An analysis of groundwater flow in an infinite region with a sinusoidal top

In this paper we give a mathematical analysis with numerical computation for a groundwater flow problem described by an elliptic equation of the form ∇ · (e^dx∇φ(x, z)) = 0, d ≥ 0 in a semi-infinite vertical region bounded on top by a sloping sinusoidal curve, under given boundary conditions. φ(x, z) represents the hydraulic head and e^dx represents the relative hydraulic conductivity (or permeability). We reduce the problem to an infinite system of linear equations using the method of separation of variables and construction of a Grammian matrix. Truncation of this system yields an approximate solution that gives the best match on the top boundary. Computational results for some typical parameters are presented.