Hello. I am having some trouble with the following PDE:
Laplacian(u(x,y,z)) = u(x,y,z) * (-2*E/h^2)*(1 + (GM/(2(x^2+y^2+z^2)^(1/2))))^4
Where, G,M,E, and h are all constants.
The problem that I'm having is that there is a nonconstant factor of (x^2+y^2+z^2)^(-1/2) appearing on the RHS of this equation, making it non-trivial to solve. I tried separation of variables, and couldn't seem to get it to work.
Could someone please take a look?
This equation does not separate in rectangular coordinates, so you need to convert to spherical coordinates. In these coordinates, the equation becomes
Laplacian(u(r,theta,phi)) = u(r,theta,phi) * (-2*E/h^2)*(1 + (GM/(2r)))^4.
Writing out the Laplacian in spherical coordinates, this equation becomes
(1/r^2) d/dr(r^2 du/dr) + 1/(r^2 sin theta) d/dtheta(sin theta du/dtheta) + 1/(r^2 sin^2 theta) d^2 u/dphi^2
= u(r,theta,phi) * ...
We show how to solve a partial differential equation by first transforming the equation to spherical coordinates and then separating variables.
Apply Separation of Variables to the PDE to get 2 ODE's
I have included a Wave Equation problem with parts a-c, that has variable tension. It involves separation of variables, the Sturm-Liouville system, and an application to the "Rayleigh Quotient" involving the Eigenvalues. I have included notes on the Sturm-Liouville system with examples and properties. Please refer to these notes to maintain continuity for the solution needed. Thank you for your time and consideration in these matters.
1.) Given the wave equation below (with variable tension τ(x) = x) and appropriate Boundary Conditions:
(a) Apply Separation of Variables to the PDE to get 2 ODE's.
(b) Show that the Eigenvalue problem is a singular Sturm-Liouville system; identify the functions p(x), w(x), and q(x). Do not try to solve it.
(c) Based on the equation below, called the "Rayleigh quotient", do you expect this problem to have any negative Eigenvalues? How about a zero Eigenvalue? Be sure to use specific p(x), w(x), q(x) and Boundary Conditions for this problem.