Exact solutions of forced Burgers equations with time variable coefficients

• Published in: Communications in nonlinear science & numerical simulation Vol. 18; no. 7; pp. 1635 - 1651
• Main Authors: Büyükaşık, Şirin A., Pashaev, Oktay K.
• Format: Journal Article
• Language: English
• Published: Elsevier B.V 01.07.2013
• Subjects: Burgers equation; C-integrable; Damping; Decay; Differential equations; Exact solutions; Forced Burgers equation; Initial value problems; Mathematical analysis; Mathematical models; Pole dynamics; Shock waves; Variable parameters; C-integrable; Shock waves; Pole dynamics; Forced Burgers equation; Exact solutions; Variable parameters
• ISSN: 1007-5704; 1878-7274
• DOI: 10.1016/j.cnsns.2012.11.027

► Exactly solvable variable parametric forced Burgers equations are considered. ► Exact explicit solutions are found for forced Burgers equations with damping and decaying diffusion coefficient. ► Shock and multi-shock waves, triangular waves and rational type solutions are found and illustrated graphically. ► Dynamics of pole singularities is described. In this paper, we consider a forced Burgers equation with time variable coefficients of the form Ut+(μ˙(t)/μ(t))U+UUx=(1/2μ(t))Uxx-ω2(t)x, and obtain an explicit solution of the general initial value problem in terms of a corresponding second order linear ordinary differential equation. Special exact solutions such as generalized shock and multi-shock waves, triangular wave, N-wave and rational type solutions are found and discussed. Then, we introduce forced Burgers equations with constant damping and an exponentially decaying diffusion coefficient as exactly solvable models. Different type of exact solutions are obtained for the critical, over and under damping cases, and their behavior is illustrated explicitly. In particular, the existence of inelastic type of collisions is observed by constructing multi-shock wave solutions, and for the rational type solutions the motion of the pole singularities is described.