Try the eigenvalue method on the advection equation du(x, t) _ ou(x, t) = -c- ", əx ? @t 2 € (0,00) (a) what are the time dependent solutions for each eigenvalue of the operator -com? (b) show that there are no eigenfunctions of cha subject to the boundary condition u(0,t) = 0 to conclude that the only solution we obtain from the eigenvalue method is the trivial solution u(x, t) = 0 and that is is impossible to satisfy the any arbitrary initial condition u(x,0) = (c) for what eigenvalues of can do the eigenfunctions satisfy the "boundary" condition lim u(x, t) = 0? 200 unlike the boundary condition from part (b), we do get non-zero solutions here. however, explain why the eigenvalue method does not allow us to write the general solution as linear combinations of solutions un(x, t). ne au ಶಿಲ --m, ze10,) c at дх