Similarity reductions of equations for river pollution O.D. Makinde a, * , R.J. Moitsheki b , B.A. Tau c a Department of Applied Mathematics, University of Limpopo Turfloop Campus, Private Bag X1106, Sovenga 0727, South Africa b Department of Mathematics, Vaal University of Technology, Private Bag X021, Vanderbijlpark 1900, South Africa c School of Modelling Sciences, North West University Vaal Traingle Campus, P.O. Box 1174, Vanderbijlpark 1900, South Africa Abstract We consider a system of coupled partial differential equations describing pollutant transport in a river system. Symme- try analysis of this system resulted in admitted large Lie algebras for a some special cases of the arbitrary constants and the source term. Furthermore, we construct the one-dimensional optimal systems of the admitted symmetries. However, sim- ilarity (invariant) solutions for the system are constructed for some more realistic source term. Ó 2006 Elsevier Inc. All rights reserved. Keywords: River pollution; Lie point symmetries; Similarity solutions 1. Introduction Industrial operations give rise to a number of serious environmental effects which among other includes river pollution. The impacts of the pollution in the river system are severe and include significant depletion of aquatic ecosystem and restriction on consumptive use of water. Mathematical models derived in terms of partial differential equations will remain important for effectively predicting the outcome of river pollution. Prediction may be made by computer or numerical schemes. However, we aim to provide analytical solutions to river pollution problem, which may not only be use to give insight to pollutant transport processes but also as bench marks for the numerical schemes. There has been a great interest in studying the fluid dynamic of pollutants in rivers, due to increasingly important environmental and engineering problems, and more and more refined model have been proposed [1,2,8,11–13]. Shulka [11] obtained the analytical solutions by the Fourier transform method for the case of unsteady transport dispersion of nonconservative pollutant/biochemical oxygen demand with first-order decay under each of the sine and cosine variation of waste discharge concentration at upstream boundary and nonzero initial condition through the river. In this paper, we focus on a system of partial differential equations derived from both Navier–Stokes and concentration equations for fluid flow. We employ the methods of group of transformations for differential 0096-3003/$ - see front matter Ó 2006 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2006.10.082 * Corresponding author. E-mail addresses: makindeo@ul.ac.za (O.D. Makinde), joelm@vut.ac.za (R.J. Moitsheki), rkwabt@puk.ac.za (B.A. Tau). Applied Mathematics and Computation 188 (2007) 1267–1273 www.elsevier.com/locate/amc