Problem 1:(75 pts) Consider the boundary value problem −u”(x)+u
0
(x)+u(x) = 3
on (0, 1) with u(0) = 1 and u(1) = 0. Divide [0, 1] into four subintervals of equal size
and apply the method of finite differences to set up the system of equations for the
unknown values.
2
Problem 2:(25 pts) Consider the boundary value problem −u”(x)+u
0
(x)+u(x) = 0
on (0, 1) with u(0) = 1 and u
0
(1) = 1. Divide [0, 1] into four subintervals of equal size
and apply the method of finite differences to set up the system of equations for the
unknown values. You may use part of the work done on Problem 1 for this question.
Problem 3:(100 pts) Consider the wave equation
utt(x, t) = 9uxx(x, t), 0 ≤ x ≤ 1, 0 ≤ t ≤ 1/8
u(0, t) = 0 and u(1, t) = 0, 0 ≤ t ≤ 1/8
u(x, 0) = sin(πx)
ut(x, 0) = 1.
Take k = 1/8 and h = 1/3. Find the finite difference solution using central difference
approximations in time and space.
Problem 4:(100 pts) Consider the boundary value problem −u”(x)+u
0
(x)+u(x) = 3
on (−1, 1) with u(−1) = 0 and u(1) = 0.
Divide (−1, 1) into two subintervals of equal size and apply the method of finite
elements to find the explicit formula for the piecewise linear approximation. You
must compute explicitly the integrals needed.
5
Problem 5:(100 pts)
Consider the boundary value problem −u”(x) + u
0
(x) + u(x) = 3 on (−1, 1) with
u(−1) = 0 and u
0
(1) = 0.
Divide (−1, 1) into two subintervals of equal size and apply the method of finite
elements to find the explicit formula for the piecewise linear approximation. You
must compute explicitly the integrals needed.
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