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Discrete Dynamics in Nature and Society
Volume 2017, Article ID 5234151, 10 pages
https://doi.org/10.1155/2017/5234151
Research Article

An Efficient Series Solution for Nonlinear Multiterm Fractional Differential Equations

Department of Mathematical Sciences, United Arab Emirates University, Al-Ain, UAE

Correspondence should be addressed to Mohammed Al-Refai; ea.ca.ueau@iaferla_m

Received 24 January 2017; Accepted 21 February 2017; Published 8 March 2017

Academic Editor: Thabet Abdeljawad

Copyright © 2017 Moh’d Khier Al-Srihin and Mohammed Al-Refai. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

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.

1. 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 [49], 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 , defined by [15, 16]provided the integral exists, where is the well-known Gamma function. The left Riemann-Liouville fractional integral, , of order , is defined byThe left Caputo derivative is related to the left Riemann-Liouville fractional integral byIt is known thatAlso, for , it holds thatThis 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.

2. Two-Term Fractional Differential Equations

We start with the nonlinear two-term fractional initial value problems of the formwithwhere , and and are nonzero constants. We assume that is continuous and smooth with respect to . We also assume that and are rational numbers with and , .

2.1. The Expansion Procedure

Let ; we have for some , .

In the following we expand the solution of problem (6)-(7) in an infinite series of the form where the coefficients : have to be determined sequentially in the following manner: From the initial condition (7) we have . Since , for being constant, we havewhere

By substituting (9) in (6) we haveApplying the well-known Taylor series method to compute the coefficients will lead to computing iterated fractional derivatives, which are not easily computed in general. To avoid this difficulty, let ; we haveShifting the index to zero yieldsTo avoid the singularity at , we multiply (13) by ; we haveNow, since , thus , and (14) has no singularity at

Let , and ; then (14) can be written asWe first determine the coefficients for . By performing the th derivative of (15) with respect to and substituting , we havewhich yieldsSince , and is smooth, then for , we have and hence , for .

We now determine , for . By performing the th derivative of (15) with respect to and substituting , we haveUsing the well-known Leibniz rule for differentiating the products, we have

Since we haveFrom the last equation we determine and thus the solution is obtained.

Remark 1. In (6), assuming , then and Thus (17) is reduced to which coincides with the coefficients obtained by the Taylor series expansion method. Same comment applies for the coefficient in (22).

Remark 2. The algorithm can be generalized for the two-term fractional differential equation (6) with , for arbitrary integer . But we have to take care of the existence of the fractional derivative (see Eq. (5)), by choosing the coefficient , for . This case is discussed in Section 3.

2.2. Numerical Results

Example 3. Consider the nonlinear two-term fractional initial value problemwithThe exact solution of the problem is .
Applying the current algorithm, we have , , , , and . We expand the solution in an infinite series of the formThe initial condition in (26) yields . We have andSince is continuous and smooth with respect to , we have Thus . The function satisfies the assumption of the proposed algorithm, and it holds thatThe computation above is made using the software Mathematica version 9. For ; substituting (30) in (22) yieldswhereApplying (31) together with , we have Thus,and the exact solution of problem (25)-(26) is obtained.

In the following we compare our results with the Adomian decomposition method (ADM). Assume that the nonlinear function and the solution can be expressed in the following series form: where are the well-known Adomian polynomials that can be derived from [17] For recent advances in the Adomian decomposition method we refer the reader to [18, 19]. For , we haveApplying the Riemann-Liouville fractional integral operator to (25) and substituting we have

Setand balancing (39) yieldsEvaluating the first 5 term of the power series solution, we obtain the approximate solution , where which is not the exact solution. Solving the problem by the ADM with yields , wherewhich is again not the exact solution. Figure 1 depicts the exact solution obtained by the current algorithm and the approximate solutions and obtained by the Adomian decomposition method.

Figure 1: Comparison between the exact and approximate solutions obtained by the ADM for problem ((25)-(26)).

Example 4. Consider the nonlinear two-term fractional initial value problemwithThis 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 to approximate the solution.
Applying the current algorithm, we have , andwhere The initial condition yields . 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 bywhere 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.

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

3. Three-Term Fractional Differential Equations

We consider the nonlinear three-term fractional initial value problem of the formwithwhereand , , and are nonzero constants. We assume that is continuous and smooth with respect to . We also assume that , and are rational numbers with , and , where . Let ; we have for , and .

Applying analogous steps for the case of the two-term fractional equations, we seek a solution of the form From the initial condition (52) we have , and thus, .

Using the fact that , for , we havewhere , , and

Substituting (55) in (51), we haveSince the coefficients , are known, we write the last equation in the formLet ; then (57) reduces toSubstituting andand shifting the indices to zero, we haveTo avoid singularity we multiply both sides in (60) by ; we haveEquation (61) can be written aswhere and

Applying analogs steps for deriving (22), we havewhere From the last recursion we determine

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.

3.1. Numerical Results

Example 6. Consider the Bagely-Torvik initial value problemwhereThis 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 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 . The initial condition in (65) yields . Let ; applying (59) we have and it holds thatUsing (63) and (69), we havewhere Since is smooth, then , for We now apply the last recursion together with , to compute , for . For , we have and thus .
Applying analogous arguments yields . For , we have which, together with , implies that . For , we have but and ; thus . Applying the same steps yields . For , we have and thus .
Proceeding in the same manner, we have for . 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).

Example 7. Consider the nonlinear three-term fractional initial value problemwithThe exact solution for this problem is .

Applying the current algorithm we have . We then expand the solution in an infinite series of the form . From the initial condition (78), we have . To guarantee the existence of the fractional derivative we choose . Let ; then (77) reduces towhere , , and .

Performing the th derivatives one can see that for . For , we have which yields .

Also,

For , we have which yields .

Following analogous steps, we have for Thus which is the exact solution.

4. Conclusion

For fractional differential equations of order , it is common to obtain a series solution in the form The question is how to obtain the coefficients . 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.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

The authors gratefully acknowledge the support of the UAE University under Grant COS/IRG-11/15.

References

  1. M. Inc, “The approximate and exact solutions of the space- and time-fractional Burgers equations with initial conditions by variational iteration method,” Journal of Mathematical Analysis and Applications, vol. 345, no. 1, pp. 476–484, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  2. L. Song and H. Zhang, “Application of homotopy analysis method to fractional KdV-Burgers-KURamoto equation,” Physics Letters A, vol. 367, no. 1-2, pp. 88–94, 2007. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  3. N. H. Sweilam, M. M. Khader, and R. F. Al-Bar, “Numerical studies for a multi-order fractional differential equation,” Physics Letters A, vol. 371, no. 1-2, pp. 26–33, 2007. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  4. V. Daftardar-Gejji and S. Bhalekar, “Solving multi-term linear and non-linear diffusion-wave equations of fractional order by Adomian decomposition method,” Applied Mathematics and Computation, vol. 202, no. 1, pp. 113–120, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  5. W. He, “Adomian decomposition method for fractional differential equations of Caputo Hadamard type,” Journal of Computational Complexity and Applications, vol. 2, no. 4, pp. 160–162, 2016. View at Google Scholar
  6. H. Jafari and V. Daftardar-Gejji, “Solving linear and nonlinear fractional diffusion and wave equations by Adomian decomposition,” Applied Mathematics and Computation, vol. 180, no. 2, pp. 488–497, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  7. S. Momani and K. Al-Khaled, “Numerical solutions for systems of fractional differential equations by the decomposition method,” Applied Mathematics and Computation, vol. 162, no. 3, pp. 1351–1365, 2005. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  8. G. C. Wu, D. Baleanu, and H. P. Xie, “Riesz Riemann-Liouville difference on discrete domains,” Chaos, vol. 26, no. 8, Article ID 084308, 2016. View at Publisher · View at Google Scholar
  9. C. Yang and J. Hou, “An approximate solution of nonlinear fractional differential equation by laplace transform and adomian polynomials,” Journal of Information and Computational Science, vol. 10, no. 1, pp. 213–222, 2013. View at Google Scholar · View at Scopus
  10. A. Elsaid, M. S. Abdel Latif, and F. M. Kamal, “Series solution for fractional Riccati differential equation and its convergence,” Journal of Fractional Calculus and Applications, vol. 6, no. 2, pp. 186–196, 2015. View at Google Scholar · View at MathSciNet
  11. H. M. Srivastava, J. A. Tenreiro Machado, and X.-J. Yang, “Approximate methods for local fractional differential equations,” in Fractional Dynamics, C. Cattani, H. M. Srivastava, and X.-J. Yang, Eds., pp. 244–258, Emerging Science Publishers (De Gruyter Open), Berlin, Germany, 2015. View at Google Scholar
  12. X.-J. Yang, D. Baleanu, and H. M. Srivastava, Local Fractional Integral Transforms and Their Applications, Academic Press (Elsevier Science Publishers), Amsterdam, The Netherlands, 2016.
  13. M. Al-Refai, M. A. Hajji, and M. Syam, “An efficient series solution for fractional differential equations,” Abstract and Applied Analysis, vol. 2014, Article ID 891837, 7 pages, 2014. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  14. M. K. Al-Srihin, Series solutions of multi-term fractional differential equations [M.S. thesis], United Arab Emirates University, Al Ain, United Arab Emirates, 2015.
  15. A. A. Kilbas, H. M. Srivastava, and J. J. Trujillo, Theory and Applications of Fractional Differential Equations, vol. 204 of North-Holland Mathematical Studies, Elsevier Science Publishers, Amsterdam, The Netherlands, 2006.
  16. I. Podlubny, Fractional Differential Equations. An Introduction to Fractional Derivatives, Fractional Differential Equations, Some Methods of Their Solution and Some of Their Applications, vol. 198 of Mathematics in Science and Engineering, Academic Press, San Diego, Calif, USA, 1999. View at MathSciNet
  17. G. Adomian, “A review of the decomposition method and some recent resuluts for nonlinear equations,” Mathematical and Computer Modelling, vol. 13, no. 7, pp. 17–43, 1990. View at Google Scholar
  18. J.-S. Duan, “Convenient analytic recurrence algorithms for the Adomian polynomials,” Applied Mathematics and Computation, vol. 217, no. 13, pp. 6337–6348, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  19. L.-L. Huang, G.-C. Wu, M. M. Rashidi, and W.-H. Luo, “Chaos analysis of the nonlinear duffing oscillators based on the new Adomian polynomials,” Journal of Nonlinear Science and its Applications. JNSA, vol. 9, no. 4, pp. 1877–1881, 2016. View at Google Scholar · View at MathSciNet
  20. M. Blasik, “Numerical scheme for a two-term sequential fractional differential equation,” Scientific Research of the Institute of Mathematics and Computer Science, vol. 2, no. 10, pp. 17–29, 2011. View at Google Scholar
  21. E. H. Doha, A. H. Bhrawy, and S. S. Ezz-Eldien, “A Chebyshev spectral method based on operational matrix for initial and boundary value problems of fractional order,” Computers & Mathematics with Applications, vol. 62, no. 5, pp. 2364–2373, 2011. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus