An Efficient Series Solution for Nonlinear Multiterm Fractional Differential Equations

In this paper, we introduce an efficient series solution for a class of nonlinear multiterm fractional differential equations of Caputo type. The approach is a generalization to our recent work for single fractional differential equations. We extend the idea of the Taylor series expansion method to multiterm fractional differential equations, where we overcome the difficulty of computing iterated fractional derivatives, which are difficult to be computed in general. The terms of the series are obtained sequentially using a closed formula, where only integer derivatives have to be computed. Several examples are presented to illustrate the efficiency of the new approach and comparison with the Adomian decomposition method is performed.


Introduction
Fractional differential equations (FDEs) are generalization to differential equations (DEs) for noninteger orders.In recent years, FDEs caught the attention of many researchers because of their appearance in modeling several phenomenon in the physical sciences.As many FDEs do not possess exact solutions on closed forms, analytical and numerical techniques have been implemented to study these equations.Iterative methods, such as the variational iteration method (VIM) in [1], the homotopy analysis method (HAM) in [2,3], the Adomian decomposition method (ADM) in [4][5][6][7][8][9], and the fractional differential transform method in [10], have been implemented to solve various types of FDEs.These methods produce a solution in a series form whose terms are determined sequentially.We refer the reader to [11,12] for a comprehensive study of series solutions of fractional differential equations.Recently, we have introduced a new series solution for single fractional differential equations [13].The new approach is a modified form of the well-known Taylor series expansion, where we overcome the difficulty of computing iterative fractional derivatives.The efficiency of the new approach has been illustrated through several examples.In this paper we extend the idea to multiterm fractional differential equations.The presented work is a part of the Master thesis [14].We consider the left Caputo fractional derivative   0 + , defined by [15,16] provided the integral exists, where Γ is the well-known Gamma function.The left Riemann-Liouville fractional integral,   0 + , of order  ∈ R + , is defined by The left Caputo derivative is related to the left Riemann-Liouville fractional integral by It is known that Also, for  − 1 <  < ,  ∈ N,  > 0, it holds that This paper is organized as follows.In Section 2, we present the series solution of nonlinear two-term fractional differential equations.We illustrate the efficiency of the presented technique through several examples.We also compare our results with the ones obtained by the Adomian decomposition method.In Section 3, we present and illustrate the efficiency of the new series solution for three-term fractional differential equations of several types.Finally, we conclude with some remarks in Section 4.

Two-Term Fractional Differential Equations
We start with the nonlinear two-term fractional initial value problems of the form with where 0 <  2 ≤  1 < 1, and  1 and  2 are nonzero constants.We assume that (, ()) is continuous and smooth with respect to ().We also assume that  1 and  2 are rational numbers with 2.1.The Expansion Procedure.Let  = lcm( 1 ,  2 ); we have  =  1 =  2 for some ,  ∈ N.
In the following we expand the solution of problem ( 6)-( 7) in an infinite series of the form where the coefficients   :  ⩾ 0 have to be determined sequentially in the following manner: From the initial condition (7) we have (0) =  =  0 .Since   0 + () = 0, for  being constant, we have where By substituting ( 9) in ( 6) we have Applying the well-known Taylor series method to compute the coefficients {  ;  ≥ 1} will lead to computing iterated fractional derivatives, which are not easily computed in general.To avoid this difficulty, let  =   ; we have Shifting the index to zero yields To avoid the singularity at  = 0, we multiply (13) by   1 −1 ; we have Now, since and (14) has no singularity at  = 0.
We now determine   , for  ≥ +1.By performing the th derivative of (15) with respect to  and substituting  = 0, we have Using the well-known Leibniz rule for differentiating the products, we have Since we have where  =  +  2 . ( From the last equation we determine   :  ≥  + 1 and thus the solution is obtained. Remark 1.In ( 6), assuming  1 =  2 = 1, then   =   = ,  = −1 and  +  2 = 0. Thus ( 17) is reduced to which coincides with the coefficients obtained by the Taylor series expansion method.Same comment applies for the coefficient  +1 in (22).

Numerical Results
Example 3. Consider the nonlinear two-term fractional initial value problem ) with The exact solution of the problem is () =  1/2 .Applying the current algorithm, we have and  = 2.We expand the solution in an infinite series of the form The initial condition in (26) yields  0 = 0. We have  =   =  10 and Since () is continuous and smooth with respect to , we have Thus  1 =  2 =  3 = 0.The function () satisfies the assumption of the proposed algorithm, and it holds that The computation above is made using the software Mathematica version 9.For  ≥ 3; substituting (30) in ( 22) yields where (32) Applying (31) together with  1 =  2 =  3 = 0, we have Thus, and the exact solution of problem (25)-( 26) is obtained.
Example 4. Consider the nonlinear two-term fractional initial value problem with This example has been discussed in [20], where the problem is transformed to a fractional integral equation, and then the Adams-Bashforth-Moulton method is used with step size ℎ = 1/50 to approximate the solution.
Applying the current algorithm, we have  1 = 4,  1 = 5,  2 = 1,  2 = 2,  = l.c.m(5, 2) = 10,  = 2,  = 5,  = 2, and where The initial condition (0) = 1 yields  0 = 1.We apply (46) to compute the first 13 nonzero terms of   , and we have Since the exact solution of problem (44)-( 45) is not available in a closed form, we define the error   () by where Tables 1 and 2 present the error   for different values of .One can see that the error decreases with  and more accuracy can be achieved by considering more terms.Also, the error increases with , as any other series solutions.

Three-Term Fractional Differential Equations
We consider the nonlinear three-term fractional initial value problem of the form with where and  1 ,  2 , and  3 are nonzero constants.We assume that (, ()) is continuous and smooth with respect to ().
Remark 5. Our derivation is based on the facts that the nonlinear function is smooth and the fractional differential equations is of constant coefficients.In case if one of these conditions does not hold, a modified treatment will be considered as we will see in Example 7.

Numerical Results
Example 6.Consider the Bagely-Torvik initial value problem where This example has been discussed in [21] using a Chebyshev spectral method, where the solution has been approximated by the shifted Chebyshev polynomials with different degrees.Then the exact solution () =  9 was obtained by considering the shifted Chebyshev polynomial of degree 9. We mention here that there are simple typos in presenting the example in (64) and in (66) and we correct them here.Applying the current algorithm we have We expand the solution in infinite series of the form () = ∑ ∞ =0    /2 .The initial condition in (65) yields  0 =  2 =  4 = 0. Let  =  2 ; applying (59) we have and it holds that Using ( 63) and (69), we have where Since () is smooth, then  +5 = 0, for  < 4. We now apply the last recursion together with  0 = ⋅ ⋅ ⋅ =  8 = 0, to compute  +5 , for  ≥ 4. For  = 4, we have and thus  9 = 0.
Proceeding in the same manner, we have  +1 = 0 for  ≥ 18.Thus, and the exact solution of problem (64)-( 65) is obtained.
In the following example we show that the current algorithm can be applied to more general multiterm fractional differential equations which are not necessary of the form in (51). with The exact solution for this problem is () =  3 .

Conclusion
For fractional differential equations of order  − 1 <  < , it is common to obtain a series solution in the form ∑ ∞ =0    () .The question is how to obtain the coefficients   ,  = 0, 1, . ... Naturally, if the problem is of fractional order, the differentiation is also of fractional order.In this paper, we presented a new algorithm for obtaining a series solution for nonlinear multiterm fractional differential equations of Caputo type, where we overcome the use of fractional differentiation.We employed a transformation that allows us to use ordinary differentiation rather than fractional differentiation to recursively compute the coefficient of the series expansion.Then the terms of the series,   , are computed sequentially using a closed form formula.We applied the new algorithm to several types of multiterm fractional differential equations, where accurate solutions as well as exact solutions in closed forms have been obtained.For one example it is noted that the current algorithm is more efficient than the ADM as it is more easier to apply and it produces the exact solution while the ADM does not.We have developed the new algorithm for two-and three-term fractional equations, while the idea can be extended to multiterm fractional equations of arbitrary order; obtaining a general formula in this case is not an easy task.The current algorithm can be modified to deal with the fractional multiterm time-diffusion equations.

Table 1 :
The error of Example 4 for  = 35 and different values of .

Table 2 :
The error of Example 4 at  = 1 and for different values of .