Symmetry analysis and conservation laws to the space-fractional Prandtl equation

Abstract

This paper studies the Lie point symmetries, conservation laws and invariant solutions to the space-fractional Prandtl equation with the Riemann–Liouville derivative. Exploiting the classical Lie symmetry analysis method extended to fractional equations, three vector fields are calculated, which were used to reduce the fractional partial differential equation into an ordinary differential equation through similarity transformation. Further, it is found that the fractional Prandtl equation is nonlinearly self-adjoint. Therefore, the nonlinear self-adjointness method not requiring the existence of a Lagrangian is utilized to explore the conservation laws for this equation. Three conservation laws are constructed, and one of them is a trivial conservation law. Finally, due to the difficulties of obtaining analytical solution for the ordinary equation, similarity solutions are presented numerically.