Generalized Taylor Series Method for Solving Nonlinear Fractional Differential Equations with Modified Riemann-Liouville Derivative

We propose an efficient analytic method for solving nonlinear differential equations of fractional order. The fractional derivative is defined in the sense of modified Riemann-Liouville derivative. A new technique for calculating the generalized Taylor series coefficients (also known as “generalized differential transforms,” GDTs) of nonlinear functions and a new approach of the generalized Taylor series method (GTSM) are presented. This new method offers a simple algorithm for computing GDTs of nonlinear functions and avoidsmassive computational work that usually arises in the standardmethod. Several illustrative examples are demonstrated to show effectiveness of the proposed method.


Introduction
Fractional differential equations are generalizations of classical differential equations of integer order and have recently proved to be valuable tools in the modeling of many phenomena in various fields of science and engineering.Apart from diverse areas of mathematics, fractional differential equations arise in rheology, viscoelasticity, chemical physics, electrical networks, fluid flows, control, and dynamical processes in self-similar and porous structures.There has appeared lots of work in which fractional derivatives are used for a better description of considered material properties; mathematical modelling based on enhanced rheological models naturally leads to differential equations of fractional order and to the necessity of the formulation of initial conditions to such equations.Several numerical methods for solving fractional differential equations have been introduced lately.The authors in [1] presented the predictor-corrector approach based on the Adam-Bashforth-Moulton type numerical method that has been successful in obtaining stable approximations for solving many fractional differential equations.Some of the semianalytic methods such as the Adomian decomposition method (ADM) [2,3], homotopy analysis method (HAM) [4][5][6], homotopy perturbation method (HPM) [7,8], variational iteration method (VIM) [9,10], and generalized differential transform method (GDTM) [11][12][13] have been introduced to provide analytic or numeric approximations.
In this paper we focus on the generalized Taylor series method (GTSM), which is based on the generalized Taylor series.The fractional derivative is defined in the sense of the modified Riemann-Liouville derivative [14].From the given fractional differential equation, the GTSM provides a simple recurrence relation of the generalized Taylor series coefficients of the solution.We obtain recurrence relations of complex nonlinear functions such as the exponential, logarithmic, and trigonometric functions.The paper is organized as follows.Section 2 introduces some preliminary results from the fractional calculus that we will use.The basic idea and some properties of GTSM are presented in Section 3. We present the recurrence relations of complex nonlinear functions in Section 4. In Section 5, numerical results of several examples are demonstrated by using new recurrence relations.Finally, we give a conclusion in Section 6.

Preliminary Results
In [14], Jumarie proposed a definition for fractional derivative which is known as the modified Riemann-Liouville derivative in the literature.Since then, many authors have investigated various applications of the modified Riemann-Liouville derivative (e.g., see [15][16][17]) including various fractional calculus formulae, the fractional variational iteration method, and the fractional subequation method for solving fractional partial differential equations.The definition and some of the key properties of modified Riemann-Liouville derivative are which do not hold for classical Riemann-Liouville and Caputo derivatives.Particularly chain rule (also known as "Faà di Bruno's formula" in fractional calculus) plays the key role in our method.

Generalized Taylor Series Method
The Taylor series method (TSM) (also known as differential transform method, DTM) that is based on the Taylor series has been successful in achieving accurate approximate solutions for the linear and nonlinear problems.In TSM, all coefficients of Taylor series of the solution can be determined by solving the recurrence equations induced from the given differential equation.The authors in [18] developed the fractional differential transform method (FDTM) which is based on the classical TSM and generalized with fractional derivative.The generalized Taylor formula is introduced in [11] and has been used to develop GDTM in [12].The authors in [19] proposed a new algorithm for calculating the differential transforms of several nonlinear functions based on the chain rule.The purpose of this paper is to obtain efficient algorithms to calculate generalized Taylor series coefficients of complex nonlinear functions.
Theorem 1 (generalized Taylor's formula [11]).Suppose that (   )  () ∈ (0, ] for  = 0, 1, . . .,  + 1, where 0 <  ≤ 1; then one has for all  ∈ (0, ] with 0 ≤  ≤ . For an analytic function (), let us define the generalized differential transform (GDT) of the th derivative as follows: where 0 <  ≤ 1,  = 0, 1, 2, . .., and the generalized differential inverse transform of () is defined as follows: Roughly speaking, coefficient of the th term in the generalized Taylor series of the function  is called the GDT of the th derivative of .In case of  = 1, the GDT reduces to the classical differential transform.Some of the fundamental properties of GDT are listed below.
The proof of these results for Caputo derivative can be found in [12] and clearly holds for modified Riemann-Liouville derivative also.

Calculating the Coefficients of Generalized Taylor Series for Complex Nonlinear Functions
In this section we will introduce an efficient algorithm to calculate generalized Taylor series coefficients (GDTs) of several complex nonlinear functions.Since the chain rule holds for the modified Riemann-Liouville derivative, the algorithms for calculating the GDTs of typical nonlinear functions given in [19] can be totally adopted to nonlinear fractional differential equations with modified Riemann-Liouville derivative.

Error Analysis.
From Theorem 1, if (   )  () ∈ (0, ] for  = 0, 1, . . .,  + 1, where 0 <  ≤ 1, then we have for all  ∈ (0, ].Furthermore, there is a value  with 0 ≤  ≤  so that the error term    () has the form So, if () is GDT of the th derivative of the analytic function , we have and the error term has the form for some  ∈ [0, ].Absolute value of the error done with approximation can be calculated by maximizing the right hand side of following inequality: for  ∈ [0, ].The accuracy of    () increases when we choose large  and decreases as the value of  moves away from the center 0. Hence, we must choose  large enough so that the error does not exceed a specified bound.

Conclusion
In this study we proposed a generalized Taylor series method (GTSM) based on the generalized Taylor formula for solving nonlinear differential equations of fractional order.We consider the fractional differential equations with modified Riemann-Liouville derivative.In GTSM, it is key to obtain recursive relationships from the given differential equation.However, the recurrence relations for the complex nonlinear functions for fractional differential equations with modified Riemann-Liouville derivative have not been derived before.The GTSM with new recurrence relations given in this study offers a simple algorithm to compute the generalized Taylor series coefficients.Thus, it is worthwhile to mention that the GTSM is a straightforward, promising, and powerful method for solving nonlinear fractional differential equations.

Figure 1 :
Figure 1: Approximate solutions obtained by GTSM in Example 1 for different order of derivatives.

Figure 2 :
Figure 2: Approximate solutions obtained by GTSM in Example 2 for different order of derivatives.

Figure 3 :
Figure 3: Approximate solutions obtained by GTSM in Example 3 for different order of derivatives.

Figure 4 :
Figure 4: Approximate solutions obtained by GTSM in Example 4 for different order of derivatives.

Table 8 :
Approximate solutions obtained by GTSM in Example 3 for different order of derivatives.