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.
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.
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.
Sample Solution