On Direct Transformation Approach to Asymptotical Analytical Solutions of Perturbed Partial Differential Equation

In this article, we will derive an equality, where the Taylor series expansion around ε=0 for any asymptotical analytical solution of theperturbed partial differential equation (PDE) with perturbing parameter ε must be admitted. By making use of theequality, we may obtain a transformation, which directly map theanalytical solutions of a given unperturbed PDE to the asymptoticalanalytical solutions of the corresponding perturbed one. The notion of Lie-Bäcklund symmetries is introduced in order to obtainmore transformations. Hence, we can directly create more transformations in virtue of known Lie-Bäcklund symmetries and recursion operators of corresponding unperturbed equation. Theperturbed Burgers equation and the perturbed Korteweg-de Vries (KdV)equation are used as examples.