On solutions of variable-order fractional differential equations

Article History: Received 11 July 2016 Accepted 22 November 2016 Available 20 January 2017 Numerical calculation of the fractional integrals and derivatives is the code to search fractional calculus and solve fractional differential equations. The exact solutions to fractional differential equations are compelling to get in real applications, due to the nonlocality and complexity of the fractional differential operators, especially for variable-order fractional differential equations. Therefore, it is significant to enhance numerical methods for fractional differential equations. In this work, we consider variable-order fractional differential equations by reproducing kernel method. There has been much attention in the use of reproducing kernels for the solutions to many problems in the recent years. We give an example to demonstrate how efficiently our theory can be implemented in practice.


Introduction
Fractional differential equations have been studied by many investigators in recent years.The notion of a variable order operator is a much more recent improvement.Different authors have presented different definitions of variable order differential operators.The kernel of the variable order operators is too complex for having a variableexponent.Therefore, to get the numerical solutions of variable order fractional differential equations is quite compelling.There are few studies of variable order fractional differential equations.Coimbra [1] applied a consistent approximation with first-order accurate for the solution of variable order differential equations.Lin et al. [2] worked the stability and the convergence of an explicit finite-difference approximation for the variable-order fractional diffusion equation with a nonlinear source term.Zhuang et al. [3] acquired explicit and implicit Euler approximations for the fractional advection-diffusion nonlinear equation of variable-order.For more details see [4][5][6].No one had tried to find the numerical solutions of the variable order fractional differential equations by the reproducing kernel method (RKM).
The aim of our work is to investigate the efficiency of RKM to solve variable-order fractional differential equations.Let us consider and subjected to the initial condition where C D α(ν) 0,ν is variable order fractional derivative of Caputo sense, f (ν) is the known continuous function, u(ν) is the unknown function, 0 < α min ≤ α(ν) ≤ α max < 1.
The theory of reproducing kernels was used for the first time at the beginning of the 20th century by Zaremba [7].Reproducing kernel theory has *Corresponding Author considerable implementations in numerical analysis, differential equations, probability and statistics [8][9][10][11].Some authors discussed fractional differential equations, nonlinear oscillators with discontinuity, singular nonlinear two-point periodic boundary value problems, integral equations and nonlinear partial differential equations [7,[12][13][14][15][16][17][18][19][20][21].This paper is arranged as follows.Some definitions and properties of the variable order fractional integrals and derivatives are presented in Section 2. Section 3 shows some useful reproducing kernel functions.The representation in W 2 2 [0, 1] and a related linear operator are given in Section 4. Section 5 gives the main results.Numerical experiments are demonstrated in Section 6.Some conclusions are given in the last section.

Reproducing kernel functions
Definition 1.We define the space G 1 2 [0, 1] by where AC denotes the space of absolutely continuous functions.The inner product and the norm in and Lemma 1 (See [23, page 17]).The space G 1 2 [0, 1] is a reproducing kernel space, and its reproducing kernel function Q y is given by Definition 2. We describe the space W 2 2 [0, 1] by The inner product and the norm in and Lemma 2 (See [23, page 148]).The space W 2 2 [0, 1] a reproducing kernel space, and its reproducing kernel function is given by
Proof.We need to show Lu 2 , where M > 0 is a positive constant.We get We obtain , by reproducing property.Therefore, we get where M 1 > 0. Therefore, we get where M 2 > 0. Therefore, we obtain , and Thus, we get where
The approximate solution u n (ν) can be acquired as:

Numerical results
To prove the efficiency and the practicability of the RKM, we give an example and find its solution.

Conclusion
We used the reproducing kernel method to solve a class of the variable order fractional differential equation in this work.We defined the method and used it in the test example in order to prove its applicability and validity in comparison with exact and other numerical solutions.The obtained results are uniformly convergent and the operator that was used is a bounded linear operator.

Table 1 .
The