Abstract
We present a new numerical method for solving fractional delay differential equations. The method is based on Taylor wavelets. We establish an exact formula to determine the Riemann–Liouville fractional integral of the Taylor wavelets. The exact formula is then applied to reduce the problem of solving a fractional delay differential equation to the problem of solving a system of algebraic equations. Several numerical examples are presented to show the applicability and the effectiveness of this method.
Similar content being viewed by others
References
Oldham KB (2010) Fractional differential equations in electrochemistry. Adv Eng Softw 41(1):9–12
Baillie RT (1996) Long memory processes and fractional integration in econometrics. J Econom 73(1):5–59
Carpinteri A, Mainardi F (eds) (2014) Fractals and fractional calculus in continuum mechanics. Springer, Berlin
Rossikhin YA, Shitikova MV (1997) Applications of fractional calculus to dynamic problems of linear and nonlinear hereditary mechanics of solids. Appl Mech Rev 50(1):15–67
Hall MG, Barrick TR (2008) From diffusion-weighted MRI to anomalous diffusion imaging. Magn Reson Med 59(3):447–455
Povstenko Y (2010) Signaling problem for time-fractional diffusion-wave equation in a half-space in the case of angular symmetry. Nonlinear Dyn 59(4):593–605
He JH (1999) Some applications of nonlinear fractional differential equations and their approximations. Bull Sci Technol 15(2):86–90
Mandelbrot B (1967) Some noises with I/f spectrum, a bridge between direct current and white noise. IEEE Trans Inf Theory 13(2):289–298
Ockendon JR, Tayler AB (1971) The dynamics of a current collection system for an electric locomotive. Proc R Soc Lond A Math Phys Sci 322(1551):447–468
Aiello WG, Freedman HI, Wu J (1992) Analysis of a model representing stage-structured population growth with state-dependent time delay. SIAM J Appl Math 52(3):855–869
Evans DJ, Raslan KR (2005) The Adomian decomposition method for solving delay differential equation. Int J Comput Math 82(1):49–54
Wang WS, Li SF (2007) On the one-leg \(\theta \)-methods for solving nonlinear neutral functional differential equations. Appl Math Comput 193(1):285–301
Yu ZH (2008) Variational iteration method for solving the multi-pantograph delay equation. Phys Lett A 372(43):6475–6479
Hafshejani MS, Vanani SK, Hafshejani JS (2011) Numerical solution of delay differential equations using Legendre wavelet method. World Appl Sci J 13:27–33
Sedaghat S, Ordokhani Y, Dehghan M (2012) Numerical solution of the delay differential equations of pantograph type via Chebyshev polynomials. Commun Nonlinear Sci Numer Simul 17(12):4815–4830
Tohidi E, Bhrawy AH, Erfani K (2013) A collocation method based on Bernoulli operational matrix for numerical solution of generalized pantograph equation. Appl Math Model 37(6):4283–4294
Moghaddam BP, Mostaghim ZS (2013) A numerical method based on finite difference for solving fractional delay differential equations. J Taibah Univ Sci 7(3):120–127
Khader MM, Hendy AS (2012) The approximate and exact solutions of the fractional-order delay differential equations using Legendre seudospectral method. Int J Pure Appl Math 74(3):287–297
Yang Y, Huang Y (2013) Spectral-collocation methods for fractional pantograph delay-integrodifferential equations. Adv Math Phys
Saeed U (2014) Hermite wavelet method for fractional delay differential equations. J Differ Equ Appl
Rahimkhani P, Ordokhani Y, Babolian E (2017) A new operational matrix based on Bernoulli wavelets for solving fractional delay differential equations. Numer Algorithms 74(1):223–245
Wang Z (2013) A numerical method for delayed fractional-order differential equations. J Appl Math
Razzaghi M, Yousefi S (2001) The Legendre wavelets operational matrix of integration. Int J Syst Sci 32(4):495–502
Beylkin G, Coifman R, Rokhlin V (1991) Fast wavelet transforms and numerical algorithms I. Commun Pure Appl Math 44(2):141–183
Zhu L, Fan Q (2012) Solving fractional nonlinear Fredholm integro-differential equations by the second kind Chebyshev wavelet. Commun Nonlinear Sci Numer Simul 17(6):2333–2341
Heydari MH, Hooshmandasl MR, Mohammadi F (2014) Legendre wavelets method for solving fractional partial differential equations with Dirichlet boundary conditions. Appl Math Comput 234:267–276
Saeedi H, Moghadam MM, Mollahasani N, Chuev GN (2011) A CAS wavelet method for solving nonlinear Fredholm integro-differential equations of fractional order. Commun Nonlinear Sci Numer Simul 16(3):1154–1163
Li Y, Zhao W (2010) Haar wavelet operational matrix of fractional order integration and its applications in solving the fractional order differential equations. Appl Math Comput 216(8):2276–2285
Keshavarz E, Ordokhani Y, Razzaghi M (2014) Bernoulli wavelet operational matrix of fractional order integration and its applications in solving the fractional order differential equations. Appl Math Model 38(24):6038–6051
Miller KS, Ross B (1993) An introduction to the fractional calculus and fractional differential equations. Willey, New York
Caputo M (1967) Linear models of dissipation whose Q is almost frequency independent-II. Geophys J Int 13(5):529–539
Keshavarz E, Ordokhani Y, Razzaghi M (2018) The Taylor wavelets method for solving the initial and boundary value problems of Bratu-type equations. Appl Numer Math 128:205–216
Luenberger DG (1997) Optimization by vector space methods. Wiley, Hoboken
Yuttanan B, Razzaghi M (2019) Legendre wavelets approach for numerical solutions of distributed order fractional differential equations. Appl Math Model 70:350–364
Stewart GW (1993) Afternotes on numerical analysis. University of Maryland at College Park
Saeed U, Rehman M (2014) Hermite wavelet method for fractional delay differential equations. J Differ Equ Appl
Yousefi S, Lotfi A (2013) Legendre multiwavelet collocation method for solving the linear fractional time delay systems. Cent Eur J Phys 11(10):1463–1469
Acknowledgements
The authors wish to express their sincere thanks to anonymous referees for their valuable suggestions that improved the final manuscript.
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
Toan, P.T., Vo, T.N. & Razzaghi, M. Taylor wavelet method for fractional delay differential equations. Engineering with Computers 37, 231–240 (2021). https://doi.org/10.1007/s00366-019-00818-w
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00366-019-00818-w