A comparison between Cole–Hopf transformation and the decomposition method for solving Burgers’ equations

Abstract

The equation under discussion in this paper is the non-linear parabolic Burgers' equation given by(1)ut+uu[email protected]xx,where u=u(x,t) in some domain and @n is a parameter (@n>0), and uux 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 Adomian 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 @n is a constant that defines the kinematic viscosity. If the viscosity @n=0, the equation is called inviscid Burgers' equation which governs gas dynamics. The aim of this paper is to study the exact solution of Eq. (1) in some domain with initial values by the method of Cole-Hopf transformation [E. Hopf, The partial differential equation ut+uux=uxx, Comm. Pure Appl. Math. 3 (1950) 201-230; P. Lagestrom, J.D. Cole, L. Trilling, Problems in the theory of viscous compressible fluids, California Institute of Technology, 1949] which transforms (1) from a non-linear partial differential equation into linear heat equation which in turns can be solved exactly. We then formally derive the exact solution of (1) with initial and boundary conditions using the Adomian decomposition method. The comparative study will be conducted to show how reliable, sufficient, and simpler is the Adomian decomposition method.