MATHS 361 Partial Differential Equations 2014
Hello, dear friend, you can consult us at any time if you have any questions, add WeChat: daixieit
MATHS 361
Partial Differential Equations
Mid-semester Test
April 30th, 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 ).
2023-03-25