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MATH704 Linear Partial Differential Equations. Assignment 2
Due on Monday 30 March 2020, 4 pm

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MATH704. Linear Partial Differential Equations. 2020.
Assignment 2
Question 1. The distribution of heat in a metal rod is modelled by the following nonhomogeneous
heat equation:
ut = 2uxx + 3 + 4 cos x
Solve the equation if the initial temperature u(x, 0) = 5 + 6 cos x + 7 cos 2x, and the ends of
the rod are insulated:
ux(0, t) = ux(π, t) = 0, t > 0.
Question 2. Solve the following initial-boundary value problem modelling the vibration of a
string with length L = 1 and fixed ends.
utt = uxx + xu(x, 0) = 3π3sin(πx)ut(x, 0) = x
u(0, t) = u(1, t) = 0, t > 0.
Question 3. Solve the following initial-boundary value problem modelling the vibration of a
string with length L = 2π and fixed ends.
utt = 4uxx + 2π − xu(x, 0) = 7 sinut(x, 0) = 0
u(0, t) = u(2π, t) = 0, t > 0.
Question 4. The following non-homogeneous Laplace equation (Poisson equation) models
the distribution of electrical potential when an outside charge is present:
uxx + uyy = π − x
Solve the equation subject to the following boundary conditions:
u(x, 0) = π − x, u(x, π) = 0,
u(0, y) = u(π, y) = 0.
Question 5. The temperature u(x, t) of a narrow metal rod of length L = π with a heat
source is modelled by the following non-homogeneous heat equation:
ut = uxx + x.
Solve the equation if the initial temperature u(x, 0) = 2π + 4x, and the ends are kept at
constant temperatures as follows:
u(0, t) = 2π, u(π, t) = 5π, t > 0.
Hint: Transform the non-homogeneous boundary conditions into homogeneous ones for
w = u − c1x − c2.2