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

Authors: Süleyman Öğrekçi
Journal: Advances in Mathematical Physics
 (2015)

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## Abstract

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 avoids massive computational work that usually arises in the standard method. Several illustrative examples are demonstrated to show effectiveness of the proposed method.


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## Body

## 1. 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) [4–6], homotopy perturbation method (HPM) [7, 8], variational iteration method (VIM) [9, 10], and generalized differential transform method (GDTM) [11–13] 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.


## 2. 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 [15–17]) 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(1)Dtαft=1Γ1-addt·∫0t‍t-ξ-αfξ-f0dξ,0<α<1fntα-n,1≤n≤α<n+1,Dtαtβ=Γ1+βΓ1+β-αtβ-α,Dtαftgt=gtDtαft+ftDtαgt,Dtαfgt=fg′gtDtαgt=Dgαfgtg′tα,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.


## 3. 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.For the fractional differential operatorDα,  m-1<α≤m, in the sense of modified Riemann-Liouville definition, let us define Dαn=Dα·Dα⋯Dα (n times).Theorem 1 (generalized Taylor’s formula [11]).
Suppose thatDtαkft∈C0,b for k=0,1,…,n+1, where 0<α≤1; then one has (2)ft=∑i=0n‍tiαΓiα+1Dtαif0+Dtαn+1fηΓn+1α+1tn+1αfor all t∈0,b with 0≤η≤t.For an analytic functionft, let us define the generalized differential transform (GDT) of the kth derivative as follows:(3)Fk=1Γαk+1Dtαkftt=0,where 0<α≤1, k=0,1,2,…, and the generalized differential inverse transform of Fk is defined as follows:(4)ft=∑k=0∞‍Fktαk.Roughly speaking, coefficient of the kth term in the generalized Taylor series of the function f is called the GDT of the kth derivative of f. In case of α=1, the GDT reduces to the classical differential transform. Some of the fundamental properties of GDT are listed below.Theorem 2.
Suppose thatFk, Gk, and Hk are the generalized differential transforms of the functions ft, gt, and ht, respectively. Then the following properties are satisfied.(1)
Ifft=gt±ht, then Fk=Gk±Hk.(2)
Ifft=agt, where a is a constant, then Fk=aGk.(3)
Ifft=gtht, then Fk=∑l=0k‍GlHk-l.(4)
Ifft=Dtαgt, then Fk=Γαk+1+1/Γαk+1Gk+1.The proof of these results for Caputo derivative can be found in [12] and clearly holds for modified Riemann-Liouville derivative also.


## 4. 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.


### 4.1. GDTs for the Exponential and Logarithmic Nonlinearity

Let us considerfyt=eayt, where a is a constant. From the definition of transform,(5)F0=eaytt=0=eay0=eaY0.Now taking fractional derivative of fyt=eayt with respect to variable t, we have(6)Dtαfyt=aeaytDtαyt=afytDtαyt.With the application of Theorem 2 to (6) to obtain the generalized Taylor series coefficients, we have (7)Γαk+1+1Γαk+1Fk+1=a∑l=0k‍Γαl+1+1Γαl+1·Yl+1Fk-l.Now replacing k+1 with k gives(8)Fk=aΓαk-1+1Γαk+1∑l=0k-1‍Γαl+1+1Γαl+1·Yl+1Fk-1-lfor k≥1. Combining (5) and (8), we obtain the recursive relationship for calculating GDTs of fyt=eayt:(9)Fk=eaY0,k=0,aΓαk-1+1Γαk+1∑l=0k-1‍Γαl+1+1Γαl+1·Yl+1Fk-1-l,k≥1.Now considerfyt=ln⁡a+byt, where a and b are constants and y is a continuous function satisfying a+by(t)>0 for all t∈R. By definition of transform, we have(10)F0=ln⁡a+bytt=0=ln⁡a+by0=ln⁡a+bY0.By taking fractional derivative of fyt=ln⁡a+byt with respect to variable t, we have(11)Dtαfyt=ba+bytDtαytor equivalently(12)aDtαfyt=bDtαyt-ytDtαfyt.With the application of the generalized differential transform to (12) we have, from Theorem 2,(13)aFk+1=bYk+1-Γαk+1Γαk+1+1∑l=0k‍·∑l=0k‍Γαl+1+1Γαl+1Fl+1Yk-l.Replacing k+1 with k gives(14)aFk=bYk-Γαk-1+1Γαk+1∑l=0k-1‍·∑l=0k-1‍Γαl+1+1Γαl+1Fl+1Yk-1-l.Substitute k=1 in (14) to get(15)F1=ba+bY0Y1.For k≥2, (14) can be rewritten as(16)Fk=ba+bY0·Yk-Γαk-1+1Γαk+1∑l=0k-2‍·∑l=0k-2‍Γαl+1+1Γαl+1Fl+1Yk-1-l.By combining (10), (15), and (16), we obtain the recursive relationship for calculating GDTM of fyt=ln⁡a+byt:(17)Fk=ln⁡a+bY0,k=0,ba+bY0Y1,k=1,ba+bY0Yk-Γαk-1+1Γαk+1∑l=0k-2‍·∑l=0k-2‍Γαl+1+1Γαl+1∑l=0k-2‍·Fl+1Yk-1-l,k≥2.


### 4.2. GDTMs for the Trigonometric and Hyperbolic Nonlinearity

Now considerfyt=sinayt and gyt=cos⁡ayt, where a and b are constants. By definition of GDT,(18)F0=sinaytt=0=sinay0=sinaY0,G0=cos⁡aytt=0=cos⁡ay0=cos⁡aY0.Further, taking fractional derivative of (18), we have(19)Dtαfyt=agytDtαyt,Dtαgyt=-afytDtαyt.Similarly, applying GDT with the aid of Theorem 2, we have(20)Γαk+1+1Γαk+1Fk+1=a∑l=0k‍Γαk-l+1+1Γαk-l+1GlYk-l+1,Γαk+1+1Γαk+1Gk+1=-a∑l=0k‍Γαk-l+1+1Γαk-l+1FlYk-l+1.Replacing k+1 with k gives(21)Fk=aΓαk-1+1Γαk+1·∑l=0k-1‍Γαk-l+1Γαk-l-1+1GlYk-l,Gk=-aΓαk-1+1Γαk+1·∑l=0k-1‍Γαk-l+1Γαk-l-1+1FlYk-l.Combine (18) and (21) to obtain recurrence relation:(22)Fk=sinaY0,k=0,aΓαk-1+1Γαk+1·∑l=0k-1‍Γαk-l+1Γαk-l-1+1GlYk-l,k≥1,(23)Gk=cos⁡aY0,k=0,-aΓαk-1+1Γαk+1·∑l=0k-1‍Γαk-l+1Γαk-l-1+1FlYk-l,k≥1.For the functions fyt=sinh⁡ayt and gyt=cosh⁡ayt, we immediately obtain the recursive relationships:(24)Fk=sinh⁡aY0,k=0,aΓαk-1+1Γαk+1·∑l=0k-1‍Γαk-l+1Γαk-l-1+1GlYk-l,k≥1,Gk=cosh⁡aY0,k=0,aΓαk-1+1Γαk+1·∑l=0k-1‍Γαk-l+1Γαk-l-1+1FlYk-l,k≥1.


### 4.3. Error Analysis

From Theorem1, if Dtαkft∈C(0,b] for k=0,1,…,n+1, where 0<α≤1, then we have (25)ft≅Sn(t)=∑i=0n‍tiαΓiα+1Dtαif0for all t∈(0,b]. Furthermore, there is a value η with 0≤η≤t so that the error term Rnα(t) has the form (26)Rnα(t)=Dtαn+1fηΓn+1α+1tn+1α.So, if F(k) is GDT of the kth derivative of the analytic function f, we have (27)ft≅Sn(t)=∑k=0n‍F(k)tαk,and the error term has the form (28)Rnα(t)=F(n+1)ηα(n+1)for some η∈[0,t]. Absolute value of the error done with approximation can be calculated by maximizing the right hand side of following inequality:(29)Rnαt=Fn+1ηα(n+1)for η∈[0,t]. The accuracy of Pkα(t) increases when we choose large k and decreases as the value of t moves away from the center 0. Hence, we must choose k large enough so that the error does not exceed a specified bound.


## 4.1. GDTs for the Exponential and Logarithmic Nonlinearity

Let us considerfyt=eayt, where a is a constant. From the definition of transform,(5)F0=eaytt=0=eay0=eaY0.Now taking fractional derivative of fyt=eayt with respect to variable t, we have(6)Dtαfyt=aeaytDtαyt=afytDtαyt.With the application of Theorem 2 to (6) to obtain the generalized Taylor series coefficients, we have (7)Γαk+1+1Γαk+1Fk+1=a∑l=0k‍Γαl+1+1Γαl+1·Yl+1Fk-l.Now replacing k+1 with k gives(8)Fk=aΓαk-1+1Γαk+1∑l=0k-1‍Γαl+1+1Γαl+1·Yl+1Fk-1-lfor k≥1. Combining (5) and (8), we obtain the recursive relationship for calculating GDTs of fyt=eayt:(9)Fk=eaY0,k=0,aΓαk-1+1Γαk+1∑l=0k-1‍Γαl+1+1Γαl+1·Yl+1Fk-1-l,k≥1.Now considerfyt=ln⁡a+byt, where a and b are constants and y is a continuous function satisfying a+by(t)>0 for all t∈R. By definition of transform, we have(10)F0=ln⁡a+bytt=0=ln⁡a+by0=ln⁡a+bY0.By taking fractional derivative of fyt=ln⁡a+byt with respect to variable t, we have(11)Dtαfyt=ba+bytDtαytor equivalently(12)aDtαfyt=bDtαyt-ytDtαfyt.With the application of the generalized differential transform to (12) we have, from Theorem 2,(13)aFk+1=bYk+1-Γαk+1Γαk+1+1∑l=0k‍·∑l=0k‍Γαl+1+1Γαl+1Fl+1Yk-l.Replacing k+1 with k gives(14)aFk=bYk-Γαk-1+1Γαk+1∑l=0k-1‍·∑l=0k-1‍Γαl+1+1Γαl+1Fl+1Yk-1-l.Substitute k=1 in (14) to get(15)F1=ba+bY0Y1.For k≥2, (14) can be rewritten as(16)Fk=ba+bY0·Yk-Γαk-1+1Γαk+1∑l=0k-2‍·∑l=0k-2‍Γαl+1+1Γαl+1Fl+1Yk-1-l.By combining (10), (15), and (16), we obtain the recursive relationship for calculating GDTM of fyt=ln⁡a+byt:(17)Fk=ln⁡a+bY0,k=0,ba+bY0Y1,k=1,ba+bY0Yk-Γαk-1+1Γαk+1∑l=0k-2‍·∑l=0k-2‍Γαl+1+1Γαl+1∑l=0k-2‍·Fl+1Yk-1-l,k≥2.


## 4.2. GDTMs for the Trigonometric and Hyperbolic Nonlinearity

Now considerfyt=sinayt and gyt=cos⁡ayt, where a and b are constants. By definition of GDT,(18)F0=sinaytt=0=sinay0=sinaY0,G0=cos⁡aytt=0=cos⁡ay0=cos⁡aY0.Further, taking fractional derivative of (18), we have(19)Dtαfyt=agytDtαyt,Dtαgyt=-afytDtαyt.Similarly, applying GDT with the aid of Theorem 2, we have(20)Γαk+1+1Γαk+1Fk+1=a∑l=0k‍Γαk-l+1+1Γαk-l+1GlYk-l+1,Γαk+1+1Γαk+1Gk+1=-a∑l=0k‍Γαk-l+1+1Γαk-l+1FlYk-l+1.Replacing k+1 with k gives(21)Fk=aΓαk-1+1Γαk+1·∑l=0k-1‍Γαk-l+1Γαk-l-1+1GlYk-l,Gk=-aΓαk-1+1Γαk+1·∑l=0k-1‍Γαk-l+1Γαk-l-1+1FlYk-l.Combine (18) and (21) to obtain recurrence relation:(22)Fk=sinaY0,k=0,aΓαk-1+1Γαk+1·∑l=0k-1‍Γαk-l+1Γαk-l-1+1GlYk-l,k≥1,(23)Gk=cos⁡aY0,k=0,-aΓαk-1+1Γαk+1·∑l=0k-1‍Γαk-l+1Γαk-l-1+1FlYk-l,k≥1.For the functions fyt=sinh⁡ayt and gyt=cosh⁡ayt, we immediately obtain the recursive relationships:(24)Fk=sinh⁡aY0,k=0,aΓαk-1+1Γαk+1·∑l=0k-1‍Γαk-l+1Γαk-l-1+1GlYk-l,k≥1,Gk=cosh⁡aY0,k=0,aΓαk-1+1Γαk+1·∑l=0k-1‍Γαk-l+1Γαk-l-1+1FlYk-l,k≥1.


## 4.3. Error Analysis

From Theorem1, if Dtαkft∈C(0,b] for k=0,1,…,n+1, where 0<α≤1, then we have (25)ft≅Sn(t)=∑i=0n‍tiαΓiα+1Dtαif0for all t∈(0,b]. Furthermore, there is a value η with 0≤η≤t so that the error term Rnα(t) has the form (26)Rnα(t)=Dtαn+1fηΓn+1α+1tn+1α.So, if F(k) is GDT of the kth derivative of the analytic function f, we have (27)ft≅Sn(t)=∑k=0n‍F(k)tαk,and the error term has the form (28)Rnα(t)=F(n+1)ηα(n+1)for some η∈[0,t]. Absolute value of the error done with approximation can be calculated by maximizing the right hand side of following inequality:(29)Rnαt=Fn+1ηα(n+1)for η∈[0,t]. The accuracy of Pkα(t) increases when we choose large k and decreases as the value of t moves away from the center 0. Hence, we must choose k large enough so that the error does not exceed a specified bound.


## 5. Numerical Illustrations

Example 1.
Consider the following nonlinear fractional differential equation:(30)Dtαyt+eyt=0for t>0 and 0<α<1, subject to initial condition y0=0. Applying GTSM to (30) with the aid of Theorem 2, we have the following recurrence relation:(31)Γαk+1+1Γαk+1Yk+1+Fk=0,k≥0,where Fk is the GDT of eyt and is given by (9) as(32)Fk=aΓαk-1+1Γαk+1∑l=0k-1‍Γαl+1+1Γαl+1·Yl+1Fk-1-lfor k≥1. From the initial condition y0=0, we have Y0=0 and F0=eY0=1.
Solving the recurrence relations (31) and (32) together gives all GDTs. Table 1 shows the values of Fk and Yk for α=0.9, t=0.05, and k≤10. We set the number of iteration n=5 which gives the approximation yt≈Snt=∑k=0n‍Yktαk and the absolute error bound |R50.9(t)|=|Y(6)|η6η∈[0,0.05]≤5.480973×10-8. We see that the speed of convergence of Yk is sufficient to obtain accurate approximations even in 5 iterations. In Table 2, we summarize these approximations of solution for α=0.9 and t=0.01,…,0.1.
Approximations of the solutionyt for different order of fractional derivatives are given in Figure 1. We set iteration number n=5 and summarize approximate solutions for α=0,6,…,0.9.Table 1
F
(
k
)  and  Y(k)  values in Example 1 for α=0.9, t=0.05, and  k≤10.
k
F
k
Y
k
0
1
0
1
−1.039754
−1.039754
2
1.241336
0.644852
3
−1.146326
−0.177535
4
0.955850
0.020540
5
−0.732814
−0.001364
6
0.521868
0.000058
7
−0.350469
-
1.7
×
1
0
-
6
8
0.223641
3.64
×
1
0
-
8
9
−0.136464
-
5.92
×
1
0
-
10
10
0.080040
7.52
×
1
0
-
12Table 2
Approximate solutions with the error bound|R50.9(t)|≤5.480973×10-8 in Example 1 obtained by GTSM for α=0.9.
t
y
t
≈
S
n
t
:
n
=
5
0.01
−0.016317
0.02
−0.030191
0.03
−0.043136
0.04
−0.055448
0.05
−0.067265
0.06
−0.078667
0.07
−0.089710
0.08
−0.100432
0.09
−0.110863
0.1
−0.121026Figure 1
Approximate solutions obtained by GTSM in Example1 for different order of derivatives.Example 2.
Consider the nonlinear fractional differential equation(33)Dtαyt=yt+ytln⁡ytfor t>0 and 0<α<1, with the initial data y0=1. Applying GTSM to (33), we have the following recurrence relation:(34)Γαk+1+1Γαk+1Yk+1=Yk+∑l=0k‍YlFk-l,where Fk is the GDT of ln⁡yt and is given by (17) as(35)Fk=ln⁡Y0,k=0,Y1Y0,k=1,1Y0Yk-Γαk-1+1Γαk+1∑l=0k-2‍·∑l=0k-2‍Γαl+1+1Γαl+1∑l=0k-2‍·Fl+1Yk-1-l,k≥2.From the initial data y0=1, we have Y0=1 and F0=ln⁡Y0=0. By using this in (34), we have (36)Y1=1Γα+1Y0+Y0Y1=1Γα+1.And by using this in (35) we obtain (37)F1=Y1Y0=1Γα+1.So we have(38)Fk=0,k=0,1Γα+1,k=1,Yk-Γαk-1+1Γαk+1∑l=0k-2‍Γαl+1+1Γαl+1·Fl+1Yk-1-l,k≥2.Solving the recurrence relations (34) and (38) together gives all GDTs. Table 3 shows the values of Fk and Yk for α=0.9, t=0.05, and k≤10. From Table 4, we see that the speed of convergence of Yk is sufficient to obtain accurate approximations with 10 iterations. We set the number of iteration n=10 which gives the approximation yt≈Snt=∑k=0n‍Yktαk and the error bound |R100.9(t)|=|Y(11)|η11η∈[0,0.05]≤|Y(11)|0.05(11). We summarize the approximations of solution for α=0.9 and t=0.01,…,0.1 in Figure 2 and Table 5.Table 3
F
(
k
)  and  Y(k)  values in Example 2 for   α=0.9, t=0.05, and  k≤10.
k
F
k
Y
k
0
0
1
1
1.039754
1.039754
2
0.596484
1.192968
3
0.239770
1.153880
4
0.074731
1.014274
5
0.019104
0.830763
6
0.004152
0.642560
7
0.000786
0.473775
8
0.0000132
0.335368
9
0.000020
0.229136
10
2.7
×
1
0
-
7
0.151740Table 4
S
n
(
t
) and |Rnα(t)| values in Example 2 for α=0.9, t=0.05.
n
y
t
≈
S
n
t
|
R
n
α
(
t
)
|
5
1.075952287
6.058317
×
1
0
-
8
10
1.075952351
1.287679
×
1
0
-
14
50
1.075952351
1.281773
×
1
0
-
73
100
1.075952351
1.378753
×
1
0
-
152
500
1.075952351
2.864704
×
1
0
-
783
1000
1.075952351
1.358354
×
1
0
-
1554Table 5
Approximate solutions obtained by GTSM in Example2 for different order of derivatives.
t
y
t
≈
S
n
t for α=0.06
y
t
≈
S
n
t for α=0.07
y
t
≈
S
n
t for α=0.08
y
t
≈
S
n
t for α=0.09
0.01
1.078558
1.047288
1.028575
1.017330
0.02
1.126255
1.078509
1.049768
1.031825
0.03
1.169182
1.107859
1.070435
1.046551
0.04
1.210235
1.136106
1.090613
1.061220
0.05
1.250579
1.163895
1.110622
1.075952
0.06
1.290840
1.191577
1.130639
1.090816
0.07
1.331417
1.219377
1.150782
1.105859
0.08
1.372593
1.247450
1.171131
1.121118
0.09
1.414586
1.275919
1.191753
1.136620
0.1
1.457579
1.304879
1.212697
1.152390Figure 2
Approximate solutions obtained by GTSM in Example2 for different order of derivatives.Example 3.
Consider the fractional differential equation(39)Dtαyt+sinyt=0for t>0 and 0<α<1, with the initial data y0=3π/4. Applying GTSM to (39), we have the following recurrence relation:(40)Γαk+1+1Γαk+1Yk+1+Fk=0,where Fk is the GDT of sinyt and is given by (22) as(41)Fk=Γαk-1+1Γαk+1∑l=0k-1‍Γαk-l+1Γαk-l-1+1GlYk-land Gk is the GDT of cos⁡yt which is given by (23) as(42)Gk=-Γαk-1+1Γαk+1∑l=0k-1‍Γαk-l+1Γαk-l-1+1·FlYk-lfor k≥1.
From the initial datay0=3π/4, we have Y0=3π/4, F0=sin3π/4=2/2, and G0=cos⁡3π/4=-2/2. Solving the recurrence relations (40), (41), and (42) together gives all GDTs. Table 5 shows the values of Fk and Yk for α=0.9, t=0.05, and k≤10. Table 6 seems a bit complicated and does not give hint about speed of convergence of approximations; however, we see from Table 7 that the speed of convergence of Snt is sufficient to obtain accurate approximations even with 5 steps of iteration. We set the number of iteration n=5 which gives the approximation yt≈Snt=∑k=0n‍Yktαk and the error bound |R50.9(t)|=|Y(6)|η6η∈[0,0.05]≤|Y(6)|0.056≤1.712146×10-10. Approximations of the solution yt for different order of fractional derivatives are given in Table 8 and Figure 3. We set iteration number n=5 and summarize approximate solutions for α=0,6,…,0.9.Table 6
F
(
k
)  and  Y(k)  values in Example 3 for   α=0.9, t=0.05, and  k≤10.
k
F
k
G
k
Y
k
0
0.707106
−0.707106
2.356194
1
0.519877
0.519877
−0.735217
2
1.14
×
1
0
-
10
0.421777
−0.298242
3
−0.228527
0.108642
4.61
×
1
0
-
11
4
−0.142652
−0.100730
0.071226
5
0.008355
−0.112319
0.036468
6
0.066994
−0.029668
−0.001815
7
0.039284
0.029484
−0.012690
8
−0.003954
0.032013
−0.006604
9
−0.019730
0.007836
0.000598
10
−0.011001
−0.008954
0.002717Table 7
S
n
(
t
) and |Rnα(t)|    values in Example 3 for   α=0.9, t=0.05.
n
y
t
≈
S
n
t
|
R
n
α
(
t
)
|
0
2.356194490
0.049600
1
2.306593692
0.001357
2
2.305236270
1.416166
×
1
0
-
14
3
2.305236270
1.475491
×
1
0
-
6
4
2.305237745
5.096609
×
1
0
-
8
5
2.305237796
1.712146
×
1
0
-
10
10
2.305237796
1.833138
×
1
0
-
16
50
2.305237796
5.118854
×
1
0
-
71
100
2.305237796
7.834481
×
1
0
-
139Table 8
Approximate solutions obtained by GTSM in Example3 for different order of derivatives.
t
y
t
≈
S
n
t for α=0.06
y
t
≈
S
n
t for α=0.07
y
t
≈
S
n
t for α=0.08
y
t
≈
S
n
t for α=0.09
0.01
2.304460
2.324576
2.336903
2.344467
0.02
2.276386
2.304185
2.322322
2.334189
0.03
2.252983
2.286388
2.308990
2.324333
0.04
2.232084
2.270016
2.296360
2.341710
0.05
2.212826
2.254587
2.284196
2.305237
0.06
2.194756
2.239842
2.272369
2.295867
0.07
2.177605
2.225624
2.260779
2.286568
0.08
2.161196
2.211829
2.249432
2.277322
0.09
2.145406
2.198384
2.238231
2.268112
0.1
2.130145
2.185236
2.227165
2.258928Figure 3
Approximate solutions obtained by GTSM in Example3 for different order of derivatives.Example 4.
Consider the nonlinear fractional differential equation(43)Dtαyt-sinh⁡yt-cos⁡yt=0for t>0 and 0<α<1, with the initial data y0=0. Applying GTSM to (43), we have the following recurrence relations:(44)Yk+1=Γαk+1Γαk+1+1F1k+G1k,where F1k and G1k are the GDTs of sinh⁡yt and cos⁡yt, which are given by(45)F1k=Γαk-1+1Γαk+1∑l=0k-1‍Γαk-l+1Γαk-l-1+1F1k=·F2lYk-l,F2k=Γαk-1+1Γαk+1∑l=0k-1‍Γαk-l+1Γαk-l-1+1F1k=·F1lYk-l,G1k=-Γαk-1+1Γαk+1∑l=0k-1‍Γαk-l+1Γαk-l-1+1G1k·G2lYk-l,G2k=Γαk-1+1Γαk+1∑l=0k-1‍Γαk-l+1Γαk-l-1+1G1k·G1lYk-lfor k≥1. From the initial data y0=0, we have Y0=0, F10=sinh⁡0=0, F20=cosh⁡⁡0=1, G10=cos⁡⁡0=1, and G20=sin0=0. Solving the recurrence relations (44), (45) together gives all GDTs.
Table9 gives the values of F1k, F2k, G1k, G2k, and Yk for α=0.9, t=0.05, and k≤10. We see from Table 10 that the speed of convergence of Snt is sufficient to obtain accurate approximations with 10 steps of iteration. Approximations of the solution yt for different order of fractional derivatives are given in Figure 4. We set iteration number n=10 and summarize approximate solutions for α=0,6,…,0.9. The numerical values are also given in Table 11.Table 9
F
1
(
k
), F2(k), G1(k), G2(k), and Y(k) values in Example 4 for   α=0.9, t=0.05, and  k≤10.
k
F
1
k
F
2
k
G
1
k
G
2
k
Y
k
0
0
1
1
0
0
1
−1.039754
0
0
−1.039754
−1.039754
2
0.569484
0.569484
0.569484
0.569484
0.569484
3
−0.239770
−0.674339
0.674339
0.239770
2.4
×
1
0
-
10
4
0.268032
0.268032
−0.118570
−0.538923
−0.135445
5
−0.285977
−0.016742
−0.317020
0.171348
−0.038209
6
0.138927
0.100162
0.181588
0.229499
0.131055
7
−0.044360
−0.198550
0.165109
−0.196168
0.060712
8
0.088423
0.134687
−0.222951
−0.085070
−0.020301
9
−0.126689
−0.039427
0.005124
0.194067
0.020357
10
0.077910
0.031339
0.140686
−0.049923
0.016741Table 10
S
n
(
t
)  values in Example 4 for   α=0.9, t=0.05.
n
y
t
≈
S
n
t
|
R
n
α
(
t
)
|
0
0
0
1
−0.07014612096
0.002714
2
−0.06743127723
7.405723
×
1
0
-
14
3
−0.06743127723
2.805799
×
1
0
-
6
5
−0.06743413643
1.235645
×
1
0
-
8
10
−0.06743412447
3.642307
×
1
0
-
15
50
−0.06743412447
4.457207
×
1
0
-
66
100
−0.06743412447
2.306171
×
1
0
-
129
1000
−0.06743412447
7.405826
×
1
0
-
1260Table 11
Approximate solutions obtained by GTSM in Example4 for different order of derivatives.
t
y
t
≈
S
n
t for α=0.06
y
t
≈
S
n
t for α=0.07
y
t
≈
S
n
t for α=0.08
y
t
≈
S
n
t for α=0.09
0.01
−0.067009
−0.042538
−0.026528
−0.016329
0.02
−0.098771
−0.067813
−0.045619
−0.030229
0.03
−0.123115
−0.088613
−0.062391
−0.043211
0.04
−0.143374
−0.106779
−0.077707
−0.055567
0.05
−0.160910
−0.123103
−0.091953
−0.067434
0.06
−0.176454
−0.138023
−0.105349
−0.078890
0.07
−0.190450
−0.151813
−0.118037
−0.089990
0.08
−0.203197
−0.164664
−0.130118
−0.100769
0.09
−0.214909
−0.176712
−0.141664
−0.111258
0.1
−0.225742
−0.186063
−0.152733
−0.121478Figure 4
Approximate solutions obtained by GTSM in Example4 for different order of derivatives.


## 6. 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.



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*Source: 507970-2015-04-08.xml*