Problem 1. In this problem we will consider an initial-boundary-value problem (IBVP) for the heat operator of the form

{ut = uxx   for t > 0, 0 < x < Π
{u(t, 0) = 0 = ux(t, π)   for t ≥ 0                                                   (1.1)
{u(0, x) = f(x)              for 0 ≤ x ≤ Π,

where k is a positive constant and f is a given function.

(a) Let a < b be real numbers. Recall that the L2((a, b)) inner product is defined for real-valued functions f, g : (a, b) → R by

(f,g)_L2((a,b)) = ∫_a^b f(x)g(x)dx.

Show that the spatial operator -d²/dx² acting on smooth functions X : [a, b] →R for which both X(a) = 0 and X'(b) = 0 is symmetric with respect to the L²((a, b)) inner product.

(b) One of the major ideas we have seen related to solving IBVP's is the importance of the eigenpairs for the spatial operator. The eigenproblem for the spatial operator corresponding to IBVP (1.1) is

{-Xⁿ = λX    for 0 < x < Π
X(0) = 0 = X'(Π)

where X = X(x) is a function of the single spatial variable x. In view of part (a), we are guaranteed that all eigenvalues in this eigenproblem are real. Solve this eigenproblem.

(c) Let λn, Xn be the eigenpairs you found in part (b)1. Show that the eigenfunctions Xn are orthogonal with respect to the L²((0, π)) inner product. For every n, compute the square L2((0, π))-length of Xn. That is, compute

(Xn, Xn)L²((0,π))

for every n.

(d) Suppose f : [0, π] → R admits a representation of the form

f(x) = _n∈J∑Xa_nX_n,

where J ⊂ Z is the index set to which n belongs2 and an ∈ R are coefficients. Use the orthogonality relations in part (c) to derive a formula for each an. Your formula for an should involve f and the eigenfunctions found in part (b). Compute the coefficients an in the special case that f(x) = x.

(e) Find a series representation for a solution u to problem (1.1) in the case that f(x) = x (I am not asking you to address convergence issues here). Hint: Take advantage of the work you have already done in parts (a) - (d).

Problem 2. Consider the non-homogeneous IBVP with Dirichlet data

{u_t - u_xx = F(t, x)    for t > 0, 0 < x < π
{u(t, 0) = 0 = u(t, π)  for t ≥ 0
u(0, x) = 0 for 0 ≤ x ≤ π.

We saw in class that the solutions to the (spatial) Dirichlet eigenproblem

{-d²/dx2 X = λX for 0 < x < π
{X(0) = 0 = X(π)

are
λn = n² , X_n(x) = sin(nx) for n = 1, 2, 3, · · · .

Assume that the non-homogeneity F admits a representation of the form

F(t, x) = _n=1∑^∞ F_n(t)X_n(x)                                        (2.2)

for some functions Fn depending only on time. Assume also that the solution u(t, x) to (2.1) admits a representation of the form

u(t, x) =  _n=1∑^∞T_n(t)X_n(x)                                        (2.3)

for some functions T_n depending only on time.

(a) Find ordinary differential equations (one for each n) that when satisfied by T_n guarantee that u(t, x) as given in equation (2.3) is a solution to the PDE in (2.1).

(b) Using standard ODE techniques, derive formulas (one for each n) for the general solutions to the ODE's of part (a). Your formulas should be of the form

T_n(t) = C_nK_n(t) + ∫_o^t K_n(t - s)F_n(s) ds                         (2.4)

for some constants Cn (which will be determined in part (c)) and some functions K_n depending only on time (which you need to determine).

Note: I am looking for the derivation of formula (2.4). Do not simply reverse-engineer this formula to see which Kn works.

(c) Use the initial data given in (2.1), formulas (2.4) (with Kn identified) and the form of u in equation (2.3) to determine the values of the constants Cn for n = 1, 2, 3, · · · . Write an updated formula for the solution u of (2.1).

(d) The function F(t, x) = t admits an expansion3 of the form in equation (2.2). Find the corresponding coefficients Fn(t) then use the formula for u obtained in part (c) to find the solution4 to (2.1) with F(t, x) = t.