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MATH3078: PDEs and Waves
发布时间:2021-10-10
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School of Mathematics and Statistics
Assignment
MATH3078: PDEs and Waves
Semester 2, 2021
1. Consider the eigenvalue problem
(a) Let
be equipped with the usual
product on
that is,
Prove that the operator
is self-adjoint on the domain U.
(b) Explain why all eigenvalues of (1) are i) real and ii) positive.
(c) Find all eigenvalues of (1) and the corresponding eigenfunctions.
(d) Determine the multiplicity of the eigenvalues.
2. Consider the nonhomogeneous wave problem
(a) Let
Find
such that v solves the homogeneous problem
subject to the homogeneous boundary conditions
(b) Using the method of separation of variables, solve the initial boundary value prob-lem for v. [Hint. Find v(x, 0) and vt(x, 0).]
(c) Hence, solve the non-homogeneous problem (2).