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Unformatted text preview: Chapter 8 Laplace’s Equation Syllabus section; 7. Laplace’s equation. Uniqueness under suitable boundary conditions. Separation of variables. Two- dimensional solutions in Cartesian and polar coordinates. Axisymmetric spherical harmonic solutions. 8.1 The Laplace and Poisson equations Laplace’s equation is simply ∇ 2 Φ ( r ) = where, as before, ∇ 2 Φ = ∇ · ∇Φ . It often occurs as an equation ∇ · F = 0 for a conservative field F (so that F = ∇Φ for some Φ ). Laplace’s equation is the simplest and most basic example of one of the three types 1 of second-order linear PDEs, known as the “elliptic” type. Laplace’s equation is a linear homogeneous equation. Going back to our gravitational and electromagnetic examples of conservative fields, and using the Di- vergence Theorem integraldisplay V ∇ · F d V = integraldisplay ∂ V F . d S to obtain Gauss’s results, we see that if ∇ · F = 0 everywhere there are no sources inside the volume, which for gravity means that there is no mass there and for electric field means that there is no (net) charge. Hence Laplace’s equation describes the gravitational potential in regions where there is no matter, and the electric potential in regions where there are no charges. If instead there is a net charge density ρ , the electric field satisfies ε ∇ · E = ρ ( r ) 1 The basic examples of the other types, the “hyperbolic” and “parabolic” equations, are the wave equation 1 c 2 ∂ 2 f ∂ t 2 = ∇ 2 f and the heat equation or diffusion equation ∂ f ∂ t = κ ∇ 2 f respectively. We met the heat equation with a single spatial variable in Example 7.8 on Fourier series. 85 (where ε is a constant). Combining this with E = − ∇Φ gives ε ∇ 2 Φ = − ρ ( r ) . This is known as Poisson’s equation . Laplace’s equation is of course a special case of Poisson’s equation, in which the function on the right-hand side happens to be zero. Laplace’s and Poisson’s equations are so important, both because of their occurrence in applications and because they are the basic examples of elliptic equations, that we are now going to spend some time considering their solutions. Solutions of Laplace’s equation are known as harmonic functions . To find the solution in a particular case we need to know some boundary conditions. We will prove that under a variety of boundary conditions the solution of Poisson’s (or Laplace’s) equation is unique . We shall then investigate what the solutions actually are in some simple cases. 8.2 Uniqueness of Solutions to Poisson’s (and Laplace’s) Equation Note that ∇ 2 is a linear operator: that is ∇ 2 ( Φ 1 + Φ 2 ) = ∇ 2 Φ 1 + ∇ 2 Φ 2 ....
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This document was uploaded on 02/10/2010.
- Spring '09