Similarity solutions of the Burgers equation with linear damping

A new similarity solution of the Burgers equation with linear damping

“…It is still difficult to find a general solution of this equation. However, it has a special solution ~ = clexp(-at) + c2 and r = -claexp(-at), which, along with ~-= c3, is exactly the one obtained by Vaganan et al [5] by means of the classical method.…”

“…(1) is invariant under this transformation. Using this approach, Vaganan et al [5] have obtained the only similarity reduction (6) and (7). In the nonclassical method, the original equation (1) is augmented with the invariant surface condition ~(x, t, u)ux + T(X, t, U)Ut = r t, U),…”

“…The reduction (6) and (7) is only the classical similarity one of Eq. (1), and it can also be obtained using the Lie group method of infinitesimal transformations [5]. When z~ _= 0, it is clear that z --z(t) in the ansatz (2) is taken as t simply without loss of generality.…”

On the generalized Burgers equation

“…Here ε is a positive constant, f (t, z), g(t, z) and φ(x) are given functions. Equation (1.1) appears in different applications, see [2,[6][7][8][9][10][11][12][13]16,17]. The goal of the present paper is to obtain an a priori estimate of |u x | independent of ε and T and on the basis of this estimate to prove the existence and the uniqueness of the classical solution of problem (1.1)-(1.3) and to investigate the behavior of this solution when ε → 0 and when T → +∞.…”

“…Example 1. The simplest case of (1.1) is the following equation (see [6][7][8][9][10][11][12][13]16])…”