Problem

Consider u(x, y) satisfying Laplace’s equation inside a rectangle (0 < x < L, 0 &l...

Consider u(x, y) satisfying Laplace’s equation inside a rectangle (0 < x < L, 0 < y ) subject to the boundary conditions

*(a) Without solving this problem, briefly explain the physical condition under which there is a solution to this problem.

(b) Solve this problem by the method of separation of variables. Show that the method works only under the condition of part (a). [Hint: You may use (2.5.16) without derivation.]

(c) The solution [part (b)] has an arbitrary constant. Determine it by consideration of the time-dependent heat equation (1.5.11) subject to the initial condition

u(x, y, 0) = g(x, y).

Reference Equation 1.5.11: