ABSTRACT
This chapter discusses some examples from ODEs for which the balancing argument has been used profitably. In the context of nonlinear partial differential equations (PDE), the balancing argument can be easily extended. One first introduces some cononical independent variables, the similarity variable, and time, say, and then uses the balancing argument in the time variable. One may shift from the full equation(s) those terms which balance in terms of power of the limiting variable, the remaining terms being smaller in comparison in the limit. The lowest-order equation is still a PDE, but simpler than the original one and can often be solved. The solution thus obtained can be improved upon but the error analysis, similar to that in the work of N. Levinson for ODEs, seems difficult. The dominant balance approach is simpler and, when combined with numerical solution, can provide a good picture of the asymptotic structure of the solution.
