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The University of British Columbia Final Examination - December, 2011 Mathematics 405/607E Instructor: Dr. Lisa Gordeliy
Closed book examination Time: 2.5 hours
Last Name: , First: Signature
Student Number
Special Instructions:
- Be sure that this examination has 11 pages. Write your name on top of each page.
- A formula sheet is provided. No calculators or notes are permitted.
- Show all your work and make your reasoning clear.
- In case of an exam disruption such as a fire alarm, leave the exam papers in the room and exit quickly and quietly to a pre-designated location.
Rules governing examinations
- Each candidate must be prepared to produce, upon request, a UBCcard for identification.
- Candidates are not permitted to ask questions of the invigilators, except in cases of supposed errors or ambiguities in examination questions.
- No candidate shall be permitted to enter the examination room after the expiration of one-half hour from the scheduled starting time, or to leave during the first half hour of the examination.
- Candidates suspected of any of the following, or similar, dishon- est practices shall be immediately dismissed from the examination and shall be liable to disciplinary action. (a) Having at the place of writing any books, papers or memoranda, calculators, computers, sound or image play- ers/recorders/transmitters (including telephones), or other mem- ory aid devices, other than those authorized by the examiners. (b) Speaking or communicating with other candidates. (c) Purposely exposing written papers to the view of other can- didates or imaging devices. The plea of accident or forgetfulness shall not be received.
- Candidates must not destroy or mutilate any examination mate- rial; must hand in all examination papers; and must not take any examination material from the examination room without permis- sion of the invigilator.
- Candidates must follow any additional examination rules or di- rections communicated by the instructor or invigilator.
Total 50
Page 1 of 11 pages
Useful formulas
- The Taylor series expansion of f (x) about x 0 to n + 1 terms is given by
f (x) =
^ n
k=
(x − x 0 ) k k!
f (k)^ (x 0 ) +
(x − x 0 ) n+ (n + 1)!
f (n+1)^ (ξ)
in which x 0 ≤ ξ ≤ x.
- The Newton’s algorithm approximates the solution of a system of n nonlinear equa- tions in n unknowns u 1 , ..., u (^) n :
R(U ) =
R 1 (U )
R (^) n (U )
,^ where^ U^ =
u (^1) .. . u (^) n
via iterations, U (k)^ = U (k−1)^ −
J
U (k−1)^
R
U (k−1)^
in which the Jacobian matrix is defined from
J(U ) =
∂R
∂U
, J(U ) (^) ij =
∂R (^) i ∂u (^) j
Problem 2 (10 points)
(a) By expanding f (x) about x 0 in a Taylor series, derive the following expression for the error involved in the midpoint approximation: (^) x 0 +h/ 2
x 0 −h/ 2
f (x)dx = h f (x 0 ) +
f (2)^ (ξ) 24
h 3 , ξ ∈ [x 0 − h/ 2 , x 0 + h/2] (2)
(b) Using the above expression (2) for the error, explain why the composite midpoint rule applied directly will not yield a very good approximation to the following integral: (^) π
0
cos(x) x 1 /^3
dx (3)
Explain what needs to be done to obtain a better approximation with the composite midpoint rule to the integral in (3). Support your explanation with derivations.
(c) The Gauss-Legendre quadrature formula with m = 2 integration points evaluates inte- grals of the form: (^) 1
− 1
f (x)dx
exactly if f (x) is a polynomial of degree 2m − 1 = 3. Use this fact to determine the integration points x (^) i and the weights w (^) i for the integration formula (^1)
− 1
f (x)dx ≈ w 1 f (x 1 ) + w 2 f (x 2 )
By symmetry you may assume in your derivation that w 1 = w 2 and that x 1 = −x 2.
Extra page
Extra page
Problem 4 (10 points) Consider the boundary value problem for u(x) with x ∈ (0, 1):
u ^ + u ^ − u 2 = 0, u(0) = 1, u(1) = 2. (5)
(a) Discretize (5) with the second-order finite difference method using a uniform mesh of nodes x (^) n = hn, where n = 0, ..., N , and h = 1/N. Write down the resulting system of nonlinear equations in the matrix form. Be explicit about the size of the matrix and how the boundary conditions are taken into account.
(b) Explain how the Newton’s algorithm can be used to solve the system of nonlinear equa- tions that you obtained in (a). Present an explicit expression for the residual and derive the Jacobian involved in the Newton’s algorithm.
Problem 5 (10 points) Consider the one dimensional wave equation:
∂u ∂t
∂u ∂x
(a) Use central differences in space on a uniform mesh x (^) n = nh, with n = 0,... , N , to arrive at a system of ODE (ordinary differential equations):
du (^) n (t) dt
= A un n = 1,... , N
for u (^) n (t) ≈ u(x (^) n , t), that approximates (6) via an operator A.
(b) Given that
e iξx^ n
are the eigenfunctions of the operator A, where −π ≤ ξh ≤ π and i =
−1, derive the eigenvalues of A. (The eigenvalues will be purely imaginary numbers.)
(c) Will the Forward Euler method be a stable algorithm to solve this system of ODE? Will the Backward Euler method be a stable algorithm to solve this system of ODE? Justify your answer.
(d) Given that the stability region for the Leapfrog method is {z : z = iv, |v| ≤ 1 }, what is the maximum timestep that can be used to solve the above system of ODE using this method? Justify your answer.
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