MATHS 361 Partial Differential Equations 2014

1.  (8 marks) This question is about the Fourier series for the function

f (α) = { ．

一π ≤ α < 0．

0 ≤ α ≤ π．

(a) This function can be represented by a complex Fourier series,

o

cn e  L

=lo

where

cn  = 1 L  f (α)e dα

(b) To what value does the Fourier series converge at α = 0? What about at α = π? Give reasons for your answers.

2.  (12 marks) Consider the following Sturm-Liouville problem:

夕\\ + λ夕 = 0  for 0 < α < 1． with 夕(0) = 0．夕\ (1) = 一3夕(1)．

(a)  Show that there are no non-trivial solutions to the Sturm-Liouville prob- lem if λ < 0.

(b)  Show that the eigenvalues of the Sturm-Liouville problem can be written λn  = Kn(2) , n = 1．2．．．．, where Kn  is a solution of the equation

tan K = 一 K．

(c) By sketching a graph, or otherwise, find an approximation for λn  that is valid for n large.

3.  (12 marks) Consider the following PDE with boundary and initial conditions: ut  = u北北 一 2． u(0．t) = 1． u北 (1．t) = 0． u(α．0) = 0．

(a) Find the steady state solution, us (α), for the PDE.

(b) Let 〇(α．t) = u(α．t) 一 us (α).  Derive a PDE plus boundary and initial conditions for 〇(α．t). Show your working.

(c) Assuming there is a solution of the form 〇(α．t) = X(α)T (t), derive a boundary value problem that is satisfied by X(α).  You do not need to solve the boundary value problem.

4.  (8 marks) Using Fourier Transforms, solve the problem

aa aa

a≠     aα．

a(α．0) = f (α)． 一& < α < &．

You may use the fact that the Fourier transform of a function g, i.e., 于{g(α)} = gˆ(① ) is defined by the formula

o

gˆ(① )   =          g(α)eliω北dα．

lo

You may also use

于{g(α 一 a)} = eliωagˆ(① )．

于{g\ (α)} = 乞① gˆ(① )．