Solution Found!
Solve Exercise 8.6.3 using one-dimensional eigenfunctions
Chapter 8, Problem 8.6.4(choose chapter or problem)
Solve (using two-dimensional eigenfunctions) \(\nabla^{2} u=Q(r, \theta)\) inside a circle of radius a subject to the given boundary condition. In what situations are there solutions?
(a) \(u(a, \theta)=0\)
(b) \(\frac{\partial u}{\partial r}(a, \theta)=0\)
(c) \(u(a, \theta)=f(\theta)\)
(d) \(\frac{\partial u}{\partial r}(a, \theta)=g(\theta)\)
Questions & Answers
QUESTION:
Solve (using two-dimensional eigenfunctions) \(\nabla^{2} u=Q(r, \theta)\) inside a circle of radius a subject to the given boundary condition. In what situations are there solutions?
(a) \(u(a, \theta)=0\)
(b) \(\frac{\partial u}{\partial r}(a, \theta)=0\)
(c) \(u(a, \theta)=f(\theta)\)
(d) \(\frac{\partial u}{\partial r}(a, \theta)=g(\theta)\)
ANSWER:Step 1 of 4
Consider
(a) To solve the equation when 
Where, by the boundary condition
Use the method of eigenfunction expansion, to get
Where,
And
