The following is a general fourier series for ux y explain

Question

Math 054 Partial Differential Equations - HW Assignment 7

1. The following is a general Fourier series for u(x, y)

u(x, y) = ½ u₀(t) + _n=1∑^∞u_n(t) cos nπx

Explain why we multiply u₀(t) by ½. A thorough explanation will involve eigenfunctions and orthogonality.

2. Using results from class, solve for u(x, t).

u_t(x, t) = Du_xx(x, t), 0 < x < 1, t > 0,

u(0, t) = u(1, t) = 0, t > 0.