A 6-cm by 5-cm rectangular silver plate has heat being uniformly generated at each point at the rate q = 1.5 cal/cm3·s. Let x represent the distance along the edge of the plate of length 6 cm and y be the distance along the edge of the plate of length 5 cm. Suppose the temperature u along the edges is kept at the following temperatures:
u(x, 0) = x(6 − x), u(x, 5) = 0,....................0 ≤ x ≤ 6,
u(0, y) = y(5 − y), u(6, y) = 0, ....................0 ≤ y ≤ 5,
where the origin lies at a corner of the plate with coordinates (0, 0) and the edges lie along the positive x- and y-axes. The steady-state temperature u = u(x, y) satisfies Poisson's equation:
∂2u / ∂x2 (x, y) + ∂2u / ∂y2 (x, y) = −q/K, 0< x < 6, 0 < y < 5,
where K, the thermal conductivity, is 1.04 cal/cm·deg·s. Approximate the temperature u(x, y) using Algorithm 12.1 with h = 0.4 and k = 1/3.