Describe the general behavior of the solution over time

Question

Assignment:

1. Find the displacement u(x, t) of a string, given:

u_tt= 4u_xx,  0≤ x ≤ 2 , 0 ≤ t

u_x(0,t) = 0 , u_x(2, t) = 0

u(x, 0) = sin(Πx), u_t(x, 0) = 3

2. The series of functions below (where it converges) defines the function f (x):

ƒ(x) = _n=1 ∑^{∞ (-1)n cos(n xn)/ x2 + n2}

(a) For what values of x does the series converge? Justify your answer.

(b) Does the series converge uniformly where it converges? Justify your answer.

(c) Is f(x) continuous? Justify your answer.

(d) Is f (x) a periodic function? If so, what is the period?

(e) Is f(x) even, odd, or neither?

3. Given the two functions

h(t)= {t, -4 ≤ t ≤ 0

        0, t < -4 & t > 0}

g(t)=  {e-^2t, 2 ≤ t

          0, t < 2}

(a) Sketch h(t) and g(t).

(b) Compute (from the definition) the convolution: (h*g)(t). [Hint: carefully consider the different cases; sketches for various t may help.]

(c) Find 1/(ƒ), the Fourier Transform of h(t), by using the definition of the Fourier Transform (i.e., evaluate the integral).

(d) Find G(ƒ), the Fourier Transform of g(t), by using the Fourier tables and proper¬ties.

(e) Using the convolution theorem and your previous answers, what is the Fourier Transform of the convolution: ƒ {(h * g)(t)}?

SEE OTHER SIDE FOR REMAINING PROBLEMS

4. Given the following problem (a variant of the heat equation):

x² ∂u/ ∂t= x ∂/ ∂x (x ∂u/ ∂x)- u , 0 ≤ x ≤ 5 , 0 ≤ t

u(0,t) & u_x(0,t) finite , u(5,t) = 0

u(x,0) = ƒ(x) = x

(a) Separate Variables (with the usual separation constant -λ) to get 2 ODE's, for X(x) and T(t). (Check your work! This step is critical.)

(b) Show that the eigenvalue problem for X(x) is a Sturm-Liouville problem (what type?); identify the functions p(x), w(x), and q(x).

(c) Use equation (3.10.54) on page 122 (also called the "Rayleigh quotient") to show there are no negative eigenvalues. Is zero an eigenvalue? [Be sure to use the specific p(x), w(x), q(x) and Boundary Conditions for this problem.]

(d) Find the eigenvalues An and eigenfunctions X_n(x). [Hint: they can be expressed in terms of Bessel functions.: (If you get stuck on this step, just call the eigen¬functions X_n(x) and move on.)

(e) Finish solving the given problem, including finding all coefficients in terms of integrals.

(f) Describe the general behavior of the solution over time: what happens?

(g) [extra credit] Evaluate the integrals.