Loading [MathJax]/jax/output/HTML-CSS/jax.js

Math Equations

Thursday, April 3, 2014

The discrete heat problem - Part 2: Solving the characteristic equation

In the last post, we came to the conclusion that E=n1j=0(j+n2j+1)E(j)0. This is an n-1'th degree inhomogenous linear equation. To solve it, we need to solve the characteristic equation n1j=0(j+n2j+1)rj=0 for r.

It seems that, in general, the solutions to the characteristic equation are ri=4cos2πi2n(1i<n) but I don't have a proof for this yet. I found the solutions by chance: by noticing that the solutions for low n involved the same square roots as cos did.

As I don't have a proof yet, I'll give you an example instead. Let n=4. Then the characteristic equation is
0=3j=0(j+42j+1)rj=(41)+(53)r+(65)r2+(77)r3=4+10r+6r2+r3=(2+r)(2+4r+r2)=(2+r)(2+2+r)(22+r)
whence we may read off the solutions r=2,22,2+2. This is just what the formula predicts:
r1=4cos2π8=412(1+12)=22r2=4cos2π4=412=2r3=4cos23π8=412(112)=2+2

In conclusion, the formula is reasonable and will do without a proof for the moment.

No comments:

Post a Comment