A silver plate in the shape of a trapezoid (see the accompanying figure) has heat being uniformly generated at each point at the rate q = 1.5 cal/cm3 · s. The steady-state temperature u(x, y) of the plate satisfies the Poisson equation
∂2u / ∂x2 (x, y) + ∂2u ∂y2 (x, y) = − q k,
where k, the thermal conductivity, is 1.04 cal/cm·deg·s. Assume that the temperature is held at 15◦C on L2, that heat is lost on the slanted edges L1 and L3 according to the boundary condition ∂u/∂n = 4, and that no heat is lost on L4; that is, ∂u/∂n = 0. Approximate the temperature of the plate at (1, 0), (4, 0), and 5/2, √3/2 by using Algorithm 12.5.
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