I've followed the method explained in Numerical Recipes in C, chapter 19, to solve a elliptic equation:
http://www.capca.ucalgary.ca/top/teaching/phys499+535/PHYS535/nrf90/pde-c19-0.pdf
I'm currently working with polar coordinates, because the problem I'm modelling is a circular furnace. So the values in the band matrix are not constant. Depend on the radius and the angle.
How can I proof that there'll always be a solution, analytically?
I've been thinking of demonstrate that it's strictly diagonally dominant, by showing that
$|-2 r_j^2 b^2 + r_j a b^2 - 2 a^2| > |a^2| + |a^2| + |r_j^2 b^2| + |r_j^2 b^2 - a r_j b^2|$
where $r_j$ is the length of a radius, $a = \Delta r$ and $b = \Delta \theta$ (both two are constant)
The subject is Numerical methods
A circular furnace is used to melt metal. The temperature inside the furnice is 1500°C. The inner side (suppose it's circular) is also 1500°. The temperature of the outer side of the furnice is registered by a set of $n$ thermocouples.
The problem consists in to find the 400°C - isotherm.
We were given this formula*
$\frac{\partial^2T(r,\theta)}{\partial r^2} + \frac{1}{r} \frac{\partial T(r,\theta)}{\partial r} + \frac{1}{r^2} \frac{\partial ^2 T(r,\theta)}{\partial \theta ^2} = 0 $
which is satisfied for every point $(r,\theta)$ in polar coordinates, for $0 < r_{inner} < r < r_{outer}$ and $0 \leq \theta \leq 2\pi$
$T(r,\theta)$ is the temperature in $(r,\theta)$
The idea is to discretize the domain (a grid of size $n \times (m+1)$), as it's shown in the pdf file.
(*)NOTE: Math in my career does not include Differential Equations (except for Optative subjects)
As a result of replacing differentials by finite differences, and adding the boundary conditions, my problem is now to solve a linear equation system of size $(m+1)n \times (m+1)n$. Each variable is the value of the temperature at a point of the grid ("the solution")
( The Dirichlet's Problem http://es.wikipedia.org/wiki/Ecuaci%C3%B3n_de_Laplace#Problema_de_Dirichlet )
The matrix ("the band matrix") I've obtained is similar to the one in page 832. But the non-zero values are not constant (they are the ones I've written at the very first)
It's not clear for me if the matrix is invertible