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Supplementary Problems

Exercise 150. Solve the heat equation with convection: u − ku xx + V u = 0
t
x
for −∞ < x < ∞ with u(x, 0) = φ(x), where V is a constant. (Hint: Go to a
moving frame of reference by substituting y = x − V t.)

−x
Exercise 151. Solve u = ku ; u(x, 0) = e ; u(0, t) = 0 on the half-line
xx
t
0 < x < ∞.
Exercise 152. Solve u = ku ; u(x, 0) = 0; u(0, t) = 1 on the half-line 0 <
xx
t
x < ∞.

Exercise 153. Derive the solution formula for the half-line Neumann problem
w − kw xx = 0 for 0 < x < ∞, 0 < t < ∞; w (0, t) = 0; w(x, 0) = φ(x).
x
t
Exercise 154. Consider the following problem with a Robin boundary condition:

DE : u = ku xx on the half-line 0 < x < ∞(and0 < t < ∞)
t

IC : u(x, 0) = x for t = 0 and 0 < x < ∞

BC : u (0, t) − 2u(0, t) = 0 for x = 0.
x
The purpose of this exercise is to verify the solution formula for (*). Let f(x) = x
for x > 0, let f(x) = x + 1 for x < 0, and let

1 Z ∞ 2
v(x, t) = √ e −(x−y) /4kt f(y)dy.
4πkt −∞

(a) What PDE and initial condition does v(x, t) satisfy for −∞ < x < ∞?
(b) Let w = v − 2v. What PDE and initial condition does w(x, t) satisfy for
x
−∞ < x < ∞?
0
(c) Show that f (x) − 2f(x) is an odd function (for x 6= 0).
(d) Deduce that v(x, t) satisfies DE, IC, BC for x>0. Assuming uniqueness,deduce
2
that the solution of (*) is given by u(x, t) = √ 1 R ∞ e −(x−y) /4kt f(y)dy.
4πkt −∞
Exercise 155. Use the method of Exercise 154 to solve the Robin problem:

DE : u = ku xx on the half-line 0 < x < ∞ (and 0 < t < ∞)
t
IC : u(x, 0) = x for t = 0 and 0 < x < ∞

BC : u (0, t) − hu(0, t) = 0 for x = 0,
x
where h is a constant. Generalize the method to the case of general initial data

φ(x).

Exercise 156. Solve the inhomogeneous diffusion equation on the half-fine with

Dirichlet boundary condition: u − ku xx = f(x, t) (0 < x < ∞, 0 < t < ∞)
t
u(0, t) = 0 u(x, 0) = φ(x) using the method of reflection.

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