## Transcribed Text

1. Derive the following formula for approximating the derivative and show
that is o(h2) by establishing its error term:
3f(x)-4f(x-h)+j(x-2h) - -
j'(x) 2
2h
2. Consider the boundary value problem
IS" =0, 0Show that the problem has no solution if O # 8. and infinite many
solutions when O = 8.
3.
(a) Write a MATLAB function that implements the finite difference
method for solving the boundary value problem
(b) Use the function written in part (a) to solve the boundary value
problem
2.x
2
=
u(0) =0, u(1) = In(2).
The true solution is u(x)
Solve the boundary value problem for h = 0.1.0.05,0.025,0.0125,
and print the errors of the numerical solution at T = 0.2,0.4.0.6.0.8.
Comment on how errors decrease when h is halved.
Plot the true solution and the numerical solution for h = 0.05 on
the same plot.
4. The general solution of the equation
xu" - (2x + 1)z/ + (x+ 1)u = 0
is 21(I) = GIE* + Find the solution of the equation with the
boundary conditions
u '(0) 1, u'(1) =0
Write down a formula for a discrete approximation of the boundary
conditions. Implement the method by modifying the function in Prob-
lem 3, and solve the problem with h = 0.1,0.05,0.025,0.0125. Print
the errors in a loglog plot and comment on how the error decreases
when h is halved.

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