Indicate whether each of the statements below are true or fa

Indicate whether each of the statements below are true or false. No justification is needed. Let u(t, x) be the solution to the wave equation IVP: u_tt - u_xx = 0, u(0, x) = 0, u_t(0, x) Then: lim |x| rightarrow + infinity u(t, x) = 0. Let u(t, x) be a solution to the heat equation (with convection) obeying periodic boundary conditions on the interval [0, pi]: u_t - u_xx + u_x = 0, u(t, 0) = u(t, pi), u_x(t, 0) = u_x(t, pi). Define the total heat to be: H(t) = u(t, x) dx. Then, for u(t, x) the total heat is conserved in the sense that: H(t) = H(0).

Solution

a)Yes,We can also solve the previous example using D’Alembert’s solution. The problem has zero initial velocity and its initial displacement has already been expanded into the required Fourier sine series.

b)Yes, the periodic boundary conditions u(L,t) = u(L,t) and ux(L,t) = ux(L,t) for all t 0.

In this case the eigenvalue problem takes the form ( X + X = 0 X(0) = X(L), X (0) = X (L)

c)Note that the Neumann boundary conditions are symmetric, because for all functions f, g such that satisfies the above given relation.

d) and (e)

An orthogonal system {_n(x)}n0 on [a, b] is complete if the fact that a function f(x) on [a, b] satisfies (f, _n) = 0 for all n 0 implies that f 0 on [a, b],

The system {1, cos(x), cos(2x), cos(3x), . . . } = {cos(k x), k 0} is orthogonal on [, ] but it is not complete on [, ]

So (d) and (e) are not true.