A comparison between Cole-Hopf transformation and the decomposition method for solving Burgers' equations

The equation under discussion in this paper is the non-linear parabolic Burgers' equation given by u(1) + uu(x) = uv(xx) where u = u(x, t) in some domain and v is a parameter (v > 0), and uu(x) is the non-linear term. Eq. (1) is similar in structure to Navier Stokes equation and has appeared in problems of aerodynamics, and was first introduced by J.M. Burgers. An equation simply related to (1) appears in the approximate theory of weak non-stationary shock wave in a real fluid [G. Adomian, Solving frontier problems of physics: The decomposition method, Kluwer, Boston, 1994, A. Wazwaz, Partial differential equations: Methods and applications, Balkema Publishers, The Netherlands, 2002, A. Wazwaz, A. Gorguis, An analytic study of Fisher's equation by using Adornian decomposition methods, Appl. Math. Comput. 154 (2004) 609-620, A. Wazwaz, A. Gorguis, Exact solutions for heat-like and wave-like equations with variable coefficients, Appl. Math. Comput. 149 (2004) 15-29]. The coefficient v is a constant that defines the kinematic viscosity. If the viscosity v = 0, the equation is called inviscid Burgers' equation which governs gas dynamics,