Math 4100 - Fourier Analysis

Homework #1 - Due 1/25

1. Find the Fourier series of f(x) = |x|.

2. Find the Fourier series of f(x) = x^2.

3. Find the Fourier series of f(x) = 1 if x >0 and f(x) = -1 if x <0.

4. Find the Fourier series of f(x) = cos(ax) where a is NOT an integer.

Homework #2 - Due 2/1

1. Find the Fourier series of f(x) = cosh(x) = [e^x + e^(-x)]/2.

2. Find the Fourier series of f(x) = sinh(x) = [e^x - e^(-x)]/2.

Ex. 2.2 - 1b,c,d,f,2,7 (use the Fourier series for cosh(bx))

Ex. 1.3 - 1

Homework #3 - Due 2/8

1. Find a formal solution of u_t = u_xx subject to u_x(0,t)= 0 =u_x(pi,t) and u(x,0) = f(x).

2. Find a formal solution of u_t = u_xx subject to u(0,t)= 0 =u_x(pi,t) and u(x,0) = f(x).

3. Find a formal solution of u_tt = uxx subject to u(0,t)=0=u(pi,t), u(x,0)=f(x), u_t(x,0) = g(x).

Homework #4 - Due 2/15

Review problems on the handout from class.

Homework #5 - Due 2/29

Ex. 7.1 - 3a (just find f*f), 4

Ex. 7.2 - 2,6, 12a

Complete the proof that the integral from 0 to infinity of sin(x)/x is pi/2.

Homework #6 - Due 3/7

1. Show that v(x,t) = [1/sqrt(4pit)] e^{ -x^2/(4t)} satisfies the heat equation: v_t = v_xx.

2. Show that v(x,y) = {1/pi}{ y/(x^2 + y^2)} satisfies the Laplace equation: v_xx + v_yy = 0.

3. Solve u_t = u_xx for t > 0 and 0 < x < infty subject to u(x,0) = f(x) and u(0,t) = 0. (Extend f to be an odd function and then solve u_t=u_xx and u(x,0) = f(x). Then rewrite so that your answer is expressed in terms of integrals only defined on 0 < x < infty).

Ex. 7.3 - 1,3

Homework #7 - Due 3/14

1. Solve u_t = -u_xxxx subject to u(x,0)=f(x) by using the Fourier transform. Find the DE the Fourier transform satisfies and solve for it. You do NOT have to calculate the inverse Fourier transform.

2. Solve u_tt = -u_xxxx subject to u(x,0)=f(x) and u_t(x,0)=g(x) by using the Fourier transform. Find the DE the Fourier transform satisfies and solve for it. You do NOT have to calculate the inverse Fourier transform.

3. Solve u_t = u_xx where t > 0 and 0 < x < infty subject to u(x,0)=f(x) and u_x(0,t) = 0. (Extend f to be an even function and then solve u_t = u_xx and u(x,0) = f(x). Then rewrite so that your answer is expressed in terms of integrals only defined on 0 < x < infty).

4. Let f(x) = 0 if x < -3 and f(x) = e^(-x) if x > -3. Determine f*f.

5. K_t(x) = [1/sqrt(4pit)] e^{ -x^2/(4t)}. Show that K_t*K_s = K_(t+s).

Homework #8 - Due 4/4

1. Begin to solve u_t = u_rr + (1/r)u_r + (1/r^2)u_theta,theta for 0 2. Find the sum of the series: (1/2) + sum_{n=1}^{\infty}[ r^n cos(nt) ] and use it to obtain the Poisson integral formula found on page 119.

3. Let J solve: J'' + (1/r) J' + (1- (nu^2\r^2)) J = 0. Let G(r) = sqrt{r} J(r). Show that G satisfies: G'' + G + [(1/4) - nu^2]G/r^2 = 0.

Homework #9 - Due 4/11

1. Use separation of variables to find a formal solution of the following differential equation: u_t = u_rr + (1/r)u_r + (1/r^2)u_theta,theta with boundary conditions: u(1, theta, t) = 0, u(r,0,t)=0, u(r, alpha, t) =0 and u(r, theta, 0 ) = f(r, theta). Here alpha is a constant between 0 and 2 pi.

2. Using spherical coordinates (r,theta,phi) show that (grad r).(grad theta) = (grad r).(grad phi) = (grad theta).(grad phi) = 0. (Here a.b represents the dot product of a and b).

3. Also show that |grad phi|^2 = 1/r^2 and Laplacian(phi) = cos(phi)/(r^2 sin(phi) ).

Homework #10 - Due 4/18

1. Use separation of variables to find a formal solution of the following differential equation: u_t = u_rr + (1/r)u_r + (1/r^2)u_theta,theta with boundary conditions: u_r(1, theta, t) = 0, u(r, theta + 2pi,t) = u(r, theta,t), and u(r, theta, 0 ) = f(r, theta).

Homework #11 - Due 5/2

Ex. 6.2 - 3

Separate variables and find the differential equations satisfied by R,T, Theta if u satisfies: -iu_t = (1/2)(u_rr + (1/r)u_r + (1/r^2)u_theta\theta) + (1/r) u. You should be able to show that R'' + (1/r) R' + [lambda + (2/r)-(m^2/r^2)]R = 0. Let lambda = -b^2. Find the equation satisfied by w if R= e^(-br)w.