Description

Linear Operators
Assignment 10
Exercise 10.1
Consider the Sturm-Liouville problem
−u′′ = λu on (0,1), u(0) = 0, u(1) + u′(1) = 0.
i) Show directly that is positive definite on
U = {u ∈ C2([a,b]): u(0) = u(1) + u′(1) = 0}⊂ L2([0,1]).
(2Marks) ii) Deduce that the eigenvalues of L are strictly positive and show that they satisfy √λ = −tan(√λ ).
(2Marks)
iii) Use a computer to obtain a numerical value for the two lowest eigenvalues λ1 and λ2. (2Marks) iv) Use the Rayleigh-Ritz method to estimate the lowest eigenvalue by taking
V1 = P2∩ U,
where(2MarksPn )denotes the space of polynomials of degree not more than n.
v) Improve your estimate by taking
V2 = P3∩ U
You are encouraged to use a computer to assist in calculating integrals and performing the minimization. (2Marks)
vi) By setting
,
find an estimate for the second eigenvalue. (2Marks)