IDEAS home Printed from https://ideas.repec.org/a/eee/chsofr/v34y2007i5p1473-1481.html
   My bibliography  Save this article

Solution of fractional differential equations by using differential transform method

Author

Listed:
  • Arikoglu, Aytac
  • Ozkol, Ibrahim

Abstract

In this study, we implement a well known transformation technique, Differential Transform Method (DTM), to the area of fractional differential equations. Theorems that never existed before are introduced with their proofs. Also numerical examples are carried out for various types of problems, including the Bagley–Torvik, Ricatti and composite fractional oscillation equations for the application of the method. The results obtained are in good agreement with the existing ones in open literature and it is shown that the technique introduced here is robust, accurate and easy to apply.

Suggested Citation

  • Arikoglu, Aytac & Ozkol, Ibrahim, 2007. "Solution of fractional differential equations by using differential transform method," Chaos, Solitons & Fractals, Elsevier, vol. 34(5), pages 1473-1481.
    • Handle: RePEc:eee:chsofr:v:34:y:2007:i:5:p:1473-1481
    DOI: 10.1016/j.chaos.2006.09.004
    as

    Download full text from publisher

    File URL: http://www.sciencedirect.com/science/article/pii/S0960077906008150
    Download Restriction: Full text for ScienceDirect subscribers only
    File URL: https://libkey.io/10.1016/j.chaos.2006.09.004?utm_source=ideas
    LibKey link: if access is restricted and if your library uses this service, LibKey will redirect you to where you can use your library subscription to access this item
    ---><---

    As the access to this document is restricted, you may want to search for a different version of it.

    References listed on IDEAS

    as
    1. Momani, Shaher & Odibat, Zaid, 2007. "Numerical comparison of methods for solving linear differential equations of fractional order," Chaos, Solitons & Fractals, Elsevier, vol. 31(5), pages 1248-1255.
    Full references (including those not matched with items on IDEAS)

    Citations

    Citations are extracted by the CitEc Project, subscribe to its RSS feed for this item.
    as


    Cited by:

    1. Moghaddam, B.P. & Machado, J.A.T. & Behforooz, H., 2017. "An integro quadratic spline approach for a class of variable-order fractional initial value problems," Chaos, Solitons & Fractals, Elsevier, vol. 102(C), pages 354-360.
    2. Skwara, Urszula & Mozyrska, Dorota & Aguiar, Maira & Stollenwerk, Nico, 2024. "Dynamics of vector-borne diseases through the lens of systems incorporating fractional-order derivatives," Chaos, Solitons & Fractals, Elsevier, vol. 181(C).
    3. Rehman, Mujeeb ur & Idrees, Amna & Saeed, Umer, 2017. "A quadrature method for numerical solutions of fractional differential equations," Applied Mathematics and Computation, Elsevier, vol. 307(C), pages 38-49.
    4. Nemati, S. & Lima, P.M., 2018. "Numerical solution of nonlinear fractional integro-differential equations with weakly singular kernels via a modification of hat functions," Applied Mathematics and Computation, Elsevier, vol. 327(C), pages 79-92.
    5. İbrahim Avcı & Nazim I. Mahmudov, 2020. "Numerical Solutions for Multi-Term Fractional Order Differential Equations with Fractional Taylor Operational Matrix of Fractional Integration," Mathematics, MDPI, vol. 8(1), pages 1-24, January.
    6. Heydari, M.H. & Hooshmandasl, M.R. & Maalek Ghaini, F.M. & Cattani, C., 2016. "Wavelets method for solving fractional optimal control problems," Applied Mathematics and Computation, Elsevier, vol. 286(C), pages 139-154.
    7. Arikoglu, Aytac & Ozkol, Ibrahim, 2009. "Solution of fractional integro-differential equations by using fractional differential transform method," Chaos, Solitons & Fractals, Elsevier, vol. 40(2), pages 521-529.
    8. Salah Abuasad & Ahmet Yildirim & Ishak Hashim & Samsul Ariffin Abdul Karim & J.F. Gómez-Aguilar, 2019. "Fractional Multi-Step Differential Transformed Method for Approximating a Fractional Stochastic SIS Epidemic Model with Imperfect Vaccination," IJERPH, MDPI, vol. 16(6), pages 1-15, March.
    9. Mahmoud, Gamal M. & Arafa, Ayman A. & Abed-Elhameed, Tarek M. & Mahmoud, Emad E., 2017. "Chaos control of integer and fractional orders of chaotic Burke–Shaw system using time delayed feedback control," Chaos, Solitons & Fractals, Elsevier, vol. 104(C), pages 680-692.
    10. Damarla, Seshu Kumar & Kundu, Madhusree, 2015. "Numerical solution of multi-order fractional differential equations using generalized triangular function operational matrices," Applied Mathematics and Computation, Elsevier, vol. 263(C), pages 189-203.
    11. Khudair, Ayad R. & Haddad, S.A.M. & khalaf, Sanaa L., 2017. "Restricted fractional differential transform for solving irrational order fractional differential equations," Chaos, Solitons & Fractals, Elsevier, vol. 101(C), pages 81-85.
    12. Eriqat, Tareq & El-Ajou, Ahmad & Oqielat, Moa'ath N. & Al-Zhour, Zeyad & Momani, Shaher, 2020. "A New Attractive Analytic Approach for Solutions of Linear and Nonlinear Neutral Fractional Pantograph Equations," Chaos, Solitons & Fractals, Elsevier, vol. 138(C).
    13. Kumar, Manoj & Daftardar-Gejji, Varsha, 2019. "A new family of predictor-corrector methods for solving fractional differential equations," Applied Mathematics and Computation, Elsevier, vol. 363(C), pages 1-1.
    14. Allahviranloo, T. & Gouyandeh, Z. & Armand, A., 2015. "Numerical solutions for fractional differential equations by Tau-Collocation method," Applied Mathematics and Computation, Elsevier, vol. 271(C), pages 979-990.
    15. Yu, Jianping & Jing, Jian & Sun, Yongli & Wu, Suping, 2016. "(n+1)-Dimensional reduced differential transform method for solving partial differential equations," Applied Mathematics and Computation, Elsevier, vol. 273(C), pages 697-705.
    16. Yang, Xiao-Jun & Tenreiro Machado, J.A. & Srivastava, H.M., 2016. "A new numerical technique for solving the local fractional diffusion equation: Two-dimensional extended differential transform approach," Applied Mathematics and Computation, Elsevier, vol. 274(C), pages 143-151.
    17. Kavyanpoor, Mobin & Shokrollahi, Saeed, 2017. "Challenge on solutions of fractional Van Der Pol oscillator by using the differential transform method," Chaos, Solitons & Fractals, Elsevier, vol. 98(C), pages 44-45.
    18. Alam, M. Shamsul & Huq, M. Ashraful & Hasan, M. Kamrul & Rahman, M. Saifur, 2021. "A new technique for solving a class of strongly nonlinear oscillatory equations," Chaos, Solitons & Fractals, Elsevier, vol. 152(C).
    19. Raja, Muhammad Asif Zahoor & Samar, Raza & Manzar, Muhammad Anwar & Shah, Syed Muslim, 2017. "Design of unsupervised fractional neural network model optimized with interior point algorithm for solving Bagley–Torvik equation," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 132(C), pages 139-158.

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. He, Ji-Huan, 2009. "Nonlinear science as a fluctuating research frontier," Chaos, Solitons & Fractals, Elsevier, vol. 41(5), pages 2533-2537.
    2. Xu, Lan, 2009. "The variational iteration method for fourth order boundary value problems," Chaos, Solitons & Fractals, Elsevier, vol. 39(3), pages 1386-1394.
    3. Roman Parovik, 2020. "Mathematical Modeling of Linear Fractional Oscillators," Mathematics, MDPI, vol. 8(11), pages 1-26, October.
    4. Deng, Hongmin & Li, Tao & Wang, Qionghua & Li, Hongbin, 2009. "A fractional-order hyperchaotic system and its synchronization," Chaos, Solitons & Fractals, Elsevier, vol. 41(2), pages 962-969.
    5. Odibat, Zaid M., 2009. "Exact solitary solutions for variants of the KdV equations with fractional time derivatives," Chaos, Solitons & Fractals, Elsevier, vol. 40(3), pages 1264-1270.
    6. Marwan Abukhaled, 2013. "Variational Iteration Method for Nonlinear Singular Two-Point Boundary Value Problems Arising in Human Physiology," Journal of Mathematics, Hindawi, vol. 2013, pages 1-4, February.
    7. Mossa Al-sawalha, M. & Noorani, M.S.M., 2009. "A numeric–analytic method for approximating the chaotic Chen system," Chaos, Solitons & Fractals, Elsevier, vol. 42(3), pages 1784-1791.
    8. Soliman, A.A., 2009. "On the solution of two-dimensional coupled Burgers’ equations by variational iteration method," Chaos, Solitons & Fractals, Elsevier, vol. 40(3), pages 1146-1155.
    9. Tuan Hoang, Manh & Nagy, A.M., 2019. "Uniform asymptotic stability of a Logistic model with feedback control of fractional order and nonstandard finite difference schemes," Chaos, Solitons & Fractals, Elsevier, vol. 123(C), pages 24-34.
    10. Goh, S.M. & Noorani, M.S.M. & Hashim, I., 2009. "Efficacy of variational iteration method for chaotic Genesio system – Classical and multistage approach," Chaos, Solitons & Fractals, Elsevier, vol. 40(5), pages 2152-2159.
    11. Tomar, Saurabh & Singh, Mehakpreet & Vajravelu, Kuppalapalle & Ramos, Higinio, 2023. "Simplifying the variational iteration method: A new approach to obtain the Lagrange multiplier," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 204(C), pages 640-644.
    12. Yu, Yongguang & Li, Han-Xiong, 2008. "The synchronization of fractional-order Rössler hyperchaotic systems," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 387(5), pages 1393-1403.
    13. Kateryna Marynets, 2021. "Successive Approximation Technique in the Study of a Nonlinear Fractional Boundary Value Problem," Mathematics, MDPI, vol. 9(7), pages 1-19, March.
    14. Odibat, Zaid M., 2009. "Computational algorithms for computing the fractional derivatives of functions," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 79(7), pages 2013-2020.
    15. Rehman, Mujeeb ur & Idrees, Amna & Saeed, Umer, 2017. "A quadrature method for numerical solutions of fractional differential equations," Applied Mathematics and Computation, Elsevier, vol. 307(C), pages 38-49.
    16. Das, S., 2009. "A note on fractional diffusion equations," Chaos, Solitons & Fractals, Elsevier, vol. 42(4), pages 2074-2079.
    17. Tien, Wei-Chung & Chen, Cha’o-Kuang, 2009. "Adomian decomposition method by Legendre polynomials," Chaos, Solitons & Fractals, Elsevier, vol. 39(5), pages 2093-2101.
    18. Gafiychuk, V. & Datsko, B. & Meleshko, V. & Blackmore, D., 2009. "Analysis of the solutions of coupled nonlinear fractional reaction–diffusion equations," Chaos, Solitons & Fractals, Elsevier, vol. 41(3), pages 1095-1104.
    19. Odibat, Zaid, 2020. "An optimized decomposition method for nonlinear ordinary and partial differential equations," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 541(C).
    20. Yu, Yongguang & Li, Han-Xiong, 2009. "Application of the multistage homotopy-perturbation method to solve a class of hyperchaotic systems," Chaos, Solitons & Fractals, Elsevier, vol. 42(4), pages 2330-2337.

    More about this item

    Statistics

    Access and download statistics

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:eee:chsofr:v:34:y:2007:i:5:p:1473-1481. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: Thayer, Thomas R. (email available below). General contact details of provider: https://www.journals.elsevier.com/chaos-solitons-and-fractals .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.