Math 257 – Assignment 6. Due: Wednesday, March 2
1. Let f (x) be a function on 0 ≤ x ≤ 2 define as
(
x, 0 ≤ x < 1 ,
f (x) =
1, 1 ≤ x ≤ 2.
(a) (2 points) Fourier sine series: Express f (x) as the series of sine functions.
(b) (2 points) Fourier cosine series: Express f (x) as the series of cosine functions plus a constant.
(c) (2 point) Convergence of Fourier series: In the above two series expressions ( in (a) and in
(b)), find where exactly (for which x) the equality holds between f (x) and the series. Justify
your answer. (For example, does the equality between f and the sine (or cosine) series hold at
x = 0 or x = 2? What about at other x?)
2. (3 points) Heat equations with zero-BC.
Consider the conduction of heat in a rod 4cm in length whose ends are maintained at 0◦ C for all t > 0.
Find an expression (in series form) for the temperature u(x, t) if the initial temperature distribution in
the rod is u(x, 0) = 50 (◦ C) for 0 < x < 4. Suppose the diffusion constant (“thermal diffusivity” in
the lecture notes) α2 = 1.
3. (3 points) Heat equation with Neumann boundary condition.
Find the solution of the heat conduction problem
ut = uxx ,
(0 < x < π, t > 0);
u(x, 0) = π − x,
ux (0, t) = 0 = ux (π, t),
(t > 0);
(0 < x < π).
4. Difference equation and numerical method.
(a) (2 points) Find the difference equation corresponding to the following (inhomogeneous) PDE:
ut = uxx + x
(1)
where u = u(x, t) is a function of two variables t and x.
Hint: For example, the difference equation corresponding to ut = uxx is
i
∆t h
u(x, t + ∆t) = u(x, t) +
u(x
+
∆x,
t)
−
2u(x,
t)
+
u(x
−
∆x,
t)
∆x2
(b) (6 points) Spreadsheet: Use spreadsheet to find an approximate value (up to 4 decimals) of
the function u(x, t) at (x, t) = (0.5, 0.3) where u(x, t) is the solution to the initial-boundary value
problem


ut = α2 uxx + x,
0<x<1

(2)
BC : u(0, t) = u(1, t) = 0,
t>0


2
IC : u(x, 0) = x ,
0 ≤ x ≤ 1.
Use the mesh size ∆x = 0.05 and ∆t = 0.004. Let α2 = 0.2. You must hand-in the print out of
the spreadsheet and the graph of (approximate solution) u(x, t) for three cases t = 0.02, t = 0.1,
t = 0.3. You can use any of the templates or examples given in the course webpage.