Iterative analytic approximation to nonlinear convection dominated systems

Abstract

In this paper we are interested in the analytic approximation of nonlinear convection dominated systems of mathematical physics. Solutions of such systems, contaminated by a small parameter, often show sharp boundary and interior layers. The problem becomes still more complex as the equation under consideration has nonlinearity together with small dissipation. Then, shock waves appear alongside boundary layers. To approximate the multi-scale solution of convection dominated problems, we present and analyze an iterative analytic method based on a Lagrange multiplier technique. The Lagrange multiplier is obtained optimally, in a general setting, using variational theory and Liouville–Green transforms. The idea of the paper is to overcome the well known difficulties associated with the numerical methods. Examples, with quadratic nonlinear convection terms and quasi-linear terms, are taken into account to show the effectiveness and accuracy of the present approach. It is observed that the method is straightforward, highly accurate, brief and can also be functional to other nonlinear evolution equations of mathematical physics.