Exam 4 Solutions, MAP 4341/5345, Spring 2015
1. Solve the eigenvalue problem in a three-dimensional ball of radius a:
u = u ,
|x| < a ,
u + 2a
u
n
|x|=a
= 0,
by separating variables in spherical coordinates, where n is the outward unit normal on the spher
Exam 2 with solutions, MAP 4341/5345, Spring 2015
1. Solve the eigenvalue problem in L2 (a, a)
u (x) = u(x) ,
a < x < a ,
u(a) = 0 ,
u(a) = 0 .
Construct an orthonormal set of eigenfunctions.
Solution. The operator is a Sturm-Liouville operator. Its eigen
L1, 01/07/2015: Basic idea of equations in partial derivatives (PDEs). Linear PDEs. Example: vibration of
an elastic string with fixed ends. The wave equation. A solution to a PDE. Initial conditions. Examples of
solutions of the wave equation. Forward an
Exam 3 with solutions, MAP 4341/5345, Spring 2015
1. Solve the eigenvalue problem in L2 () by separating variables
u(x, y) = u(x, y) ,
(x, y) = (0, a) (b, b) ,
u
n
= 0.
Prove that an orthonormal set of eigenfunctions is complete in L2 (). Use this fact to