Lie group classifications and exact solutions for time-fractional Burgers equation

Lie group method provides an efficient tool to solve nonlinear partial differential equations. This paper suggests a fractional Lie group method for fractional partial differential equations. A time-fractional Burgers equation is used as an example to illustrate the effectiveness of the Lie group method and some classes of exact solutions are obtained.

💡 Research Summary

The paper introduces a systematic extension of the classical Lie‑group method to partial differential equations that contain time‑fractional derivatives, and demonstrates the approach on the time‑fractional Burgers equation. The authors adopt the Caputo definition for the fractional time derivative, which preserves the usual initial‑value interpretation while allowing the use of standard calculus tools. Recognizing that the traditional prolongation formulas are not directly applicable to fractional derivatives, they construct a “fractional prolongation” operator that correctly accounts for the non‑local nature of the Caputo derivative.

Starting from an infinitesimal point transformation
(x^{}=x+\varepsilon\xi(x,t,u),; t^{}=t+\varepsilon\tau(x,t,u),; u^{*}=u+\varepsilon\eta(x,t,u)),
they apply the fractional prolongation to the time‑fractional Burgers equation
({}^{C}D_{t}^{\alpha}u+u,u_{x}= \nu u_{xx}) (with (0<\alpha\le 1) and viscosity (\nu)). By demanding invariance of the equation under the transformation, a system of determining equations for the coefficient functions (\xi,\tau,\eta) is derived. Solving this overdetermined system yields three non‑trivial symmetry generators:

1. A time‑scaling symmetry that mixes the temporal variable with the spatial coordinate through the fractional order (\alpha).
2. A combined space‑translation/scale symmetry that also acts linearly on the dependent variable.
3. A nonlinear exponential symmetry acting solely on (u).

These generators form a Lie algebra that can be used to construct similarity variables. For example, using the first generator leads to the similarity variable (\zeta = x,t^{-\alpha}) and a reduced dependent variable (v(\zeta)=u(x,t)). Substituting this ansatz eliminates the fractional derivative and reduces the original 1+1‑dimensional PDE to an ordinary differential equation (ODE) of the form
(\nu v’’ + v v’ + \alpha \zeta v’ = 0).

Further exploitation of the other symmetries reduces the ODE to a Riccati‑type or Bernoulli‑type equation, which can be integrated analytically. The authors present explicit exact solutions expressed through the Mittag‑Leffler function, e.g.
(v(\zeta)= -2\nu \frac{d}{d\zeta}\ln!\bigl