Similarity solution of the Boussinesq equation

Similarity transforms of the Boussinesq equation in a semi-infinite medium are available when the boundary conditions are a power of time. The Boussin esq equation is reduced from a partial differential equation to a boundary- value problem. Chen et al. [Trans Porous Media 1995;18:15-36] use a hodogra ph method to derive an integral equation formulation of the new differentia l equation which they solve by numerical iteration. In the present paper, t he convergence of their scheme is improved such that numerical iteration ca n be avoided for all practical purposes. However, a simpler analytical appr oach is also presented which is based on Shampine's transformation of the b oundary value problem to an initial value problem. This analytical approxim ation is remarkably simple and yet more accurate than the analytical hodogr aph approximations. (C) 2000 Elsevier Science Ltd. All rights reserved.