MATH3121 Assignment One (Version Two)
Due: 5pm Friday 4th March 2022
1. Consider the PDE,
3uxx + 7uxy + 2uyy = 0
i) Show that this PDE is hyperbolic.
ii) Show that it has the general solution u(x, y) = F (3y − x) + G(2x − y).
Where F and G are arbitrary functions.
iii) Find a solution to the equation that satisfies the following boundary con-
ditions,
u(x, 0) = x and uy(x, 0) = 0 for x ∈ R.
iv) Find a solution to the equation with the following general boundary condi-
tions,
u(x, 0) = g0(x) and uy(x, 0) = g1(x) for x ∈ R.
2. Consider the PDE,
∂2u
∂t2
= αu
∂u
∂x
,
over the domain x, t > 0 with α > 0, subject to the initial condition,
u(x, 0) = x2
and the boundary condition,
u(0, t) = 0.
i) Using the similarity solution method, write u in terms of a function of a
single variable f(η), for some suitably defined variable η.
ii) Obtain the ordinary differential equation and boundary conditions govern-
ing f(η). You do NOT have to solve this boundary value problem for f(η).
3. Consider the quasi-linear PDE,
∂u
∂t
+ u
∂u
∂x
= −2u,
with the boundary condition u(0, t) = e−t, on the domain x ≥ 0, t ≥ 0.
Show, using the method of characteristics, that the solution of this BVP is,
u(x, t) =
e−2t
x+
√
x2 + e−2t
.
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