We investigate the symmetry properties of a variable coefficient space-time fractional potential Burgers’ equation. Fractional Lie symmetries and corresponding infinitesimal generators are obtained. With the help of the infinitesimal generators, some group invariant solutions are deduced. Further, some exact solutions of fractional potential Burgers’ equation are generated by the invariant subspace method.

1. Introduction

Partial differential equations (PDEs) have a wide range of applications in many fields, such as physics, engineering, and chemistry, which are fundamental for the mathematical formulation of continuum models [1, 2]. Burgers’ equation is a one-dimensional nonlinear partial differential equation, which is a simple form of the one-dimensional Navier-Stokes equation. It was presented for the first time in a paper in 1940s by Burgers. Later, Burgers’ equation was studied by Cole [3] who gave a theoretical solution, based on Fourier series analysis, using the appropriate initial and boundary conditions. Burgers’ equation has a large variety of applications in the modelling of water in unsaturated soil, dynamics of soil water, statistics of flow problems, mixing and turbulent diffusion, cosmology, and seismology [3, 4].

Recently, fractional differential equations have found extensive applications in many fields. Many important phenomena in viscoelasticity, electromagnetics, material science, acoustics, and electrochemistry are elegantly described with the help of fractional-order differential equations [4–9]. It has been revealed that the nonconservative forces can be described by fractional differential equations. Therefore, as most of the processes in the real physical world are nonconservative, the fractional calculus can be used to describe them. Fractional integrals and derivatives also appear in the theory of control of dynamical systems, when the controlled system and/or the controller is described by a fractional differential equation [10, 11]. In the last few decades, the subject of the fractional calculus has caught the consideration of many researchers who contributed to its development. Recently, some analytical and numerical methods have been introduced to solve a fractional-order differential equation [7, 8, 11–24]. However, all the methods have insufficient development as they allow one to find solutions only in case of linear equations and for some isolated examples of nonlinear equations [4, 7, 8, 11, 23, 24].

It is very well known that the Lie group method is the most effective technique in the field of applied mathematics to find exact solutions of ordinary and partial differential equations [25, 26]. However, this approach is not yet applied much to investigate symmetry properties of fractional differential equations (FDEs). To the best of our knowledge, there are a few papers (e.g., [17, 26–32]) in which Lie symmetries and similarity solutions of some fractional differential equations have been discussed by some researchers. More recently, Jumarie [33] proposed the modified Riemann-Liouville derivative and Jumarie-Lagrange method [34], after which a generalized fractional characteristic method and a fractional Lie group method have been introduced by Wu [32, 35] in order to solve a fractional-order partial differential equation.

In this paper, we intend to apply Lie group method to solve the space-time fractional potential Burgers’ equation of the form

(1)

where

and

are the modified Riemann-Liouville derivatives with respect to time and space variables, respectively, and f(t) and g(t) are arbitrary functions of t. Equation (1) is connected to the fractional Burgers’ equation by the well-known Hopf-Cole transformations and is a generalization of the time-fractional Burgers’ equation with constant coefficients examined by Wu [32]. This work is based on some basic elements of fractional calculus, with special emphasis on the modified Riemann-Liouville type derivative. The paper is organized as follows. In Section 2, we briefly describe some definitions and properties of fractional calculus. In Section 3, we obtain the symmetries for Burgers’ equation having six-dimensional Lie algebra. In Section 4, we analyze the reduced systems and find some invariant solutions of (1). Section 5 contains application of invariant subspace method on fractional Burgers’ equation (1). Finally, a conclusion is given in Section 6.

2. Some Concepts from Fractional Calculus

In this paper, the modified Riemann-Liouville derivative proposed by Jumarie [33] has been adopted. Some definitions are given which have been used in this work.

2.1. Fractional Riemann-Liouville Integral

The fractional Riemann-Liouville integral of a continuous (but not necessarily differentiable) real valued function f(x) with respect to (dx) ^α is defined as [7, 33, 36]

(2)

The fractional integral with respect to (dt) ^α was introduced by Jumarie [36] in order to study the fractional derivative of nondifferentiable functions in modified Riemann-Liouville sense. Here we are fully in Leibniz framework; that is to say (dx) ^α denote finite increment in fractional sense. As a result, we shall be able to duplicate, in a straightforward manner, most of the known standard formulae by merely making the substitution (dx) ^α → dx.

2.2. Riemann-Liouville Fractional Derivative

The fractional Riemann-Liouville derivative of f(x) is defined as [7]

(3)

2.3. Modified Riemann-Liouville Derivative

Through the fractional Riemann-Liouville integral, Jumarie [33] proposed the modified Riemann-Liouville derivative of f(x) as

(4)

2.4. Some Useful Formulae

Here, some properties of modified Riemann-Liouville derivative are given which have been used in this paper:

(i)
.
(ii)
.
(iii)
, given that df/dx exists.
(iv)
, x > 0, and β > −1.
(v)
∫ (dx) ^β = x^β, 0 < β ≤ 1.
(vi)
Γ(1 + β)dx = d^βx.

The above formulae and details thereof along with the scope of applications and limitations can be found in [33, 37].

2.5. Characteristic Method for Fractional-Order Differential Equations

Applying the fractional chain rule proposed by Jumarie [33], we have

(5)

Wu [32, 35] extended the characteristic method of first-order linear partial differential equation to a linear fractional differential equation of the form

(6)

and introduced the fractional characteristic equations [32] as given herein:

(7)

On solving the characteristic equations (7) for various infinitesimal generators of (1), one can obtain the similarity variables and reduction of (1) to an ODE.

3. Symmetry Classification of (1)

Herein, we investigate the symmetries and reductions of space-time fractional potential Burgers’ equation (1). A fractional Lie symmetry of (1) is a continuous group of point transformations of independent and dependent variables which leaves (1) invariant.

Let us assume that (1) admits the Lie symmetries of the form

(8)

where ϵ is the group parameter and ξ, τ, and η are the infinitesimals of the transformations for the independent and dependent variables, respectively. A group invariant solution of space-time fractional potential Burgers’ equation (1) is a solution which can be mapped into another solution of (1) under the point transformations (8). The associated Lie algebra of infinitesimal symmetries of (1) is then the fractional vector field of the form

(9)

The fractional second-order prolongation of (9) is given by

(10)

Now, for the invariance of (1) according to [26] under (8), we must have

(11)

where

or equivalently where

(12)

The generalized fractional prolongation vector fields η^x, η^t, and η^xx are given by

(13)

Now, using the above generalized fractional prolongation vector fields in (12) and equating the coefficient of various derivative terms to zero, we get the following simplified set of determining equations:

(14)

(15)

(16)

(17)

(18)

(19)

(20)

On solving (20) by using fractional Lie group method, we obtain a particular solution as

(21)

where

and a₁, a₂, and a₃ are arbitrary constants. Using this value of η in (17) and (18), we get

(22)

which brings forth the following possibilities:

(i)
.
(ii)
g(t) = f(t).

Case (i). In this case, from the determining equations, we get

(23)

where

, and a₁, a₂, …, a₆ are six arbitrary constants. Using (23) in (17), we also get g(t) = kf(t), where k is an arbitrary constant. This covers case (ii); as for f(t) = g(t) on solving the determining equations, we get

. Further, for f(t) = g(t) = 1 and β = 1, the infinitesimals can be reduced to those reported in [32] by setting the coefficients a₁ = −c₅, a₂ = −2c₆, a₃ = c₃, a₄ = c₄, a₅ = c₁, and a₆ = c₂.

Hence, the fractional point symmetry generators admitted by (1) are given by

(24)

These infinitesimal generators can be used to determine a six-parameter fractional Lie group of point transformation acting on (x, t, u)-space. It can be verified easily that the set {V₁, V₂, V₃, V₄, V₅, V₆} forms a six-dimensional Lie algebra under the Lie bracket [X, Y] = XY − YX, which reduces to the well-known generalized Galilea algebra [26] for α = β = 1. The commutator table is as given in Table 1.

	V₁	V₂	V₄	V₅	V₆
V₁	0	0	V₁	−V₃	2V₅
V₂	0	0	2V₂	2V₁	4V₄ − 2V₃
V₃	0	0	0	0	0
V₄	−V₁	−2V₂	0	−V₅	2V₆
V₅	V₃	−2V₁	V₅	0	0
V₆	−2V₅	2V₃ − 4V₄	−2V₆	0	0

Further, from the commutator table, it can be seen that the sets {V₃} and {V₁, V₂, V₃} form solvable subalgebra. Also, V₃ is the centre of the six-dimensional Lie algebra as it commutes with every element of the Lie algebra. The group transformation generated by the infinitesimal generators V_i, i = 1,2, …, 6, is obtained by solving the system of ordinary differential equations:

(25)

with the initial conditions

(26)

Exponentiating the infinitesimal symmetries of (1), we get the one-parameter groups g_i(ϵ) generated by V_i, i = 1,2, …, 6:

(27)

Now, since g_i is a symmetry, if u = χ(x^β/Γ(1 + β), t^α/Γ(1 + α)) is a solution of (1), following u_i are also solutions of (1):

(28)

4. Some Exact Solutions of the Space-Time Fractional Burgers’ Equation

In this section, we investigate some exact solutions of (1) corresponding to following infinitesimal generators:

(i)
V₁
(ii)
V₄
(iii)
nV₅ + mV₃
(iv)
rV₅ + V₆ and
(v)
sV₃ + V₆

r, s, m, and n are nonzero arbitrary constant parameters.

Theorem 1. Under the group of transformations T(x, t) = t^α/Γ(1 + α) and ϕ(T) = x^2β/Γ(1 + 2β) + 2G(t)u(x, t), Burgers’ equation (1) reduces to a linear differential equation of first order: ϕ^′(T) − H₁(T)ϕ(T) = −H₂(T), where and , which admits a solution given by , where k₁ is an arbitrary constant.

Proof. Consider the infinitesimal generator V₁, given by

(29)

We find the resulting invariant solution by reducing (1) to a linear ordinary differential equation (32) using differential invariants. The fractional characteristic equations for V₁ are

(30)

On solving the above fractional characteristic equations, we obtain two functionally independent invariants as T = t^α/Γ(1 + α) and ν = x^2β/Γ(1 + 2β) + 2G(t)u.

Now the solution of the fractional characteristic equations will be of the form ν = ϕ(T); therefore

(31)

Substituting this value of u in (1), we get the reduced linear ordinary differential equation as

(32)

where T = t^α/Γ(1 + α),

, and

. On solving (32), we obtain

, where k₁ is an arbitrary constant. This gives

(33)

Theorem 2. The similarity transformations u(x, t) = ψ(X) along with the similarity variable reduce fractional Burgers’ equation (1) to a nonlinear ordinary differential equation ψ^′′(X) + (g(t)/f(t))(ψ^′(X)) ² + (1/4)ψ^′(X) + (1/2X)ψ^′(X) = 0, which leads to the solution , where k₂ and k₃ are arbitrary constants.

Proof. Let us consider the infinitesimal generator

(34)

Here the fractional characteristic equations give the invariants

and u(x, t) = ψ(X). On substituting the above invariants in (1), it becomes a nonlinear ordinary differential equation of second order:

(35)

which leads to the solution

(36)

Theorem 3. Under the transformations ζ(x, t) = t^α/Γ(1 + α) and φ(ζ) = x^β/Γ(1 + β) − (n/m)u(x, t), fractional Burgers’ equation (1) reduces to an ordinary fractional differential equation , which has the general solution as u(x, t) = (m/n)[x^β/Γ(1 + β) + (m/n)G(t) − k₄], with k₄ being an arbitrary constant.

Proof. In this case, we study the infinitesimal generator

(37)

The following invariants can be derived easily: ζ(x, t) = t^α/Γ(1 + α) and ν = x^β/Γ(1 + β) − (n/m)u(x, t) and the reduced form of (1) as

(38)

This easily yields the solution

(39)

Theorem 4. Under the transformations and ω(μ) = u(x, t), fractional Burgers’ equation (1) reduces to a nonlinear ordinary differential equation ω^′′(μ) + kω^′2(μ) + (r²/Γ(1 + α))ω^′(μ) = 0, which admits the solution − , where c₁ and c₂ are arbitrary constants.

Proof. Results can be easily derived by solving the fractional characteristic equations for the infinitesimal generator rV₅ + V₆.

Theorem 5. The similarity transformations γ(x, t) = x^β/Γ(1 + β) and reduce Burgers’ equation (1) to a nonlinear ordinary differential equation ρ^′′(γ) − (ks/Γ(1 + α))(ρ^′(γ)) ² + 1 = 0, which has the general solution as , where c₃ and c₄ are arbitrary constants.

Proof. The proof is very much similar to the previous theorems.

5. Some Exact Solutions of Fractional Potential Burgers’ Equation by the Invariant Subspace Method

The invariant subspace method was introduced by Galaktionov [38] in order to discover exact solutions of nonlinear partial differential equations. The method was further applied by Gazizov and Kasatkin [39] and Sahadevan [40] to some nonlinear fractional-order differential equations. Here, we give a brief description of the method.

Consider the fractional evolution equation , where u = u(x, t) and F[u] is a nonlinear differential operator.

The n-dimensional linear space W_n = 〈f₁(x), …, f_n(x)〉 is called invariant under the operator F[u], if F[u] ∈ W_n for any u ∈ W_n, which means there exist n functions ϕ₁, …, ϕ_n such that F[C₁f₁(x)+⋯+C_nf_n(x)] = ϕ₁(C₁, …, C_n)f₁(x)+⋯+ϕ_n(C₁, …, C_n)f_n(x), where C₁, …, C_n are arbitrary constants and {ϕ_n} are the expansion coefficients of F[u] ∈ W_n in the basis {f_i}. It follows that an exact solution of fractional evolution equation can be obtained as

, where the coefficient functions a₁(t), a₂(t), …, a_n(t) satisfy a system of fractional ODEs:

(40)

For further details, the reader is referred to [40].

For (1),

. We have the space W₃ = 〈1, x^β/Γ(1 + β), x^2β/Γ(1 + 2β)〉 as invariant under F[u], if and only if, for any u(x, t) = a₁(t) + a₂(t)(x^β/Γ(1 + β)) + a₃(t)(x^2β/Γ(1 + 2β)) ∈ W₃, F[u] ∈ W₃ or equivalently if

(41)

where b₁, b₂, and b₃ are arbitrary constants. This allows us to consider an exact solution of (1) as

(42)

Substituting the value of u(x, t) from (42) into (1) and equating the coefficients of the elements of space W₃ = 〈1, x^β/Γ(1 + β), x^2β/Γ(1 + 2β)〉, we get the following system of fractional differential equations:

(43)

Equations (43) can be readily solved to yield

(44)

where the functions f(t) and g(t) are integrable in the sense of Riemann-Liouville and

On solving (44), we get

(45)

where B = −(Γ(1 + β)) ²/Γ(1 + 2β).

Using (42) and (45), one can easily obtain an exact solution of (1).

6. Conclusion

The main purpose of Lie symmetry method is to reduce PDEs to ODEs by introducing suitable similarity variable and similarity solutions. Here, in this paper, the authors show that the fractional potential Burgers’ equation possesses similarity solutions exactly as its counterparts with integer-order derivatives. By using conveniently defined similarity variables, fractional potential Burgers’ equation reduces to ordinary differential equations which are further solved to derive some group invariant solutions. The authors also utilised the invariant subspace method to deduce some exact solutions of fractional potential Burgers’ equation. Software like Mathematica and Maple has been utilised in solving some ordinary differential equations.

Competing Interests

The authors declare that they have no competing interests.

Acknowledgments

One of the authors (Manoj Gaur) would like to thank the University Grants Commission (UGC), New Delhi, India, for providing Research Fellowship under the scheme UGC-CSIR NET JRF in Science, Humanities & Social Sciences.

References

1 Brezis H. and Browder F., Partial differential equations in the 20th century, Advances in Mathematics. (1998) 135, no. 1, 76–144, https://doi.org/10.1006/aima.1997.1713, MR1617413, 2-s2.0-0013181662.
10.1006/aima.1997.1713
Web of Science® Google Scholar
2 Debtnath L., Nonlinear Partial Differential Equations for Scientist and Engineers, 1997, Birkhäuser, Boston, Mass, USA.
10.1007/978-1-4899-2846-7
Google Scholar
3 Cole J. D., On a quasi-linear parabolic equation occurring in aerodynamics, Quarterly of Applied Mathematics. (1951) 9, 225–236, MR0042889.
10.1090/qam/42889
Web of Science® Google Scholar
4 Barkai E., Fractional Fokker-Planck equation, solution, and application, Physical Review E. (2001) 63, no. 4, 46118, 2-s2.0-0035305938.
10.1103/PhysRevE.63.046118
CAS PubMed Web of Science® Google Scholar
5 He J. H., Some applications of nonlinear fractional differential equations and their approximations, Bulletin of Science, Technology & Society. (1999) 15, no. 2, 86–90.
Google Scholar
6 Jaradat H. M., Awawdeh F., Al-Shara S., Alquran M., and Momani S., Controllable dynamical behaviors and the analysis of fractal Burgers hierarchy with the full effects of inhomogeneities of media, Romanian Journal of Physics. (2015) 60, no. 3-4, 324–343, 2-s2.0-84928991580.
Web of Science® Google Scholar
7 Miller K. S. and Ross B., An Introduction to the Fractional Calculus and Fractional Differential Equations, 1993, John Wiley & Sons, New York, NY, USA, MR1219954.
Google Scholar
8 Oldham K. B. and Spanier J., The Fractional Calculus, 1974, Academic Press, New York, NY, USA, MR0361633.
Google Scholar
9 West B., Bologna M., and Grigolini P., Physics of Fractal Operators, 2012, Springer Science & Business Media, New York, NY, USA.
Google Scholar
10 Caponetto R., Dongola G., Fortuna L., and Petr I., Fractional Order Systems (Modelling and Control Applications), 2010, World Scientific, Singapore.
10.1142/7709
Google Scholar
11 Podlubny I., Fractional Differential Equations: An Introduction to Fractional Derivatives, Fractional Differential Equations, to Methods of Their Solution and Some of Their Applications, 1998, 198, Academic press, Cambridge, Mass, USA.
Google Scholar
12 Al-Khaled K. and Alquran M., An approximate solution for a fractional model of generalized Harry Dym equation, Mathematical Sciences. (2014) 8, no. 4, 125–130, https://doi.org/10.1007/s40096-015-0137-x, MR3317540.
10.1007/s40096-015-0137-x
Google Scholar
13 Alquran M., Al-Khaled K., Sardar T., and Chattopadhyay J., Revisited Fisher′s equation in a new outlook: a fractional derivative approach, Physica A. Statistical Mechanics and Its Applications. (2015) 438, 81–93, https://doi.org/10.1016/j.physa.2015.06.036, MR3384270, 2-s2.0-84937060549.
10.1016/j.physa.2015.06.036
Web of Science® Google Scholar
14 Alquran M., Analytical solution of time-fractional two-component evolutionary system of order 2 by residual power series method, The Journal of Applied Analysis and Computation. (2015) 5, no. 4, 589–599, MR3367454.
Web of Science® Google Scholar
15 Arikoglu A. and Ozkol I., Solution of fractional differential equations by using differential transform method, Chaos, Solitons & Fractals. (2007) 34, no. 5, 1473–1481, https://doi.org/10.1016/j.chaos.2006.09.004, 2-s2.0-34250215244.
10.1016/j.chaos.2006.09.004
Web of Science® Google Scholar
16 Chen Y. and An H.-L., Numerical solutions of coupled Burgers equations with time- and space-fractional derivatives, Applied Mathematics and Computation. (2008) 200, no. 1, 87–95, https://doi.org/10.1016/j.amc.2007.10.050, MR2421626, 2-s2.0-43049157795.
10.1016/j.amc.2007.10.050
Web of Science® Google Scholar
17 Gazizov R. K., Kasatkin A. A., and Yu Lukashchuk S., Continuous transformation groups of fractional differential equations, Vestnik USATU. (2007) 9, no. 3 (21), 125–135.
Google Scholar
18 Klimek M. and Agrawal O. P., Space- and time-fractional legendre-pearson diffusion equation, Proceedings of the ASME International Design Engineering Technical Conferences and Computers and Information in Engineering Conference (IDETC/CIE ′13), August 2013, https://doi.org/10.1115/detc2013-12604, 2-s2.0-84899116869.
10.1115/detc2013-12604
Google Scholar
19 Li Z.-B. and He J.-H., Fractional complex transform for fractional differential equations, Mathematical & Computational Applications. (2010) 15, no. 5, 970–973, https://doi.org/10.3390/mca15050970, MR2777702.
10.3390/mca15050970
Web of Science® Google Scholar
20 Merdan M. and Gökdoğan A., Solution of nonlinear oscillators with fractional nonlinearities by using the modified differential transformation method, Mathematical and Computational Applications. (2011) 16, no. 3, 761–772, https://doi.org/10.3390/mca16030761, MR2839364.
10.3390/mca16030761
Google Scholar
21 Momani S. and Al-Khaled K., Numerical solutions for systems of fractional differential equations by the decomposition method, Applied Mathematics and Computation. (2005) 162, no. 3, 1351–1365, https://doi.org/10.1016/j.amc.2004.03.014, MR2113975, ZBL1063.65055, 2-s2.0-12244291039.
10.1016/j.amc.2004.03.014
Web of Science® Google Scholar
22 Odibat Z. and Momani S., Numerical solution of Fokker-Planck equation with space- and time-fractional derivatives, Physics Letters, Section A: General, Atomic and Solid State Physics. (2007) 369, no. 5-6, 349–358, https://doi.org/10.1016/j.physleta.2007.05.002, 2-s2.0-34548650141.
10.1016/j.physleta.2007.05.002
CAS Google Scholar
23 Ross B., Fractional Calculus and Its Applications: Proceedings of the International Conference Held at the University of New Haven, June 1974, 1975, Springer, Berlin, Germany.
10.1007/BFb0067095
Google Scholar
24 Samko S. G., Kilbas A. A., and Marichev O. I., Fractional Integrals and Derivatives, 1993, Gordon and Breach, London, UK, MR1347689.
Google Scholar
25 Ibragimov N. H., CRC Handbook of Lie Group Analysis of Differential Equations, 1995, 3, CRC Press.
Google Scholar
26 Olver P. J., Applications of Lie Groups to Differential Equations, 2000, 107, Springer Science & Business Media, New York, NY, USA, Graduate Texts in Mathematics.
Google Scholar
27 Atanacković T. M., Konjik S., Pilipović S., and Simić S., Variational problems with fractional derivatives: invariance conditions and Nöther′s theorem, Nonlinear Analysis: Theory, Methods & Applications. (2009) 71, no. 5-6, 1504–1517, https://doi.org/10.1016/j.na.2008.12.043, 2-s2.0-67349097718.
10.1016/j.na.2008.12.043
Web of Science® Google Scholar
28 Djordjevic V. D. and Atanackovic T. M., Similarity solutions to nonlinear heat conduction and Burgers/Korteweg-de Vries fractional equations, Journal of Computational and Applied Mathematics. (2008) 222, no. 2, 701–714, https://doi.org/10.1016/j.cam.2007.12.013, MR2474657, 2-s2.0-53549087275.
10.1016/j.cam.2007.12.013
Web of Science® Google Scholar
29 Gaur M. and Singh K., On group invariant solutions of fractional order Burgers-Poisson equation, Applied Mathematics and Computation. (2014) 244, 870–877, https://doi.org/10.1016/j.amc.2014.07.053, MR3250626, 2-s2.0-84907359365.
10.1016/j.amc.2014.07.053
Web of Science® Google Scholar
30 Liu H., Complete group classifications and symmetry reductions of the fractional fifth-order KdV types of equations, Studies in Applied Mathematics. (2013) 131, no. 4, 317–330, https://doi.org/10.1111/sapm.12011, MR3125928, 2-s2.0-84886998657.
10.1111/sapm.12011
Web of Science® Google Scholar
31 Hashemi M. S., Group analysis and exact solutions of the time fractional Fokker-Planck equation, Physica A: Statistical Mechanics and its Applications. (2015) 417, 141–149, https://doi.org/10.1016/j.physa.2014.09.043, MR3274721, 2-s2.0-84907993246.
10.1016/j.physa.2014.09.043
Web of Science® Google Scholar
32 Wu G.-C., A fractional Lie group method for anomalous diffusion equations, Communications in Fractional Calculus. (2010) 1, 27–31.
PubMed Web of Science® Google Scholar
33 Jumarie G., Modified Riemann-Liouville derivative and fractional Taylor series of nondifferentiable functions further results, Computers and Mathematics with Applications. (2006) 51, no. 9-10, 1367–1376, https://doi.org/10.1016/j.camwa.2006.02.001, 2-s2.0-33745742268.
10.1016/j.camwa.2006.02.001
Web of Science® Google Scholar
34 Guy J., Lagrange characteristic method for solving a class of nonlinear partial differential equations of fractional order, Applied Mathematics Letters. (2006) 19, no. 9, 873–880, https://doi.org/10.1016/j.aml.2005.10.016, MR2240477, 2-s2.0-33745108563.
10.1016/j.aml.2005.10.016
Web of Science® Google Scholar
35 Wu G.-C., Lie group classifications and non-differentiable solutions for time-fractional Burgers equation, Communications in Theoretical Physics. (2011) 55, no. 6, 1073–1076, https://doi.org/10.1088/0253-6102/55/6/23, 2-s2.0-79959391555.
10.1088/0253-6102/55/6/23
Web of Science® Google Scholar
36 Jumarie G., From self-similarity to fractional derivative of non-differentiable functions via Mittag-Leffler function, Applied Mathematical Sciences. (2008) 2, no. 37–40, 1949–1962, MR2443892.
Google Scholar
37 Jumarie G., Fractional Differential Calculus for Non-Differentiable Functions: Mechanics, Geometry, Stochastics, Information Theory, 2013, Lambert Academic Publishing, Saarbrucken, Germany.
Google Scholar
38 Galaktionov V. A., Posashkov S. A., and Svirshchevskii S. R., Exact Solutions and Invariant Subspaces of Nonlinear Partial Differential Equations in Mechanics and Physics, 2006, CRC Press, Boca Raton, Fla, USA.
10.1201/9781420011623
Google Scholar
39 Gazizov R. K. and Kasatkin A. A., Construction of exact solutions for fractional order differential equations by the invariant subspace method, Computers & Mathematics with Applications. (2013) 66, no. 5, 576–584, https://doi.org/10.1016/j.camwa.2013.05.006, 2-s2.0-84881477753.
10.1016/j.camwa.2013.05.006
Web of Science® Google Scholar
40 Sahadevan R. and Bakkyaraj T., Invariant subspace method and exact solutions of certain nonlinear time fractional partial differential equations, Fractional Calculus and Applied Analysis. (2015) 18, no. 1, 146–162, https://doi.org/10.1515/fca-2015-0010, MR3316534.
10.1515/fca-2015-0010
Web of Science® Google Scholar

Citing Literature

All articles

Symmetry Classification and Exact Solutions of a Variable Coefficient Space-Time Fractional Potential Burgers’ Equation

Abstract

1. Introduction

2. Some Concepts from Fractional Calculus

2.1. Fractional Riemann-Liouville Integral

2.2. Riemann-Liouville Fractional Derivative

2.3. Modified Riemann-Liouville Derivative

2.4. Some Useful Formulae

2.5. Characteristic Method for Fractional-Order Differential Equations

3. Symmetry Classification of (1)

4. Some Exact Solutions of the Space-Time Fractional Burgers’ Equation

5. Some Exact Solutions of Fractional Potential Burgers’ Equation by the Invariant Subspace Method

6. Conclusion

Competing Interests

Acknowledgments

References

Citing Literature

References

Information

About Wiley Online Library

Help & Support

Opportunities

Connect with Wiley

Symmetry Classification and Exact Solutions of a Variable Coefficient Space-Time Fractional Potential Burgers’ Equation

Abstract

1. Introduction

2. Some Concepts from Fractional Calculus

2.1. Fractional Riemann-Liouville Integral

2.2. Riemann-Liouville Fractional Derivative

2.3. Modified Riemann-Liouville Derivative

2.4. Some Useful Formulae

2.5. Characteristic Method for Fractional-Order Differential Equations

3. Symmetry Classification of (1)

4. Some Exact Solutions of the Space-Time Fractional Burgers’ Equation

5. Some Exact Solutions of Fractional Potential Burgers’ Equation by the Invariant Subspace Method

6. Conclusion

Competing Interests

Acknowledgments

References

Citing Literature

References

Related

Information