Closed book examination
Time: 21 hours 2
Be sure that this examination has 3 pages.
The University of British Columbia
Special Instructions: A two-sided single page of notes is allowed. Marks
[20] 1.
Consider the ODE boundary value problem for u(x) on 0 ≤ x ≤ 1: ′′′ ′
[20] 2.
adjoint problem has no nontrivial solution.
(ii) Show how to represent u(x) in terms of a Green’s function G(ξ,x) as u(x) = 01 G(ξ,x)f(ξ)dξ. Write the conditions that G(ξ,x) must satisfy. (Do not cal- culate G explicitly).
(iii) Show that upon multiplying the ODE for u by some function p(x) the resulting problem can be made self-adjoint. Calculate the appropriate function p(x).
The free-space Green’s function for ∆u−u = δ(x−x0) in 3-D is u = Ae−r/r for some constant A, where r = |x − x0|.
(i) Derive the value of the constant A.
(ii) Let x ≡ (x, y, z). Use the method of images to determine an integral representa-
tion for the solution to the following PDE on a half-space:
∆u − u = 0 , −∞ < x < ∞ , −∞ < y < ∞ , 0 ≤ z < ∞ ,
uz(x,y,0)=f(x,y); u→0 as (x2 +y2 +z2)→∞.
(iii) Find a leading order approximation for u that is valid for x2 + y2 + z2 → +∞.
Final Examinations - April 2015
Mathematics 401
M. Ward
u −xu −u=f(x), u(0)=0, u(1)=0.
(i) Determine the homogeneous adjoint problem, and show analytically that this
Continued on page 2
[20] 3.
Let u = u(x) and consider the functional
April 2015 Mathematics 401
Page 2 of 3 pages
L
′ ′′
I(u)=
over all four times continuously differentiable functions u(x) with u(0) = u(L) = 0.
(i) Show that the Euler-Lagrange equation associated with I(u) is ∂F−d∂F+d2 ∂F=0.
∂u dx ∂u′ dx2 ∂u′′
What are the natural boundary conditions for u at x = 0, L?
(ii) Suppose that
′ ′′ 1 ′′2 1 ′2 σ F(x,u,u,u )=2 u +2 u −(1+u),
where σ is a positive constant, and that u(0) = u(L) = 0 is prescribed. Write the associated Euler-Lagrange equation and natural boundary conditions for u explicitly. (This problem models the deflection of a beam in a micro-electrical- mechanical system).
(iii) Next, consider the eigenvalue problem
0
F(x,u,u,u )dx
[20] 4.
with p(x) > 0 and r(x) > 0 on 0 ≤ x ≤ L. Find a variational principle, together with a simple trial function, that can be used to give an upper bound on the first eigenvalue λ1. (Do not calculate this bound explicitly).
Consider the following diffusion equation for u(x, t):
ut =uxx +f(x,t), 0≤x<∞, t>0,
ux(0,t) = h(t); u(x,0) = 0,
whereweassumethatf issuchthatu→0andux →0asx→∞foranyfixedt>0.
(i) Show how to represent u(x,t) in terms of an appropriate Green’s function by deriving the PDE problem for the Green’s function.
(ii) Give an analytical expression for the Green’s function that is needed in (i).
(iii) If h(t) = 0 for t ≥ 0 and f(x,t) = δ(x−(x0 +vt)) for some constants x0 > 0 and v > 0, where δ(z) is the Dirac delta function, find u(x, t) using your integral representation from (i). (This problem models temperature distribution due to a localized heat source that moves with constant speed v > 0 along the positive x-axis.)
′′
′′ ′′
p(x)u = λr(x)u , 0 ≤ x ≤ L ; u(0) = u(L) = u (0) = u (L) = 0 ,
Continued on page 3
[20] 5.
Consider the following eigenvalue problem for φ(x,y) in the 2-D elliptical-shaped domain Ω = {(x, y)| x2 + y2/9 ≤ 1}:
∇ · [p∇φ] + λφ = 0 , φ = 0 on (x, y) ∈ ∂Ω . Here p(x, y) = 1 + x2 + y2.
(i) Derive a simple upper and a lower bound for the first eigenvalue λ0 of this problem by bounding p and the domain Ω. In bounding the domain use appropriate circular domains. State carefully the bounding principle that you are using.
(ii) Can you get tighter bounds for λ0 by bounding Ω with rectangular domains? (Hint: it is an easy Calculus exercise to find the largest rectangle that can be inscribed within Ω.)
(iii) Now consider the time-dependent problem for u(x,y,t) in Ω, where we instead impose a non-flux condition on ∂Ω. The problem is formulated as
ut =∇·[p∇u], ∂nu=0 on (x,y)∈∂Ω; u(x,y,0)=u0(x,y), where p(x, y) = 1 + x2 + y2. Show that for t → +∞, the solution to this problem
has the approximate form
u(x, y, t) ≈ A0 + A1e−λ1tφ1(x, y) + · · · ,
for some constants A0, A1, λ1 > 0, and φ1(x,y). Calculate A0 explicitly. Also, formulate a variational principle for λ1 and provide a simple trial function that can be used to provide an upper bound for λ1 (do not calculate the bound explicitly).
April 2015 Mathematics 401 Page 3 of 3 pages
[100] Total Marks
The End