Abstract
This paper presents a new computational technique for solving fractional pantograph differential equations. The fractional derivative is described in the Caputo sense. The main idea is to use Müntz-Legendre wavelet and its operational matrix of fractional-order integration. First, the Müntz-Legendre wavelet is presented. Then a family of piecewise functions is proposed, based on which the fractional order integration of the Müntz-Legendre wavelets are easy to calculate. The proposed approach is used this operational matrix with the collocation points to reduce the under study problem to a system of algebraic equations. An estimation of the error is given in the sense of Sobolev norms. The efficiency and accuracy of the proposed method are illustrated by several numerical examples.
Similar content being viewed by others
References
Lakshmikantham, V., Leela, S.: Differential and integral inequalities. Academic Press, New York (1969)
Dehghan, M., Shakeri, F.: The use of the decomposition procedure of Adomian for solving a delay differential equation arising in electrodynamics. Phys. Scr. 78, 1–11 (2008)
Bagley, R.L., Torvik, P.J.: A theoretical basis for the application of fractional calculus to viscoelasticity. J. Rheol. 27, 201–210 (1983)
Magin, R.L.: Fractional calculus in bioengineering. Crit. Rev. Biomed. Eng. (2004)
Chow, T.S.: Fractional dynamics of interfaces between soft-nanoparticles and rough substrates. Phys. Lett. A. 342, 148–155 (2006)
He, J.H.: Some applications of nonlinear fractional differential equations and their approximations. Bull. Sci. Tech. 15(2), 86–90 (1999)
Panda, R., Dash, M.: Fractional generalized splines and signal processing. Signal Process. 86, 2340–2350 (2006)
Bohannan, G.W.: Analog fractional order controller in temperature and motor control applications. J. Vib. Control. 14, 1487–1498 (2008)
Diethelm, K., Walz, G.: Numerical solution of fractional order differential equations by extrapolation. Numer. Algor. 16(3), 231–253 (1997)
Gaul, L., Klein, P., Kemple, S.: Damping description involving fractional operators. Mech. Syst. Signal Process 5, 81–88 (1991)
Suarez, L., Shokooh, A.: An eigenvector expansion method for the solution of motion containing fractional derivatives. J. Appl. Mech. 64, 629–735 (1997)
Podlubny, I.: Fractional differential equations: an introduction to fractional derivatives, fractional differential equations, to methods of their solution and some of their applications. Academic Press, New York (1998)
Momani, S., Al-Khaled, K.: Numerical solutions for systems of fractional differential equations by the decomposition method. Appl. Math. Comput. 162, 1351–1365 (2005)
Odibat, Z., Momani, S.: Application of variational iteration method to nonlinear differential equations of fractional order. Int. J. Nonl. Sci. Numer. Simul. 7, 27–34 (2006)
Odibat, Z., Shawagfeh, N.: Generalized Taylor’s formula. Appl. Math. Comput. 186(1), 286–293 (2007)
Meerschaert, M., Tadjeran, C.: Finite difference approximations for two-sided space-fractional partial differential equations. Appl. Numer. Math. 56, 80–90 (2006)
Feng, L.B., Zhuang, P., Liu, F., Turner, I., Gu, Y.T.: Finite element method for space-time fractional diffusion equation. Numer. Algor. 72(3), 749–767 (2016)
Arikoglu, A., Ozkol, I.: Solution of fractional integro-differential equations by using fractional differential transform method. Chaos Sol. Frac. 40, 521–529 (2009)
Rawashdeh, E.A.: Numerical solution of fractional integro-differential equations by collocation method. Appl. Math. Comput. 176, 1–6 (2006)
Bhrawy, A.H.: A Jacobi spectral collocation method for solving multi-dimensional nonlinear fractional sub-diffusion equations. Numer. Algor. 73(1), 91–113 (2016)
Dehghan, M., Manafian, J., Saadatmandi, A.: Solving nonlinear fractional partial differential equations using the homotopy analysis method. Numer. Meth. Part. Differ. Equ. 26(2), 448–479 (2009)
Saadatmandi, A., Dehghan, M.: A new operational matrix for solving fractional-order differential equations. Comput. Math. Appl. 59, 1326–1336 (2010)
Rahimkhani, P., Ordokhani, Y., Babolian, E.: Fractional-order Bernoulli wavelets and their applications. Appl. Math. Model. 40, 8087–8107 (2016)
Bhalekar, S., Daftardar-Gejji, V.: A predictor-corrector scheme for solving nonlinear delay differential equations of fractional order. J. Fract. Calcul. Appl. 1(5), 1–9 (2011)
Morgado, M.L., Ford, N.J., Lima, P.M.: Analysis and numerical methods for fractional differential equations with delay. J. Comput. Appl. Math. 252, 159–168 (2012)
Khader, M.M., Hendy, A.S.: The approximate and exact solutions of the fractional-order delay differential equations using Legendre pseudo-spectral method. Int. J. Pure Appl. Math. 74(3), 287–297 (2012)
Wang, Z.: A numerical method for delayed fractional-order differential equations. J. Appl. Math. 2013, 1–7 (2013)
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)
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)
Yang, Y., Huang, Y.: Spectral-collocation methods for fractional pantograph delay-integro-differential equations. Adv. Math. Phys. 2013, 1–14 (2013)
Yousefi, S.A., Lotfi, A.: Legendre multiwavelet collocation method for solving the linear fractional time delay systems. Cent. Eur. J. Phys. 11(10), 1463–1469 (2013)
Saeed, U., Rehman, M.: Hermite wavelet method for fractional delay differential equations. J. Differ. Equ. 2014, 1–8 (2014)
Saeed, U., Rehman, M., Iqbal, M.A.: Modified Chebyshev wavelet methods for fractional delay-type equations. Appl. Math. Comput. 264, 431–442 (2015)
Iqbal, M.A., Saeed, U., Mohyud-Din, S.T.: Modified Laguerre wavelets method for delay differential equations of fractional-order. Egy. J. Basic Appl. Sci. 2, 50–54 (2015)
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.: A new operational matrix based on Bernoulli wavelets for solving fractional delay differential equations. Numer. Algor. 74(1), 223–245 (2017)
Strömberg, J.O.: A modified Franklin system and higher order spline systems on Rn as unconditional bases for Hardy spaces Proceedings of Harmonic Analysis, pp 475–494. University of Chicago (1981)
Grossman, A., Morlet, J.: Decomposition of Hardy functions into square integrable wavelets of constant shape. SIAM J. Math. Anal. 15, 723–736 (1984)
Meyer, Y.: Principe d’incertitude bases hilbertiennes et algèbres d’opérateurs. Sé,minaire N Bourbaki. 662, 209–223 (1985)
Yuanlu, L., Weiwei, Z.: Haar wavelet operational matrix of fractional order integration and its applications in solving the fractional order differential equations. Appl. Math. Comput. 216, 2276–2285 (2010)
Yuanlu, L.: Solving a nonlinear fractional differential equation using Chebyshev wavelets. Commun. Nonl. Sci. Num. Simulat. 15, 2284–2292 (2010)
Saeedi, H., Mohseni Moghadam, M., Mollahasani, N., Chuev, G.N.: A CAS wavelet method for solving nonlinear Fredholm integro-differential equations of fractional order. Commun. Nonl. Sci. Num. Simulat. 16, 1154–1163 (2011)
Rehman, M., Ali Khan, R.: The Legendre wavelet method for solving fractional differential equations. Communm. Nonl. Sci. Numer. Simulat. 16, 4163–4173 (2011)
Keshavarz, E., Ordokhani, Y., Razzaghi, M.: Bernoulli wavelet operational matrix of fractional order integration and its applications in solving the fractional order differential equations. Appl. Math. Model. 38, 6038–6051 (2014)
Mokhtary, P., Ghoreishi, F., Srivastava, H.M.: The müntz-legendre tau method for fractional differential equations. Appl. Math. Model. 40(2), 671–684 (2016)
Esmaeili, S.H., Shamsi, M., Luchkob, Y.: Numerical solution of fractional differential equations with a collocation method based on müntz polynomials. Comput. Math. Appl. 62, 918–929 (2011)
Krishnasamy, V.S., Razzaghi, M.: The numerical solution of the Bagley-Torvik equation with fractional Taylor method. J. Comput. Nonl. Dyn. 11(5), 1–6 (2016)
Sedaghat, S., Nemati, S., Ordokhani, Y.: Application of the hybrid functions to solve neutral delay functional differential equations. Int. J. Comput. Math. 94(3), 503–514 (2017)
Canuto, C., Hussaini, M.Y., Quarteroni, A., Zang, T.A.: Spectral methods. Fundamentals in single domains. Springer, Berlin (2006)
Marzban, H.R., Tabrizidooz, H.R., Razzaghi, M.: A composite collocation method for the nonlinear mixed Volterra-Fredholm-Hammerstein integral equations. Commun. Nonl. Sci. Numer. Simul. 16, 1186–1194 (2011)
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)
Sezer, M., Yalcinbas, S., Sahin, N.: Approximate solution of multi-pantograph equation with variable coefficients. J. Comput. Appl. Math. 214, 406–416 (2008)
Karimi Vanani, S., Aminataei, A.: On the numerical solution of delay differential equations using multiquadric approximation scheme. Funct. Differ. Equ. 17(3), 391–399 (2010)
Evans, D.J., Raslan, K.R.: The Adomian decomposition method for solving delay differential equation. Int. J. Comput. Math. 81(2), 49–54 (2005)
Rangkuti, Y.M., Noorani, M.S.M.: The exact solution of delay differential equations using coupling variational iteration with Taylor series and small term. Bull. Math. 4(1), 1–15 (2012)
Acknowledgments
Authors are very grateful to one of the reviewers for carefully reading the paper and for his(her) comments and suggestions which have improved the paper.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Rahimkhani, P., Ordokhani, Y. & Babolian, E. Müntz-Legendre wavelet operational matrix of fractional-order integration and its applications for solving the fractional pantograph differential equations. Numer Algor 77, 1283–1305 (2018). https://doi.org/10.1007/s11075-017-0363-4
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11075-017-0363-4