Mathematical Methods for Physics and Engineering - Matematica.NET

PDES: SEPARATION OF VARIABLES AND OTHER METHODSsuperposing solutions corresponding to different allowed values of the separationconstants. To take a two-variable example: ifu λ1 (x, y) =X λ1 (x)Y λ1 (y)is a solution of a linear PDE obtained by giving the separation constant the valueλ 1 , then the superpositionu(x, y) =a 1 X λ1 (x)Y λ1 (y)+a 2 X λ2 (x)Y λ2 (y)+···= ∑ a i X λi (x)Y λi (y)i(21.16)is also a solution for any constants a i , provided that the λ i are the allowed valuesof the separation constant λ given the imposed boundary conditions. Note thatif the boundary conditions allow any of the separation constants to be zero thenthe form of the general solution is normally different and must be deduced byreturning to the separated ordinary differential equations. We will encounter thisbehaviour in section 21.3.The value of the superposition approach is that a boundary condition, say thatu(x, y) takes a particular form f(x) wheny = 0, might be met by choosing theconstants a i such thatf(x) = ∑ a i X λi (x)Y λi (0).iIn general, this will be possible provided that the functions X λi (x) form a completeset – as do the sinusoidal functions of Fourier series or the spherical harmonicsdiscussed in subsection 18.3.◮A semi-infinite rectangular metal plate occupies the region 0 ≤ x ≤∞and 0 ≤ y ≤ b inthe xy-plane. The temperature at the far end of the plate and along its two long sides isfixed at 0 ◦ C. If the temperature of the plate at x =0is also fixed and is given by f(y), findthe steady-state temperature distribution u(x,y) of the plate. Hence find the temperaturedistribution if f(y) =u 0 ,whereu 0 is a constant.The physical situation is illustrated in figure 21.1. With the notation we have used severaltimes before, the two-dimensional heat diffusion equation satisfied by the temperatureu(x, y, t) is( )∂ 2 uκ∂x + ∂2 u= ∂u2 ∂y 2 ∂t ,with κ = k/(sρ). In this case, however, we are asked to find the steady-state temperature,which corresponds to ∂u/∂t = 0, and so we are led to consider the (two-dimensional)Laplace equation∂ 2 u∂x + ∂2 u2 ∂y =0. 2We saw that assuming a separable solution of the form u(x, y) =X(x)Y (y) led tosolutions such as (21.14) or (21.15), or equivalent forms with x and y interchanged. Inthe current problem we have to satisfy the boundary conditions u(x, 0)=0=u(x, b) andso a solution that is sinusoidal in y seems appropriate. Furthermore, since we requireu(∞,y) = 0 it is best to write the x-dependence of the solution explicitly in terms of718