Direct Similarity Analysis of Generalized Burgers Equations and Perturbation Solutions of Euler–Painlevé Transcendents

Abstract

Similarity reductions of the generalized Burgers equation , where α, β, and γ are non‐negative constants, n a positive integer and j= 0, 1, 2, are obtained by the direct method of Clarkson and Kruskal [1]. This is the first work to report the similarity variables as an incomplete gamma function and also as a power of , and to provide a perturbation solution of an Euler–Painlevé transcedent.

Exact Linearization and Invariant Solutions of the Generalized Burgers Equation with Linear Damping and Variable Viscosity

“…In the present work we observe in general that the damping term αu in (14) helps in the linearization of (14).…”

“…Lie's classical method is applied to the GBE with linear damping and variable viscosity (14) to reduce it to either a linear first-order ODE or to an EPT when (t) is given by any one of the following five forms:…”

“…The scheme of this paper is as follows. Section 2 contains the detailed account of the similarity analysis of (14). The results and conclusions of the present study constitute Section 3.…”

“…for its similarity solutions and reported yet another new similarity variable for the family of Burgers equations. Vaganan and Asokan [14] studied the GBE…”