ESC384 Assignment 1 Due Thursday 2025-09-18 at 16:10
The solution to the assignment must be uploaded to Quercus as a single PDF file at the specified
time. Please adhere to the collaboration policy: the final write up must be prepared individually
without consulting others. (See the syllabus for details.)
Problem 1. Classification of PDEs (36%)
Consider the following PDEs:
∂u ∂2u
(i) ∂t − ∂x2
= cos(4x) in R × R>0
∂u ∂2u u
(ii) ∂t − ∂x2
= − |1+x| in R × R>0
∂2u ∂2u ∂2u
(iii) ∂t2
− ∂x2
− ∂y 2
+ sin(u) = 0 in R2 × R>0
2 2 2
(iv) −2 ∂∂xu2 + 2 ∂x∂y
∂ u
− 2 ∂∂yu2 = 0 in R2
For each PDE, answer the following questions and briefly justify your answers:
(a) (4%) State the order.
(b) (8%) State if the PDE is linear homogeneous, linear nonhomogeneous, or nonlinear.
(c) (8%) If the equation is second-order, classify the equation as elliptic, parabolic, or hyperbolic.
Hint: the standard analysis of the “A matrix” applies even if the equation is nonlinear.
(d) (8%) State whether the following statement (S1) holds: if v and w satisfies the PDE, then v +w
also satisfies the PDE.
(e) (8%) State whether the following statement (S2) holds: if u satisfies the PDE, then w obtained
by shifting u by π in the x direction (i.e., w(x, t) = u(x − π, t) in R × R>0 and w(x, y, t) =
u(x − π, y, t) in R2 × R>0 ) also satisfies the PDE.
Problem 2. Odd, even, and periodic functions (34%)
State if each of the following statements is true or false, and then prove the statement or provide
a counterexample.
(a) (7%) If f is odd and g is even, then f + g is odd.
(b) (8%) If f : R → R is even and continuously differentiable, then g(x) ≡ f ′ (x) is odd.
Rx
(c) (8%) If f : R → R is even and integrable, then g(x) ≡ ξ=0 f (ξ)dξ is odd.
R y+z Rz
(d) (11%) If f : R → R is z-periodic and integrable, then x=y f (x)dx = x=0 f (x)dx for any y ∈ R.
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Problem 3. Fourier series solution of ODE (30%)
We wish to solve an ODE boundary value problem
−u′′ + u = f in (0, L),
u(x = 0) = u(x = L) = 0,
where f is some square-integrable function, and LP> 0 is the length of the domain. Our goal is
to express the solution u as a series; i.e., u(x) ∼ ∞ ∞
n=1 ûn ϕn (x) where (ϕn )n=1 is an appropriate
sequence of functions. Answer the following questions:
(a) (5%) Choose either (i) ϕn (x) = sin(λn x) or (ii) ϕn (x) = cos(λn x) and an appropriate λn so
that the boundary conditions are satisfied for any set of coefficients ûn , n = 1, 2, . . . . (From
hereon, ϕn (x) should be replaced by sin(λn x) or cos(λn x) with an appropriate λn .)
(b) (15%) Substitute the expression found in (a) to the ODE, perform the necessary differentiation
(assuming the solution is sufficiently smooth), and invoke the orthogonality relationship to find
RL
the expression for the coefficient ûn in terms of the coefficient fˆn ≡ L2 0 f (x)ϕn (x)dx.
(c) (10%) Let f = x. Find coefficients ûn , n = 1, 2, . . . , associated with the solution u.
