Math 307: Problems for section 1.3 1. Write down the vector approximating f (x) at interior points, the vector approximating xf (x) at interior points, and the nite dierence matrix equation for the nite dierence approximation with N = 4 for the dierential equation f (x) + xf (x) = 0 for 1 x 3 subject to f (1) = 5, f (3) = 1. 2. Write down the matrix equation to solve in order to nd the nite dierence approximation with N = 4 for the same dierential equation f (x) + xf (x) = 0 for 1 x 3 but now subject to f (1) = 5, f (3) = 1 3. Use MATLAB/Octave to solve the matrix equations you derived in the last two problems for the vector F that approximates the solution (i.e., with N = 4). Then redo the calculation with N = 50 and plot the resulting functions. 4. a) What are the solutions to the dierential equation f (x) = 0? Let us compare this to the solutions to the nite dierence approximation. Recall that we represent the discrete approximation to the derivative on the interval [0, 1] with the N by N + 1 matrix 1 DN x 1 1 0 1 1 . .. = . . x . 0 0 0 1 .. 0 0 . 0 0 .. .. . . 1 1 0 1 0 0 . . . 0 1 where x = 1/N . Let F = [F0 , F1 , . . . , Fn ] be our discrete representation of f . What are the solutions to the matrix equation 1 DN F = 0? x 1 b) Now let's add a boundary condition and change the equation. What is the solution to the dierential equation f (x) = 1, f (0) = 10? What is the solution to matrix equation 1 DN F = [1, 1, . . . , 1] , x F0 = 10? Recall that we use Fi as a discrete approximation of f (xi ) = f (i/N ). How does the approximation compare to the real thing? 2