Exact solutions of forced Burgers equations with time variable coefficients

Abstract

a b s t r a c tIn this paper, we consider a forced Burgers equation with time variable coefficients of the form2 ðtÞx, 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.

Riccati–Ermakov systems and explicit solutions for variable coefficient reaction–diffusion equations

“…In this work we also generalized the transformation for a vcBE presented in [67] (see our proposition 2) where Langevin equations and the Hill equation were used to express the transformation. Instead of these approaches, in proposition 2 we will use what we have called the Riccati system, extending the results presented in [12] and [63]; further, our solutions will show multiparameters and we present a new symmetry for Burgers equation . Tables 3 and 4 shows several examples of families with explicit solutions.…”

“…They also considered random terms. Our approach is similar to [63] and [12] with the advantage of using our Riccati system that allows multiparameters. As an application of our multiparameter approach we present a new symmetry for Burgers equation, see proposition 2.…”

Generalized heat diffusion equations with variable coefficients and their fractalization from the Black-Scholes equation

“…where f (T) and g(T) are arbitrary smooth functions. It can be proved that equation (28) may be reduced via a change of the variable T to the well-known variable coefficient generalized Burgers' equations with linear damping of the form [58,59]:…”

Green’s functions and the Cauchy problem of the Burgers hierarchy and forced Burgers equation

“…Recently Büyükaşik and Pashaev [33] and Schulze-Halberg [34] detailed a linearising transform of the Burgers equation with general forcing term and time dependent coefficients. Using results from these aforementioned studies, we focus on generating solutions to the Cauchy problem of the two following systems: 1) the Burgers hierarchy with general time dependent coefficients,…”