Approximate similarity solutions to the Boussinesq equation

We derive a new cubic polynomial approximation to the similarity solution of the Boussinesq equation for one-dimensional, unconfined groundwater flow in a horizontal, initially dry aquifer with a power law inlet condition. This derivation builds upon the work of Lockington et al. (Adv. Wat. Resour. 23 (7) (2000) 725), where a quadratic approximation is constructed, and uses an additional condition for the determination of the (approximate) scaling function. The approximate analytical solutions are compared with the direct numerical solution obtained by Shampineʹs method (ZAMM 53 (1973) 421). In all the tests performed, the new cubic approximation gives better results than the quadratic one, while it is only slightly harder to use.