Symmetry solution on fractional equation

Article History: Received 29 June 2017 Accepted 23 October 2017 Available 25 October 2017 As we know nearly all physical, chemical, and biological processes in nature can be described or modeled by dint of a differential equation or a system of differential equations, an integral equation or an integro-differential equation. The differential equations can be ordinary or partial, linear or nonlinear. So, we concentrate our attention in problem that can be presented in terms of a differential equation with fractional derivative. Our research in this work is to use symmetry transformation method and its analysis to search exact solutions to nonlinear fractional partial differential equations.

Lie symmetry analysis is powerful and universal tool for searching solution of linear and nonlinear partial differential equations and it has been widely applied for studying the invariance properties of partial differential equation (PDE) [15].A symmetry of a PDE is any transformation that each solution surface of the PDE is mapped to another solution surface of the same PDE, i.e. leaves invariant its solution space.So, by using the Lie symmetry, the equation can be transformed into a nonlinear fractional ODE.
For construction a symmetry reductions of a fractional equation we investigated the symmetry properties by using the group analysis method and presented the vector fields the equation based on the point symmetry [13,14].It is shown that our equation could be transformed into a nonlinear fractional ordinary differential equation with the new independent variable.
In this work by using the Lie group, we investigate the symmetry properties of fractional partial differential equation (FPDE) and find the correspondence infinitesimal operators and then construct some exact solution of these equations, in particulary the solution for fractional linear KdV equation.
The outline of this paper is as follows: in section 2 we will give general definitions and formulas of fractional derivative and symmetry analysis, also we show the application of symmetry group to *Corresponding Author fractional differential Eq. (1).In section 3 presented some exact solutions by using symmetry reductions.

Lie symmetry analysis of fractional PDE
Consider a time FPDE with two independent variables and 0 < α ≤ 1 is given as following: here fractional derivative are considered in the Riemann-Liouville terms.Suppose f be integrable on [a, b] ∈ R and n − 1 < α < n, n ∈ N. Then Riemann-Liouville fractional derivative is defined as Let f be integrable on [0, ∞), and piecewise continuous function on (0, ∞) and Reα > 0, t > 0.
Then Riemann-Liouville fractional integral is defined by One parameter Lie symmetry transformations are determined as where ǫ > 0 is a infinitesimals parameter with ξ = dx dǫ | ǫ=0 , τ = d t dǫ | ǫ=0 and η = dū dǫ | ǫ=0 which will be determined.After applying transformation (5) to usual partial derivatives u x , u xx and u xxx it gives the following extensions [15]: Here η x 1 , η x 2 and η x 3 are defined by formulae where D x is the total derivative And the αth extended infinitesimal related to Riemann-Liouville fractional time derivative is [16] as Here η t α has following form: and the operator D α t is the total fractional derivative operator.Using the generalized Leibnitz rule [2] Thus infinitesimal η t α is modified to The corresponding vector field V associated with transformations (5) can be written as Applying the third prolongation pr (3) V to Eq.
(2), we will get where the operator pr (3) V takes the following form: Our equation ( 2) can be written in the form Substitution of transformations ( 5), ( 6) and ( 8) into (11) we get So we find that the functions ξ(x, t, u), τ (x, t, u) and η(x, t, u) must satisfy the symmetry condition Solving the Eq. ( 12) along with Eq. ( 2) and substituting the extended infinitesimal ( 7), ( 9) into the Eq. ( 12) we get following characteristic system: Solving these equations we investigate generating infinitesimal operators as following.
Case 1: For arbitrary g(u) and 0 < α ≤ 1 there are three infinitesimal operators Case 2: For g(u) = 1 and 0 < α ≤ 1 there are two additional infinitesimal operators where the function h(t, x) satisfies the linear fractional KdV equation D α t h = h xxx .Case 3: For g(u) = u b with b = 0 and 0 < α ≤ 1 there are two additional infinitesimal operators Case 4: For g(u) = u b with b = −3 and 0 < α ≤ 1 there is one additional infinitesimal operator Case 5: For g(u) = e u with integer α (α = 1) there are two additional infinitesimal operators Theorem 1.The equation D α t u = (g(u)u xx ) x with g(u) = e u and 0 < α < 1 has no additional symmetries.
Thus by finding the corresponding derivatives and putting them to equation 2. we find that D is constant, but from 8. equation ∂ α D ∂t α = 0, which gives us D = c 5 t α−1 thereby we have obtained a contradiction.It means that for 0 < α < 1 and g(u) = e u there is not any additional symmetries.

Symmetry reductions and some exact solutions
3.1.The exact solution for g(u) = 1 For g(u) = 1 we have linear fractional KdV equation D α t u = u xxx with infinitesimal operators By composition of X 1 and X 4 we get generator where k ∈ R. Then solution under the group has the form u(t, x) = e kx where φ(t) satisfies the equation and thus Here E α,β (x) is a Mittag-Leffler function x m Γ(αm + β) .

The exact solution for
∂u solution under the group has the form where φ(t) satisfies the equation If b = 3 then we derive D α t φ(t) = 0, which gives ) .

Conclusion
For construction a symmetry reductions of the fractional equation ( 1) we investigated the symmetry properties by using the symmetry analysis method and presented different infinitesimal operators.We obtained solutions for two particular equations with some generator operators.Also we showed that the equation D α t u = (e u u xx ) x for 0 < α < 1 has only general symmetries with ∂t infinitesimal operators.The symmetry analysis or Lie group analysis is a very powerful method and is worthy of studying further to searching the solutions and symmetry properties of nonlinear partial differential equations and fractional nonlinear partial differential equations.

Gulistan
Iskandarova is a Ph.D. Student at Istanbul Commerce University.She received her M.Sc. in Eurasian National University in Kazakhstan, Astana.Her research interests are nonlinear ordinary differential equations partial differential equations, fractional derivative theory, analytical methods for the nonlinear differential equations, Painleve transients, Lie groups theory.She is author and co-author of several papers, some of which appeared in Journal Republican Student Scientific Conference on Mathematics, Mechanics and Computer Science and Journal International Scientific Conference of students, graduate students and young scientists "Lomonosov".Dogan Kaya is currently professor in Department of Mathematics and Computer Science, Istanbul Commerce University.He received his PhD degree from University of Newcastle-upon-Tyne (England) in 1995.His research area includes numerical analysis, nonlinear ordinary differential equations partial differential equations, analytical methods for nonlinear differential equations and numerical solutions of the partial differential equations, mathematical programming.An International Journal of Optimization and Control: Theories & Applications (http://ijocta.balikesir.edu.tr)This work is licensed under a Creative Commons Attribution 4.0 International License.The authors retain ownership of the copyright for their article, but they allow anyone to download, reuse, reprint, modify, distribute, and/or copy articles in IJOCTA, so long as the original authors and source are credited.To see the complete license contents, please visit http://creativecommons.org/licenses/by/4.0/.