Approximate Symmetries and Conservation Laws of Fractional Differential Generalization of the Burgers Equation

Abstract

The classical Burgers equation is well studied and is often used in problems of hydrodynamics and nonlinear acoustics. In recent years, there has been a significantly increasing interest in mathematical models that take into account the power-law memory effect of the medium. Such models are described by equations in which the time derivative is replaced by a fractional derivative. The study object is a generalized Burgers equation with a fractional Riemann-Liouville time derivative. A memory effect of the medium is assumed to be small, so a small parameter, with respect to which the fractional derivative is expanded into a series, is extracted from the fractional-order differentiation. As a result, the initial fractional differential generalization of the Burgers equation is approximated by an equation with a small parameter. The aim of the work is to study the symmetry properties of such a partial differential equation with a small parameter and to construct conservation laws for it. To achieve the goal, methods of modern group analysis are used, as well as widely known methods of integrating systems and partial differential equations of first order. The group classification of the equation under study is carried out according to the function standing at the first derivative with respect to the spatial variable. It is shown that if this function is of arbitrary form, then the admissible approximate group of point transformations is three-parameter. For power-law and linear functions, the admissible approximate group of point transformations extends to five- and seven-parameter groups, respectively. Examples of approximately invariant solutions for some admissible operators are constructed. It is proved that the studied equation with a small parameter is approximately nonlinearly self-adjoint. Based on the principle of nonlinear self-adjointness, conservation laws are constructed for each group generator. It is shown that all conservation laws are either trivial or have the form of the original equation. The results develop the theory of approximate transformation groups for fractional differential equations. The obtained symmetries can be used to construct approximate invariant solutions of the equation in question.