Symmetry determination and nonlinearization of a nonlinear time-fractional partial differential equation

Abstract

<jats:p>We first show that the infinitesimal generator of Lie symmetry of a time-fractional partial differential equation (PDE) takes a unified and simple form, and then separate the Lie symmetry condition into two distinct parts, where one is a linear time-fractional PDE and the other is an integer-order PDE that dominates the leading position, even completely determining the symmetry for a particular type of time-fractional PDE. Moreover, we show that a linear time-fractional PDE always admits an infinite-dimensional Lie algebra of an infinitesimal generator, just as the case for a linear PDE and a nonlinear time-fractional PDE admits, at most, finite-dimensional Lie algebra. Thus, there exists no invertible mapping that converts a nonlinear time-fractional PDE to a linear one. We illustrate the results by considering two examples.</jats:p>

Symmetry structure of multi-dimensional time-fractional partial differential equations

“…In particular, knowing the symmetry structure in advance will greatly facilitate the computations [11,12]. In [17], we show that the infinitesimal generators of Lie symmetries of a scalar time-fractional PDE have a simple and unified expression and are completely determined by two elegant conditions which facilitates the whole procedure to be performed with the known solvers on the computer.…”

“…Lie group theory provides widely applicable techniques to study integer-order PDEs, for example, constructing similarity solutions and linearization mappings, investigating integrability, analyzing stability and global behavior of solutions, finding potential variables and nonlocally related systems, etc [11][12][13]. Concerning Lie group theory of fractional PDEs, the fractional derivatives greatly affect the admitted Lie symmetry and related symmetry properties, such as symmetry classification, symmetry reductions and exact solutions [14][15][16][17]. Buckwar and Luchko first established the invariance of a fractional diffusion equation under scaling transformations [18].…”

Local symmetry structure and potential symmetries of time‐fractional partial differential equations

“…More particularly, a linear system of time-fractional PDEs always possesses an infiniteparameter Lie group of point transformations, which provide a possibility to discriminate whether a system of nonlinear time-fractional PDEs can be linearized via an invertible mapping 35,37 .…”

An alternative technique for the symmetry reduction of time‐fractional partial differential equation

“…Furthermore, though the method is illustrated by the generalized time-fractional Burgers equation and also effective for some fractional PDEs, such as the nonlinear anomalous diffusion equations, 15,16 it also works for the following more general time-fractional PDE involving the Riemann-Liouville fractional derivative…”