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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).