Methods 3

Question 1

Heat equation - describe what happens to the solution as t→∞

Answer 1

As t→∞ ; the higher Fourier modes decay exponentially and the temperature distribution approaches φ₀
e^-n²t

Question 2

Why is the space of solutions to the Helmholtz equation (for a fixed K) a vector space? What is its dimension?

Answer 2

The space of solutions is a vector space since it is linear
In general, the dimension of the space of solutions for K=-Nπ² is the number of ways of writing N as a sum of 2 positive square numbers m²+n² where the order of m and n matters so m = 1, n = 2 is not the same as m = 2, n =1