IDEAS home Printed from https://ideas.repec.org/a/gam/jmathe/v8y2020i10p1675-d422395.html
   My bibliography  Save this article

Numerical Solutions of Fractional Differential Equations by Using Fractional Explicit Adams Method

Author

Listed:
  • Nur Amirah Zabidi

    (Institute for Mathematical Research, Universiti Putra Malaysia, UPM Serdang 43400, Malaysia)

  • Zanariah Abdul Majid

    (Institute for Mathematical Research, Universiti Putra Malaysia, UPM Serdang 43400, Malaysia
    Department of Mathematics, Faculty of Science, Universiti Putra Malaysia, UPM Serdang 43400, Malaysia)

  • Adem Kilicman

    (Institute for Mathematical Research, Universiti Putra Malaysia, UPM Serdang 43400, Malaysia
    Department of Mathematics, Faculty of Science, Universiti Putra Malaysia, UPM Serdang 43400, Malaysia)

  • Faranak Rabiei

    (School of Engineering, Monash University Malaysia, Jalan Lagoon Selatan, Bandar Sunway, Selangor 47500, Malaysia)

Abstract

Differential equations of fractional order are believed to be more challenging to compute compared to the integer-order differential equations due to its arbitrary properties. This study proposes a multistep method to solve fractional differential equations. The method is derived based on the concept of a third-order Adam–Bashforth numerical scheme by implementing Lagrange interpolation for fractional case, where the fractional derivatives are defined in the Caputo sense. Furthermore, the study includes a discussion on stability and convergence analysis of the method. Several numerical examples are also provided in order to validate the reliability and efficiency of the proposed method. The examples in this study cover solving linear and nonlinear fractional differential equations for the case of both single order as α ∈ ( 0 , 1 ) and higher order, α ∈ 1 , 2 , where α denotes the order of fractional derivatives of D α y ( t ) . The comparison in terms of accuracy between the proposed method and other existing methods demonstrate that the proposed method gives competitive performance as the existing methods.

Suggested Citation

  • Nur Amirah Zabidi & Zanariah Abdul Majid & Adem Kilicman & Faranak Rabiei, 2020. "Numerical Solutions of Fractional Differential Equations by Using Fractional Explicit Adams Method," Mathematics, MDPI, vol. 8(10), pages 1-23, October.
    • Handle: RePEc:gam:jmathe:v:8:y:2020:i:10:p:1675-:d:422395
    as

    Download full text from publisher

    File URL: https://www.mdpi.com/2227-7390/8/10/1675/pdf
    Download Restriction: no
    File URL: https://www.mdpi.com/2227-7390/8/10/1675/
    Download Restriction: no
    ---><---

    References listed on IDEAS

    as
    1. Bonab, Zahra Farzaneh & Javidi, Mohammad, 2020. "Higher order methods for fractional differential equation based on fractional backward differentiation formula of order three," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 172(C), pages 71-89.
    2. Odibat, Zaid & Momani, Shaher, 2008. "Modified homotopy perturbation method: Application to quadratic Riccati differential equation of fractional order," Chaos, Solitons & Fractals, Elsevier, vol. 36(1), pages 167-174.
    3. Mehmet Merdan, 2012. "On the Solutions Fractional Riccati Differential Equation with Modified Riemann-Liouville Derivative," International Journal of Differential Equations, Hindawi, vol. 2012, pages 1-17, May.
    4. Adel Al-Rabtah & Shaher Momani & Mohamed A. Ramadan, 2012. "Solving Linear and Nonlinear Fractional Differential Equations Using Spline Functions," Abstract and Applied Analysis, Hindawi, vol. 2012, pages 1-9, March.
    Full references (including those not matched with items on IDEAS)

    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. Fathy, Mohamed & Abdelgaber, K.M., 2022. "Approximate solutions for the fractional order quadratic Riccati and Bagley-Torvik differential equations," Chaos, Solitons & Fractals, Elsevier, vol. 162(C).
    2. S. Balaji, 2014. "Legendre Wavelet Operational Matrix Method for Solution of Riccati Differential Equation," International Journal of Mathematics and Mathematical Sciences, Hindawi, vol. 2014, pages 1-10, June.
    3. Jim Gatheral & Radoš Radoičić, 2019. "Rational Approximation Of The Rough Heston Solution," International Journal of Theoretical and Applied Finance (IJTAF), World Scientific Publishing Co. Pte. Ltd., vol. 22(03), pages 1-19, May.
    4. Shloof, A.M. & Senu, N. & Ahmadian, A. & Salahshour, Soheil, 2021. "An efficient operation matrix method for solving fractal–fractional differential equations with generalized Caputo-type fractional–fractal derivative," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 188(C), pages 415-435.
    5. Abolvafaei, Mahnaz & Ganjefar, Soheil, 2020. "Maximum power extraction from wind energy system using homotopy singular perturbation and fast terminal sliding mode method," Renewable Energy, Elsevier, vol. 148(C), pages 611-626.
    6. M. Motawi Khashan & Rohul Amin & Muhammed I. Syam, 2019. "A New Algorithm for Fractional Riccati Type Differential Equations by Using Haar Wavelet," Mathematics, MDPI, vol. 7(6), pages 1-12, June.
    7. H. X. Mamatova & Z. K. Eshkuvatov & Sh. Ismail, 2023. "A Hybrid Method for All Types of Solutions of the System of Cauchy-Type Singular Integral Equations of the First Kind," Mathematics, MDPI, vol. 11(20), pages 1-30, October.
    8. S M, Sivalingam & Kumar, Pushpendra & Govindaraj, V., 2023. "A novel numerical scheme for fractional differential equations using extreme learning machine," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 622(C).
    9. Antony Vijesh, V. & Roy, Rupsha & Chandhini, G., 2015. "A modified quasilinearization method for fractional differential equations and its applications," Applied Mathematics and Computation, Elsevier, vol. 266(C), pages 687-697.
    10. Cveticanin, L., 2009. "Application of homotopy-perturbation to non-linear partial differential equations," Chaos, Solitons & Fractals, Elsevier, vol. 40(1), pages 221-228.
    11. 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.
    12. Waleed Mohamed Abd-Elhameed & Badah Mohamed Badah, 2021. "New Approaches to the General Linearization Problem of Jacobi Polynomials Based on Moments and Connection Formulas," Mathematics, MDPI, vol. 9(13), pages 1-28, July.
    13. Hallaji, Majid & Dideban, Abbas & Khanesar, Mojtaba Ahmadieh & kamyad, Ali vahidyan, 2018. "Optimal synchronization of non-smooth fractional order chaotic systems with uncertainty based on extension of a numerical approach in fractional optimal control problems," Chaos, Solitons & Fractals, Elsevier, vol. 115(C), pages 325-340.
    14. Siow Woon Jeng & Adem Kiliçman, 2021. "SPX Calibration of Option Approximations under Rough Heston Model," Mathematics, MDPI, vol. 9(21), pages 1-11, October.
    15. Endah R. M. Putri & Lutfi Mardianto & Amirul Hakam & Chairul Imron & Hadi Susanto, 2021. "Removing non-smoothness in solving Black-Scholes equation using a perturbation method," Papers 2104.07839, arXiv.org, revised Apr 2021.
    16. 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.
    17. Meng, Zhijun & Yi, Mingxu & Huang, Jun & Song, Lei, 2018. "Numerical solutions of nonlinear fractional differential equations by alternative Legendre polynomials," Applied Mathematics and Computation, Elsevier, vol. 336(C), pages 454-464.
    18. Rajarama Mohan Jena & Snehashish Chakraverty & Dumitru Baleanu, 2019. "On New Solutions of Time-Fractional Wave Equations Arising in Shallow Water Wave Propagation," Mathematics, MDPI, vol. 7(8), pages 1-13, August.
    19. Ahmed Farooq Qasim & Almutasim Abdulmuhsin Hamed, 2019. "Treating Transcendental Functions in Partial Differential Equations Using the Variational Iteration Method with Bernstein Polynomials," International Journal of Mathematics and Mathematical Sciences, Hindawi, vol. 2019, pages 1-8, March.
    20. Md. Habibur Rahman & Muhammad I. Bhatti & Nicholas Dimakis, 2023. "Employing a Fractional Basis Set to Solve Nonlinear Multidimensional Fractional Differential Equations," Mathematics, MDPI, vol. 11(22), pages 1-15, November.

    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:gam:jmathe:v:8:y:2020:i:10:p:1675-:d:422395. 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: MDPI Indexing Manager (email available below). General contact details of provider: https://www.mdpi.com .

    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.