Question

Please do all four parts for positive feedback. Solve the boundary-value problem for a string of unit length, subject to the given conditions.

1. u(0,t) = 0, u(1,t) = 0, u(x, 0) = f(x), ut(x, 0) = g(x) f(x) = 1/20 sin πx, g(x) = 0, α = 1/π
2. u(0,t) = 0, u(1,t) = 0, u(x, 0) = f(x), ut(x, 0) = g(x) f(x) = sin πx cos πx, g(x) = 0, α = 1/π
3. u(0,t) = 0, u(1,t) = 0, u(x, 0) = f(x), ut(x, 0) = g(x) f(x) = sin πx + 3 sin 2πx − sin 5πx, g(x) = 0, α = 1
4. g(x) = 0, α = 4, u(0,t) = 0, u(1,t) = 0, u(x, 0) = f(x), ut(x, 0) = g(x)

                                             f(x) = { 2x              0 ≤ x ≤ 1/2

                                                         2(1 − x) 1/2 ≤ x ≤ 1

Answer 1

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