ASSIGNMENT 1
PDE-MTH 405
(1) Solve the following quasilinear first order equation
ut + divF (u) = 0 in U = Rn × (0, ∞)
u=g
on Γ = Rn × {t = 0},
where F : R → Rn and g are taken to be smooth.
(2) Solve using method of characteristics:
2
(
= 0 in R × (0, ∞)
ut + u2
u=g
x
on R × {t = 0},
with the initial data

if x ≤ 0
 1
1 − x if 0 ≤ x ≤ 1
g(x) =

0
if x ≥ 1.
(3) Is it possible to solve the following transport equation
ut + aux = 0 in R2
u=0
on Γ,
where Γ := {(x, t) ∈ R2 ; x = at} and a is constant? Justify.
(4) Solve the following equation
uux + uy = 1 in R2
u = 21 s
on Γ,
where Γ := {(x, y) ∈ R2 ; x = y = s} and 0 < s < 1. Is Γ non-characteristics?
(5) Find the solution of the Cauchy problem
aux + ut = 0 in R2
u=g
on R × {t = 0},
with the initial data
g(x) =
− x3
if x ≤ 0
2x + 3 if x > 0.
Does the solution continuous? Is the curve Γ := {(x, t) ∈ R2 ; t = 0} characteristics?
(6) Prove or disprove: Let γ be a curve in the xy plane and assume that u is continuous
everywhere. Show that first derivatives of the generalised solution u to the linear
first order equation
a(x, y)ux + b(x, y)uy = u(x, y) + c(x, y)
is discontinuous along γ if and only if γ is a characteristic curve. Here we assume
that a, b, c are regular.
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