In this section we consider the special case where d x

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Unformatted text preview: z If we wait long enough, so that the temperature is no longer changing, the “steady-state” temperature u(x, y, z ) must satisfy Laplace’s equation ∂2u ∂2u ∂2u + 2 + 2 = 0. ∂x2 ∂y ∂z 117 If the temperature is independent of z , the function u(x, y ) = u(x, y, z ) must satisfy the two-dimensional Laplace equation ∂2u ∂2u + 2 = 0. ∂x2 ∂y Exercises: 5.1.1. For which of the following differential equations is it true that the superposition principle holds? a. ∂2u ∂2u ∂2u + 2 + 2 = 0. ∂x2 ∂y ∂z b. ∂2u ∂2u ∂2u + 2 + 2 + u = 0. ∂x2 ∂y ∂z c. ∂2u ∂2u ∂2u + 2 + 2 + u2 = 0. ∂x2 ∂y ∂z d. ∂2u ∂2u ∂2u + 2 + 2 = ex . ∂x2 ∂y ∂z e. ∂2u ∂2u ∂2u + 2 + 2 = ex u. ∂x2 ∂y ∂z Explain your answers. 5.1.2. Suppose that a chemical reaction creates heat at the rate λ(x, y, z )u(x, y, z, t) + ν (x, y, z ), per unit volume. Show that in this case the equation governing heat flow is ρ(x, y, z )σ (x, y, z ) 5.2 ∂u = λ(x, y, z )u(x, y, z, t) +...
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This document was uploaded on 01/12/2014.

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