Question: Consider the Laplace equation ?^2u/(?x^2) + ?^2u/(?y^2) = 0 that models the steady-state heat distribution u(x, y) in a rectangular plate 0 = x = a and 0 = y = b. In physics, the Fourier's law of heat conduction states that the heat flux density (heat energy that flows through a surface per unit time) is proportional to the temperature gradient. From multivariable calculus, we know that the gradient vector is given by ?u = (?u/?x, ?u/?y ) Suppose that the plate is held at zero degrees at y = 0 and y = b, and that it is insulated at x = 0 so that the heat flux there is zero. These three boundary conditions can be expressed mathematically as: u(x, 0) = u(x, b) = ux(0, y) = 0 . Given a heat flux profile at x = a such that u_x(a, y) = f(y), solve the Laplace equation using separation of variables.