Hint: Some of the possible solutions to these problems involve using Lanczos iteration, so make sure you are familiar with this algorithm.
Hint: I highly recommend that you avoid trying to solve for the eigenvalues of a non-hermetian matrix in any of the following problems. If your approach requires this, consider doing something else.
1. Consider a box with length, width and height given by L. The box encompasses the region described by 0<=x<=L, 0<=y<=L, and 0<=z<=L. A scalar field inside the box satisfies the differential equation:
(1)
Note that the differential operator here is known as the Laplacian. An equivalent of writing the equation is:
(2)
Here a is a positive constant that is equal to .
The field is 0 on the surfaces y=0, y=L, x=0, x=L and finally the surface z=L. On the surface z=0, the field has the functional form:
(3)
Solve for y as a function of x, y, and z inside the box. Your final solution must be an analytic expression, though it can involve an infinite sum.
2. Consider the system of nonlinear partial differential equations:
(1)
Here a is a small positive constant, such that a <<1. The boundary conditions are:
(2)
(3)
(4)
(5)
Solve for f(x,y) and g(x,y) for 0<=x<=L and y>=0 to first order in a. Your solution should be a finite sum of elementary functions. Numerical approximations are not to be used.