Exam Resources/Courses/MATH307/December 2012/Question 02 (b)

MATH307 December 2012

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Question 02 (b)

The boundary value problem ${\begin{aligned}&f''(x)+xf(x)=1,\quad 0<x<1\\&f(0)=1,\quad f'(1)=1\end{aligned}}$ can be approximated by an (N + 1) x (N + 1) system of linear equations of the form $(L+(\Delta x)^{2}Q)\mathbf {F} =\mathbf {b}$ (b) How would you use MATLAB/Octave to compute approximations to $\displaystyle f(1/2)$ and $\displaystyle f'(1/2)$? Assume that N has been defined and write code that uses this value of N.

Make sure you understand the problem fully: What is the question asking you to do? Are there specific conditions or constraints that you should take note of? How will you know if your answer is correct from your work only? Can you rephrase the question in your own words in a way that makes sense to you?

If you are stuck, check the hint below. Consider it for a while. Does it give you a new idea on how to approach the problem? If so, try it!

Hint

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Solution

Found a typo? Is this solution unclear? Let us know here. Please rate my easiness! It's quick and helps everyone guide their studies. Set dx, L, Q and b in Matlab to be equal to values above: dx = 1/N L = 1/dx2*[[dx2 0 0 0]; diag(ones(3),-1) + diag(-2*ones(3)) + diag(ones(3),1); [0 0 -dx dx]] for m = 1:N qVector(m) = m-1 end qVector(N+1) = 0 Q = 1/dx*diag(qVector) f = (L + dx2*Q) Find index of value corresponding that has x closest to 1/2: Find points corresponding to x values on either side or equal to 1/2 and approximate f(1/2) as point on line joining both points: Get f(1/2): xi = floor((1/2)/dx) xf = ceil((1/2)/dx) m = (f(xf) - f(xi))/dx b = f(x0) - m(x0) ans = m(1/2) + b