Abstract
We give an efficient numerical approach to solve variable-order fractional differential equations (VO-FDEs) by applying fractional-order generalized Chelyshkov wavelets (FOGCW). The beta function is used to determine the exact value for the Riemann-Liouville fractional integral operator of the FOGCW. The exact value and the given wavelets are used to solve the VO-FDEs. Six examples are included to demonstrate the effectiveness of this method. In the last example, we show the application of our method to the variable-order fractional relaxation model.
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Levy, D.: Introduction to Numerical Analysis. Depart. Math. Maryland University, CSCAMM (2010)
Oldham, K.B.: Fractional differential equations in electrochemistry. Adv. Eng. Softw. 41(1), 9–12 (2010)
Wong, J., Abilez, O.J., Kuhl, E.: Computational optogenetics: A novel continuum framework for the photoelectrochemistry of living systems. J. Mech. Phys. Solids. 60(6), 1158–1178 (2012)
Zaky, M.A., Machado, J.A.T.: On the formulation and numerical simulation of distributed-order fractional optimal control problems. Commun. Nonlinear Sci. Numer. Simul. 52, 177–189 (2017)
Naber, M.: Distributed order fractional sub-diffusion. Fractals 12(1), 23–32 (2004)
Meerschaert, M.M., Nane, E., Vellaisamy, P.: Distributed-order fractional diffusions on bounded domains. J. Math. Anal. Appl. 379(1), 216–228 (2011)
Agarwal, R.P.: Dynamical Systems and Applications. World Scientific (1995)
Atanackovic, T.M., Pilipovic, S., Zorica, D.: Distributed-order fractional wave equation on a finite domain: Creep and forced oscillations of a rod. Conti. Mech. Thermodyna 23(4), 305–318 (2011)
Kot, M.: Elements of Mathematical Ecology. Cambridge University Press (2001)
Meng, X., Chen, L., Wu, B.: A delay sir epidemic model with pulse vaccination and incubation times. Nonlinear. Anal. Real. World. Appl. 11(1), 88–98 (2010)
Wang, W., Lin, C.: A new algorithm for integral of trigonometric functions with mechanization. Appl. Math. Comput. 164(1), 71–82 (2005)
Dehghan, M., Shakeri, F.: Solution of parabolic integro-differential equations arising in heat conduction in materials with memory via he’s variational iteration technique. Int. J. Numer. Method. Biomed. Eng. 26(6), 705–715 (2010)
Jerri, A.: Introduction to Integral Equations with Applications. Wiley (1999)
Patnaik, S., Hollkamp, J.P., Semperlotti, F.: Applications of variable-order fractional operators: A review. Proc. R. Soc. A 476(2234), 20190498 (2020)
Glöckle, W.G., Nonnenmacher, T.F.: A fractional calculus approach to self-similar protein dynamics. Biophys. J. 68, 46–53 (1995)
Sun, H.G., et al.: A comparative study of constant-order and variable-order fractional models in characterizing memory property of systems. E.J.P.S.T. 193. 1, 185–192 (2011)
Coimbra, C.: Mechanics with variable-order differential operators. Ann. Phys. 12, 692–703 (2003)
Chechkin, A.V., Gorenflo, R., Sokolov, I.M.: Fractional diffusion in inhomogeneous media. J. Phys. A.: Math. Gen. 38(42), 679 (2005)
Sun, H., Chen, W., Chen, Y.: Variable-order fractional differential operators in anomalous diffusion modeling. Physica A Stat. Mech. Appl. 388, 4586–4592 (2009)
Usman, M., Hamid, M., Haq, R.U., Wang, W.: An efficient algorithm based on Gegenbauer wavelets for the solutions of variable-order fractional differential equations. Eur. Phys. J. Plus. 133(8), 327 (2018)
Chen, Y.M., Wei, Y.Q., Liu, D.Y., Yu, H.: Numerical solution for a class of nonlinear variable order fractional differential equations with legendre wavelets. Appl. Math. Lett. 46, 83–88 (2015)
Heydari, M.H., Avazzadeh, Z., Haromi, M.F.: A wavelet approach for solving multi-term variable-order time fractional diffusion-wave equation. Appl. Math. Comput. 341, 215–228 (2019)
Yuttanan, B., Razzaghi, M.: Legendre wavelets approach for numerical solutions of distributed order fractional differential equations. Appl. Math. Model. 70, 350–364 (2019)
Ur Rehman, M., Khan, R.A.: The legendre wavelet method for solving fractional differential equations. Commun. Nonlinear. Sci. Numer. Simul. 16(11), 4163–4173 (2011)
Li, Y.: Solving a nonlinear fractional differential equation using Chebyshev wavelets. Commun. Nonlinear. Sci. Numer. Simul. 15(9), 2284–2292 (2010)
Akyüz-Daşcıoğlu, A., Sezer, M.: Chebyshev polynomial solutions of systems of higher-order linear Fredholm–Volterra integro-differential equations. J. Franklin Institute 342(6), 688–701 (2005)
Phan, T.T., Vo, N.T., Razzaghi, M.: Taylor wavelet method for fractional delay differential equations. Eng. Comput., 1–10 (2019)
Vichitkunakorn, P., Vo, T.N., Razzaghi, M.: A numerical method for fractional pantograph differential equations based on Taylor wavelets. T. I. Meas Control 42, 1334–1344 (2020)
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(24), 6038–6051 (2014)
Oguz, C., Sezer, M.: Chelyshkov collocation method for a class of mixed functional integro-differential equations. Appl. Math. Comput. 259, 943–954 (2015)
Moradi, L., Mohammadi, F., Baleanu, D.: A direct numerical solution of time-delay fractional optimal control problems by using Chelyshkov wavelets. J. Vib. Control 25(2), 310–324 (2019)
Ghoreishi, F., Yazdani, S.: An extension of the spectral Tau method for numerical solution of multi-order fractional differential equations with convergence analysis. Comput. Math. Appl. 61, 30–43 (2011)
Mohammadi, F., Cattani, C.: Fractional-order Legendre wavelet Tau method for solving fractional differential equations. J. Comput. Appl. Math. 339, 306–316 (2018)
Kazem, S., Abbasbandy, S., Kumar, S.: Fractional-order Legendre functions for solving fractional-order differential equations. Appl. Math. Model. 37(7), 5498–5510 (2013)
Bhrawy, A.H., Alhamed, A.Y., Baleanu, D.: New spectral techniques for systems of fractional differential equations using fractional-order generalized Laguerre orthogonal functions. Fract. Calc. Appl. Anal. 17, 1138–1157 (2014)
Rahimkhani, P., Ordokhani, Y., Babolian, E.: Fractional-order Bernoulli wavelets and their applications. Appl. Math. Model. 40, 8087–8107 (2016)
Mashayekhi, S., Razzaghi, M.: Numerical solution of the fractional Bagley-Torvik equation by using hybrid functions approximation. Math. Methods. Appl. Sci. 39(3), 353–365 (2016)
Mashayekhi, S., Razzaghi, M.: Numerical solution of distributed order fractional differential equations by hybrid functions. J Comput. Phys. 315, 169–181 (2016)
Nagy, A.M., Sweilam, N.H., El-Sayed, A.A.: New operational matrix for solving multiterm variable order fractional differential equations. J. Comput. Nonlinear. Dyn., 13(1) (2018)
Razminia, A., Dizaji, A.F., Majd, V.J.: Solution existence for non-autonomous variable-order fractional differential equations. Math. Comput. Model. 55(3–4), 1106–1117 (2012)
Xu, Y., He, Z.: Existence and uniqueness results for Cauchy problem of variable-order fractional differential equations. J. Appl. Math. Comput. 43(1), 295–306 (2013)
Zheng, X., Wang, H., Fu, H.: Well-posedness of fractional differential equations with variable-order Caputo-Fabrizio derivative. Chaos, Solitons & Fractals 138, 109966 (2020)
Abramowitz, M., Stegun, I.A.: Handbook of Mathematical Functions. Dover (1973)
Chelyshkov, V.S.: Alternative orthogonal polynomials and quadratures. Elect. Trans. Numer. Anal. 25(7), 17–26 (2006)
Mohammadi, F.: Numerical solution of systems of fractional delay differential equations using a new kind of wavelet basis. Comput. Appl. Math. 37 (4), 4122–4144 (2018)
Ngo, H.T.B., Vo, T.N., Razzaghi, M.: An effective method for solving nonlinear fractional differential equations. Engineering with Computers. https://doi.org/10.1007/s00366-020-01143-3 (2020)
Bhrawy, A.H., Zaky, A.M.: Numerical algorithm for the variable-order Caputo fractional functional differential equation. Nonlinear. Dyn. 85 (3), 1815–1823 (2016)
Hassani, H., Dahaghin, M.S., Heydari, H.: A new optimized method for solving variable-order fractional differential equation. J. Math. Ext. 11, 85–98 (2017)
Zaky, M.A., et al.: New recursive approximations for variable-order fractional operators with applications. arXiv:1804.01198 (2018)
Sun, H.G., Chen, W., Wei, H., Chen, Y.Q.: A comparative study of constant-order and variable-order fractional models in characterizing memory property of systems. Eur. Phys. J. 193(1), 185–192 (2011)
Acknowledgements
A part of this paper was completed when the corresponding author was working as a researcher at Vietnam Institute for Advance Study in Mathematics (VIASM). The author would like to thank the VIASM for providing a fruitful research environment and extending support and hospitality during their visit. The authors wish to express their sincere thanks to the anonymous referee for valuable suggestions that improved the final version of the manuscript.
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Ngo, H.T.B., Razzaghi, M. & Vo, T.N. Fractional-order Chelyshkov wavelet method for solving variable-order fractional differential equations and an application in variable-order fractional relaxation system. Numer Algor 92, 1571–1588 (2023). https://doi.org/10.1007/s11075-022-01354-3
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DOI: https://doi.org/10.1007/s11075-022-01354-3