Mathematical Methods for Physics and Engineering - Matematica.NET

20.5 THE DIFFUSION EQUATIONterm is a little less obvious. It can be viewed as representing the accumulatedtransverse displacement at position x due to the passage past x of all parts ofthe initial motion whose effects can reach x within a time t, both backward andforward travelling.The extension to the three-dimensional wave equation of solutions of the typewe have so far encountered presents no serious difficulty. In Cartesian coordinatesthe three-dimensional wave equation is∂ 2 u∂x 2 + ∂2 u∂y 2 + ∂2 u∂z 2 − 1 ∂ 2 uc 2 =0. (20.32)∂t2 In close analogy with the one-dimensional case we try solutions that are functionsof linear combinations of all four variables,p = lx + my + nz + µt.It is clear that a solution u(x, y, z, t) =f(p) will be acceptable provided that) (l 2 + m 2 + n 2 − µ2 d 2 f(p)c 2 dp 2 =0.Thus, as in the one-dimensional case, f can be arbitrary provided thatl 2 + m 2 + n 2 = µ 2 /c 2 .Using an obvious normalisation, we take µ = ±c and l, m, n as three numberssuch thatl 2 + m 2 + n 2 =1.In other words (l,m,n) are the Cartesian components of a unit vector ˆn thatpoints along the direction of propagation of the wave. The quantity p can bewritten in terms of vectors as the scalar expression p = ˆn · r ± ct, and the generalsolution of (20.32) is thenu(x, y, z, t) =u(r,t)=f(ˆn · r − ct)+g(ˆn · r + ct), (20.33)where ˆn is any unit vector. It would perhaps be more transparent to write ˆnexplicitly as one of the arguments of u.20.5 The diffusion equationOne important class of second-order PDEs, which we have not yet consideredin detail, is that in which the second derivative with respect to one variableappears, but only the first derivative with respect to another (usually time). Thisis exemplified by the one-dimensional diffusion equationκ ∂2 u(x, t)∂x 2695= ∂u∂t , (20.34)