Applied Mathematics and Computation 264 (2015) 218–222 Contents lists available at ScienceDirect Applied Mathematics and Computation journal homepage: www.elsevier.com/locate/amc Analytical solutions for water pollution problems using quasi-conformal mappings Jorge Zabadal a , Bardo Bodmann a , Vinicius Ribeiro b,c,* a Department of Mechanical Engineering, Nuclear Engineering Group, Universidade Federal do Rio Grande do Sul, Av. Osvaldo Aranha, 99, 90046-900-Porto Alegre, RS, Brazil b Computer Science Faculty, Rua Orfanotrófio, Centro Universitário Ritter dos Reis, rua Orfanotrófio, 555, Alto Teresópolis, 90.850-440-Porto Alegre, RS, Brazil c ESPM-Sul, Rua Guilherme Shell, 350, Santo Antônio, 90.640-440-Porto Alegre, RS, Brazil article info Keywords: Non-linear PDEs Symmetries Perturbation scheme Split Burgers-type equation Exact solutions abstract In this work a straightforward procedure to find exact solutions and point symmetries admit- ted by nonlinear partial differential equations is presented. This method avoids the explicit computation of the infinitesimals via determining equations, which in some cases are often more difficult to solve than the target equation itself. A preliminary result is obtained by solving the Burgers equation without using the Cole–Hopf transformation. © 2015 Elsevier Inc. All rights reserved. 1. Introduction The methods based on Lie group analysis, conceived about the 1870s and widely exploited by the Russian school along more than 6 decades, mainly by L. Ovsyannikov, N. Ibragimov and coworkers [5], have been still more successfully employed in the last 20 years to solve nonlinear partial differential equations using software for symbolical calculus. The symbolical computation packages were initially employed to handle large amount of analytical operations required to deduce and solve the so-called determining equations, an auxiliary system of linear partial differential equations whose solution furnishes the coefficients of the Lie group generators [3]. It occurs that this task, which is the most expensive step in Lie group analysis, recently reveals to be not obligatory for some practical purposes. After the developments introduced by Nikitin [1,2,6,7], and other works which establish relevant connections between symmetries, commutation relations, Bäcklund transformations and differential constraints [4,9], the original Lie method has suffered some interesting simplifications and improvements from the operational point of view. The computational codes produced from these new formulations are small, and the time processing required to obtain exact solutions to partial differential equations have been drastically reduced. In certain cases for which the determining equations required to obtain the coefficients of the corresponding generators are much more difficult to solve than the own target equation, such as the unsteady two dimensional Helmholtz equation, the application of the original Lie method becomes prohibitive. Fortunately, in practice, many nonlinear problems in transport phenomena can be easily solved by means of coarse mesh formulations. Hence, it is often more convenient to prescribe for each node an exact solution containing a small number of arbitrary parameters than to search for a more general solution which eventually dispenses the discretization of the domain. In these cases, it is not necessary to find all the generators of the corresponding symmetry group, but only some symmetry expressed in explicit form, in order to obtain an exact solution whose arbitrary elements fulfill the restrictions locally prescribed. * Corresponding author: Tel.: +55 51 32303323; fax: +55 51 32303398. E-mail address: vinicius@uniritter.edu.br, vinicius.gadis@gmail.com, alternativo.vinicius@gmail.com (V. Ribeiro). http://dx.doi.org/10.1016/j.amc.2015.04.082 0096-3003/© 2015 Elsevier Inc. All rights reserved.