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Unformatted text preview: (x, y, z ) + ∇ · (κ(x, y, z )(∇u)) . ∂t The Dirichlet problem The reader will recall that the space of solutions to a homogeneous linear second order ordinary differential equation, such as d2 u du + qu = 0 +p dt2 dt is two-dimensional, a particular solution being determined by two constants. By contrast, the space of solutions to Laplace’s partial differential equation ∂2u ∂2u + 2 =0 ∂x2 ∂y 118 (5.3) is infinite-dimensional. For example, the function u(x, y ) = x3 − 3xy 2 is a solution to Laplace’s equation, because ∂u = 3x2 − 3y 2 , ∂x ∂u = −6xy, ∂y and hence ∂2u = 6x, ∂x2 ∂2u = −6x, ∂y 2 ∂2u ∂2u + 2 = 6x − 6x = 0. ∂x2 ∂y Similarly, u(x, y ) = 7, u(x, y ) = x4 − 6x2 y 2 + y 4 , and u(x, y ) = ex sin y are solutions to Laplace’s equation. A solution to Laplace’s equation is called a harmonic function. It is not difficult to construct infinitely many linearly independent harmonic functions of two variables. Thus to pick out a particular solution to Lap...
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This document was uploaded on 01/12/2014.

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