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Unformatted text preview: for the diffusion equation. See [ 35 ; p. 369] for a precise statement and proof of the general theorem. Qualitative Properties Before tackling examples in which we are able to construct explicit formulae for the eigenfunctions and eigenvalues, let us see what the eigenfunction series solution (12.22) can tell us about general diffusion processes. Based on our experience with the case of a one-dimensional bar, the final conclusions will not be especially surprising. Indeed, they also apply, word for word, to diffusion processes in three-dimensional solid bodies. A reader who is impatient to see the explicit solution formulae may wish to skip ahead to the following section, returning here as needed. Keep in mind that we are still dealing with the solution to the homogeneous boundary value problem. The first observation is that all terms in the series solution (12.22), with the possible exception of a null eigenfunction term that appears in the semi-definite Neumann case, are tending to zero exponentially fast. Since most eigenvalues are large, all the higher order terms in the series become almost instantaneously negligible, and hence the solution can be accurately approximated by a finite sum over the first few eigenfunction modes. As time goes on, more and more of the modes can be neglected, and the solution decays to thermal equilibrium at an exponentially fast rate. The rate of convergence to thermal equilibrium is, for most initial data, governed by the smallest positive eigenvalue 1 > for the Helmholtz boundary value problem on the domain. In the positive definite cases of homogeneous Dirichlet or mixed boundary conditions, thermal equilibrium is u ( t, x, y ) u ( x, y ) 0. Thus, in these cases, the equilibrium temperature is equal to the boundary temperature even if this temperature is only fixed on a small part of the boundary. The initial heat is eventually dissipated away through the non-insulated part of the boundary. In the semi-definite Neumann case, corresponding to a completely insulated plate, the general solution has the form u ( t, x, y ) = c + summationdisplay k = 1 c k e k t v k ( x, y ) , (12 . 26) where the sum is over the positive eigenmodes,...
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