## Heat equation and Wave equation problems

1.
Solve the Heat equation u_t = ku_xx for 0 < x < ∏, t > 0 with the initial condition

u(x, 0) = 1 + 2sinx

and the boundary conditions u(0, t) = u(∏, t) = 1

(Notice that the boundary condition is not homogeneous)

3.
Find the solution of the Wave equation u_tt = 4 u_xx with

u(0, t) = u (π/2, t) = 0, t > 0

u(x, 0) = sin (2x) - 2sin(6x), u_x (x, 0) = -3 sin4x, 0 ≤ x ≤ ∏/2

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Try a Fourier series solution of the form $$\sum A_n(t) sin(nx/4)+ B)n(t)cos(nx/4)$$