Lie group classifications and exact solutions for time-fractional Burgers equation
ABSTRACT 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. Comment: 9 pp, accepted
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ABSTRACT: An existence and uniqueness theorem is proved for a quasilinear stochastic evolution equation with an additive noise in the form of a stochastic integral with respect to a Hilbert space-valued fractional Borwnian motion. Ideas of the finite-dimensional approximation by the Galerkin method are used.Statistics [?] Probability Letters 01/1999; 41(4):337-346. · 0.53 Impact Factor
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ABSTRACT: It is shown that, by using Taylor’s series of fractional order, the stochastic differential equation , where b(t,a) is a fractional Brownian motion of order a, can be converted into an equation involving fractional derivative, therefore a solution expressed in terms of the Mittag–Leffler function.Applied Mathematics Letters. 01/2005;
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ABSTRACT: The paper gives some results and improves the derivation of the fractional Taylor's series of nondifferentiable functions obtained recently in the form f (χ + h) = Eα (hαDχα)f(χ), 0 α ≤ 1, where Eα is the Mittag-Leffier function. Here, one defines fractional derivative as the limit of fractional difference, and by this way one can circumvent the problem which arises with the definition of the fractional derivative of constant using Riemann-Liouville definition. As a result, a modified Riemann-Liouville definition is proposed, which is fully consistent with the fractional difference definition and avoids any reference to the derivative of order greater than the considered one's. In order to support this F-Taylor series, one shows how its first term can be obtained directly in the form of a mean value formula. The fractional derivative of the Dirac delta function is obtained together with the fractional Taylor's series of multivariate functions. The relation with irreversibility of time and symmetry breaking is exhibited, and to some extent, this F-Taylor's series generalizes the fractional mean value formula obtained a few years ago by Kolwantar.Computers & Mathematics with Applications. 05/2006;
arXiv:1011.3692v1 [math-ph] 16 Nov 2010
Lie group classifications and exact solutions for time-fractional
College of Textile, Donghua University, Shanghai 201620, P.R. China;
Modern Textile Institute, Donghua University, Shanghai 200051, P.R. China.
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.
Lie group method; Fractional Burgers equation; Fractional characteristic method
Many methods of mathematical physics have been developed to solve differential equations,
among which Lie group method is an efficient approach to derive the exact solution of nonlinear
partial differential equations.
Since Sophus Lie’s group analysis work more than 100 years ago, Lie group theory has
become more and more pervasive in its influence on other mathematical disciplines [1, 2]. There
are, however, there few applications of Lie method in fractional calculus. Then a question may
naturally arise: is there a fractional Lie group method for fractional differential equations?
Some researchers investigated Lie group method for fractional differential equations in sense
of the Caputo derivative and derived scaling transformation and similarity solutions [3–5]. Con-
sidering the classical Lie group method, method of characteristic is used to solve symmetry
equations. Recently, with the modified Riemann-Liouville derivative [6–8], we first propose a
∗Corresponding author, E-mail: email@example.com. (G.C. Wu)
more generalized fractional characteristic method  than Jumarie’s Lagrange method . Us-
ing our fractional characteristic method, the generalized symmetry equations generating by the
prolongation technique can be solved, and a fractional Lie group method was presented for an
anomalous diffusion equation .
In this study, we investigate a simplified version of the fractional Burgers equations 
= uxx+ u2
x, x ∈ (0, ∞), 0<t, 0<α < 1,
and derive its group classifications. In order to investigate the local behaviors of the above
equation, the fractional derivative is in the sense of the modified Riemann-Liouville [6–8].
2 Fractional Calculus and Some Properties
From the viewpoint of Brown motion, Jumarie proposed the modified Riemann-Liouville deriva-
Γ(n − α)
(x − ξ)n−α−1(f(ξ) − f(0)) dξ, n − 1<α < n,
where the derivative on the right-hand side is the Riemann-Liouville fractional derivative and
n ∈ Z+.
(a) Fractional Taylor series
Recently, Jumarie-Taylor series  was proposed
(kα)!f(kα)(x), 0<α < 1.
Here f(x) is a kα-differentiable function and k is an arbitrary positive integer.
Taking k = 1, f(x) is a α-differentiable function. We can derive that
Γ(1 + α)
(b) Fractional Leibniz product law
If we set Dα
xu(x) and Dα
xv(x) exist, we can readily find that
x(uv) = u(α)v + uv(α).
The properties of Jumarie’s derivative were summarized in . The extension of Jumarie’s
fractional derivative and integral to variations approach by Almeida et al. [12, 13]. Fractional
variational interactional method and Adomian decomposition method are proposed for fractional
differential equations [14, 15].
(c) Integration with respect to (dx)α
(x − ξ)α−1f(ξ)dξ =
Γ(α + 1)
f(ξ)(dξ)α,0 < α ≤ 1.
(d) Generalized Newton-Leibniz Law
xf(x) is an integrable function in the interval [0,a]. Obviously,
Γ(1 + α)
xf(x)(dx)α= f(a) − f(0),0 < α < 1,
Γ(1 + α)
ξf(ξ)(dξ)α= f(x) − f(a),
Γ(1 + α)
f(ξ)(dξ)α= f(x), 0 < α < 1.
(e) Some other useful properties
dxx(α)(t), 0 < α < 1,
Γ(1 + β)
Γ(1 + β − α)xβ−α, 0 < β < 1,
The above properties (a)–(d) can be found in Ref. . We must point out that f(x) should
be differentiable w.r.t x in Eq. (10), and x
βis an α order function in Eq. (11).
3A Characteristic Method for Fractional Differential Equations
It is well known that the method of characteristics has played a very important role in math-
ematical physics. Preciously, the method of characteristics is used to solve the initial value
problem for general first order. With the modified Riemann-Liouville derivative, Jumaire ever
gave a Lagrange characteristic method , in which the time-fractional order equals to the
space-fractional order. We present a more generalized fractional method of characteristics and
use it to solve linear fractional partial equations.
Consider the following first order equation,
The goal of the method of characteristics is to change coordinates from (x, t) to a new coor-
dinate system (x0, s) in which the partial differential equation becomes an ordinary differential
equation along certain curves in the x−t plane. The curves are called the characteristic curves.
More generally, we consider to extend this method to linear space-time fractional differential
With the fractional Taylor’s series in two variables 
= c(x,t), 0 < α,β < 1.
Γ(1 + β)∂xβ(dx)
Γ(1 + α)∂tα(dt)α, 0 < α, β < 1,
similarly, we derive the generalized characteristic curves
Γ(1 + β)ds= a(x,t),
Γ(1 + α)ds= b(x,t).
Eqs. (16)–(18) can be reduced as Jumaire’s Lagrange method of characteristic if α = β in .
4 A Fractional Lie Group Method
In the classical Lie method for partial differential equations, the one-parameter Lie group of
transformations in (x, t, u) is given by
˜ x = x + εξ(x,t,u) + O(ε2),
˜t = t + ετ(x,t,u) + O(ε2),
˜ u = u + εφ(x,t,u) + O(ε2),
where ε is the group parameter.
Use the set of fractional vector fields instead of the one of integer order
V = ξ(x,t,u)Dβ
t+ φ(x,t,u)Du, 0 < α < 1, 0 < β < 1.
For the fractional second order prolongation Pr(2β)V of the infinitesimal generators, we
Pr(2β)V = V + φ[t] ∂φ
tu+ φ[x] ∂φ
As a result, we can have
Pr(2β)V (∆[u]) = 0,
on∆[u] = 0.
In the time-fractional Burgers equation, Eq. (1), we only need to consider the case of the
fractional order of space β = 1. Thus, the corresponding Lie algebra of infinitesimal symmetries
is the set of fractional vector fields in the form
V = ξ(x,t,u)Dx+ τ(x,t,u)Dα
We assume the one-parameter Lie group of transformations in (x, t, u) given by
˜ x = x + εξ(x,t,u) + O(ε),
˜ u = u + εφ(x,t,u) + O(ε),
Γ(1+α)+ ετ(x,t,u) + O(ε),
where ε is the group parameter.
The generalized second prolongation satisfies
Pr(2)V = V + φt ∂φ
Using the following condition
Pr(2)V (∆[u]) = 0, ∆[u] = 0,
we can have
(φt− φxx− 2uxφx)??∆[u]=0= 0.
The generalized prolongation vector fields are reduced as
φx= Dxφ − (Dxξ)Dxu − (Dxτ)Dα
tφ − (Dα
tξ)Dxu − (Dα
xφ − 2(Dxξ)D2
xu − (D2
xξ)Dxu − 2(Dxτ)DxDα
tu − (D2
Substituting Eq. (27) into Eq. (26) and setting the coefficients of uxu(α)
and 1 to zero. Solve the equations with maple software, we can have
xt, uxxux, ux
ξ(x,t,u) = c1+ c4x + 2c5
τ(x,t,u) = c2+ 2c4
Γ(1+α)− c6x2+ a(x,t)eu,
φ(x,t,u) = c3− c5x +
The Lie algebra of infinitesimal symmetries of Eq. (1) is spanned by the vector field
∂u, V4= x∂