Abstract
This paper introduces a new computational technique for dealing with fractional generalized pantograph-delay differential equations (PDDEs). The main idea of the proposed method is based on the fractional-order hybrid Bessel functions (FHBFs), which are constructed by the combination of fractional Bessel functions with block-pulse functions. The fractional derivative is applied in the Caputo sense. In the numerical technique, the operational matrix of fractional derivative is applied to reduce fractional generalized PDDEs to a system of algebraic equations with unknown fractional-order hybrid Bessel coefficients. In addition, the estimation of the error is analyzed. The error analysis indicates that when the number of FHBFs bases is increased, the obtained results convergent to the exact solution. Illustrative examples will be examined to demonstrate the validity and applicability of the presented technique. Also, the outcomes in comparison with the existing results in other methods illustrate the preference of the presented method.
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Ajello, W.G., Freedman, H.I., Wu, J.: A model of stage structured population growth with density depended time delay. SIAM J. Appl. Math. 52, 855–869 (1992)
Andrews, L.C.: Special Functions for Engineers and Applied Mathematicians. Macmillan Publishing Co., New York (1985)
Bagley, R.L., Torvik, P.J.: Fractional calculus in the transient analysis of viscoelastically damped structures. J. AIAA 23, 918–925 (1985)
Balachanran, K., Kiruthika, S.: Existence of solution of nonlinear fractional pantograph equations. Acta Math. Sci. 33(3), 712–720 (2013)
Balci, M.A., Sezer, M.: Hybrid Euler–Taylor matrix method for solving of generalized linear Fredholm integro-differential difference equations. Appl. Math. Comput. 273, 33–41 (2016)
Behbahani, Z.J., Roodaki, M.: Two-dimensional Chebyshev hybrid functions and their applications to integral equations. Beni-Suef Univ. J. Basic Appl. Sci. 4, 134–141 (2015)
Bhrawy, A.H., Alhamed, Y.A., Baleanu, D., Al-zahrani, A.A.: New spectral techniques for systems of fractional differential equations using fractional-order generalized Laguerre orthogonal functions. Fract. Calc. Appl. Anal. 17, 1138–1157 (2014)
Brzdek, J., Eghbali, N.: On approximate solutions of some delayed fractional differential equations. Appl. Math. Lett. 54, 31–35 (2016)
Buhmann, M.D., Iserles, A.: Stability of the discretized pantograph differential equation. Math. Comput. 60, 575–589 (1993)
Darani, M.A., Saadatmandi, A.: The operational matrix of fractional derivative of the fractional-order Chebyshev functions and its applications. Comput. Methods Differ. Equ. 5(1), 67–87 (2017)
Davis, A.R., Karageorghis, A., Phillips, T.N.: Spectral Galerkin methods for the primary two-point bour problem in modeling viscoelastic flows. Int. J. Numer. Methods Eng. 26, 647–662 (1988)
Dehestani, H., Ordokhani, Y., Razzaghi, M.: Fractional-order Legendre–Laguerre functions and their applications in fractional partial differential equations. Appl. Math. Comput. 336, 433–453 (2018)
Dehestani, H., Ordokhani, Y., Razzaghi, M.: On the applicability of Genocchi wavelet method for different kinds of fractional-order differential equations with delay. Numer. Linear Algebra Appl. 26, e2259 (2019)
Dehestani, H., Ordokhani, Y., Razzaghi, M.: Application of the modified operational matrices in multiterm variable-order time-fractional partial differential equations. Math. Methods Appl. Sci. 42, 7296–7313 (2019)
Doha, E.H., Bhrawy, A.H., Baleanu, D., Hafez, R.M.: A new Jacobi rational-Gauss collocation method for numerical solution of generalized pantograph equations. Appl. Numer. Math. 77, 43–54 (2014)
Fox, L., Mayers, D.F., Ockendon, J.A., Tayler, A.B.: On a functional differential equation. J. Inst. Math. Appl. 8, 271–307 (1971)
He, J.H.: Nonlinear oscillation with fractional derivative and its applications. Int. Conf. Vib. Eng. 98, 288–291 (1998)
Heydari, M., Loghmani, G.B., Hosseini, S.M., Karbassi, S.M.: Application of hybrid functions for solving duffing-harmonic oscillator. Adv. Differ. Equ. 2014, 210754 (2014)
Hou, J., Yang, C.: Numerical method in solving fredholm integro-differential equations by using hybrid function operational matrix of derivative. J. Inf. Comput. Sci. 10(9), 2757–2764 (2013)
Iqbal, M.A., Saeed, U., Mohyud-Din, S.T.: Modified Laguerre wavelets method for delay differential equations of fractional-order. Egypt. J. Basic Appl. Sci. 2, 50–54 (2015)
Javadi, S., Babolian, E., Taheri, Z.: Solving generalized pantograph equations by shifted orthonormal Bernstein polynomials. J. Comput. Appl. Math. 303, 1–14 (2016)
Kazem, S., Abbasbandy, S., Kumar, S.: Fractional-order Legendre functions for solving fractional-order differential equations. Appl. Math. Model. 37, 5498–5510 (2013)
Kulish, V.V., Lage, J.L.: Application of fractional calculus to fluid mechanics. J. Fluids Eng. 124(3), 803–806 (2002)
Lederman, C., Roquejoffre, J.M., Wolanski, N.: Mathematical justification of a nonlinear integro-differential equation for the propagation of spherical flames. Ann. Mat. Pura Appl. 183, 173–239 (2004)
Mainardi, F.: Fractional calculus: some basic problems in continuum and statistical mechanics. In: Carpinteri, A., Mainardi, F. (eds.) Fractals and Fractional Calculus in Continuum Mechanics, pp. 291–348. Springer, New York (1997)
Mashayekhi, S., Razzaghi, M., Wattanataweekul, M.: Analysis of multi-delay and piecewise constant delay systems by hybrid functions approximation. Differ. Equ. Dyn. Syst. 24(1), 1–20 (2016)
Meral, F.C., Royston, T.J., Magin, R.: Fractional calculus in viscoelasticity: an experimental study. Commun. Nonlinear Sci. Numer. Simul. 15, 939–945 (2010)
Moghaddam, B.P., Mostaghim, Z.S.: A numerical method based on finite difference for solving fractional delay differential equations. J. Taibah Univ. Sci. 7, 120–127 (2013)
Mohammadi, F.: Numerical solution of systems of fractional delay differential equations using a new kind of wavelet basis. Comput. Appl. Math. 37, 4122 (2018)
Ockendon, J.R., Tayler, A.B.: The dynamics of a current collection system for an electric locomotive. Proc. R. Soc. Lond. A 322(1551), 447–468 (1971)
Odibat, Z., Momani, S., Erturk, V.S.: Generalized differential transform method: application to differential equations of fractional order. Appl. Math. Comput. 197(2), 467–477 (2008)
Odibat, Z., Shawagfeh, N.T.: Generalized Taylor’s formula. Appl. Math. Comput. 186(1), 286–293 (2007)
Ozturk, Y., Gulsu, M.: Approximate solution of linear generalized pantograph equations with variable coefficients on Chebyshev–Gauss grid. J. Adv. Res. Sci. Comput. 4, 36–51 (2012)
Phillips, G.M., Taylor, P.J.: Theory and Application of Numerical Analysis. Academic Press, New York (1973)
Pimenov, V.G., Hendy, A.S.: Numerical studies for fractional functional differential equations with delay based on BDF-type shifted Chebyshev approximations. Abstr. Appl. Anal. 2015, 1–12 (2015)
Rahimkhani, P., Ordokhani, Y., Babolian, E.: Muntz–Legendre wavelet operational matrix of fractional-order integration and its applications for solving the fractional pantograph differential equations. Numer. Algorithms 77(4), 1–23 (2018)
Rahimkhani, P., Ordokhani, Y., Babolian, E.: A new operational matrix based on Bernoulli wavelets for solving fractional delay differential equations. Numer. Algorithms 74(1), 223–245 (2017)
Saeed, U., Rehmana, M.: Hermite wavelet method for fractional delay differential equations. J. Differ. Equ. 2014, 1–8 (2014)
Saeed, U., Rehmana, M., Iqbal, M.A.: Modified Chebyshev wavelet methods for fractional delay-type equations. Appl. Math. Comput. 264, 431–442 (2015)
Sansone, G.: Orthogonal Functions. Interscience Publishers Inc., New York (1959)
Sakar, M.G., Saldır, O., Akgul, A.: Numerical solution of fractional Bratu type equations with Legendre reproducing kernel method. Int. J. Appl. Comput. Math. 4, 126 (2018)
Sedaghat, S., Ordokhani, Y., Dehghan, M.: Numerical solution of delay differential equations of pantograph type via Chebyshev polynomials. Commun. Nonlinear Sci. Numer. Simul. 17, 4815–4830 (2012)
Sezer, M., Akyuz-Das-cioglu, A.: A Taylor method for numerical solution of generalized pantograph equations with linear function alargument. J. Comput. Appl. Math. 200, 217–225 (2007)
Thomas, D.C., Ilinskii, Y.A., Hamilton, M.F.: Delay differential equation models for single and coupled bubble dynamics in a compressible liquid. (2013). arXiv:1310.2895
Wang, Z., Huang, X., Zhou, J.: A numerical method for delayed fractional-order differential equations: based on G–L definition. Appl. Math. Inf. Sci. 7(2), 525–529 (2013)
Yuzbasi, S., Sezer, M.: An exponential approximation for solutions of generalized pantograph-delay differential equations. Appl. Math. Model. 37, 9160–9173 (2013)
Yuzbasi, S.: Numerical solutions of fractional Riccati type differential equations by means of the Bernstein polynomials. Appl. Math. Comput. 219, 6328–6343 (2013)
Zhao, J., Cao, Y., Xu, Y.: Sinc numerical solution for pantograph Volterra delay-integro-differential equation. Int. J. Comput. Math. 94(5), 853–865 (2017)
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Dehestani, H., Ordokhani, Y. & Razzaghi, M. Numerical Technique for Solving Fractional Generalized Pantograph-Delay Differential Equations by Using Fractional-Order Hybrid Bessel Functions. Int. J. Appl. Comput. Math 6, 9 (2020). https://doi.org/10.1007/s40819-019-0756-2
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DOI: https://doi.org/10.1007/s40819-019-0756-2
Keywords
- Fractional-order hybrid Bessel functions
- Operational matrix of fractional derivative
- Fractional pantograph-delay differential equations
- Error estimate