Abstract
We give a novel method to solve systems of nonlinear fractional differential equations (NFDEs). We first introduce a new class of basis functions called fractional-order generalized Taylor wavelets. The Riemann–Liouville fractional integral operator, of the fractional-order generalized Taylor wavelets, is determined. An exact formula for this operator will be obtained by using the regularized beta function. By applying this exact formula we reduce the given system of NFDEs to a system of algebraic equations. The method is applied to the fractional models in human respiratory syncytial virus infection. We also give numerical examples to show the effectiveness and high accuracy of the present method.
![](http://media.springernature.com/m312/springer-static/image/art%3A10.1007%2Fs00500-021-06436-3/MediaObjects/500_2021_6436_Fig1_HTML.png)
![](http://media.springernature.com/m312/springer-static/image/art%3A10.1007%2Fs00500-021-06436-3/MediaObjects/500_2021_6436_Fig2_HTML.png)
Similar content being viewed by others
Explore related subjects
Discover the latest articles, news and stories from top researchers in related subjects.References
Abramowitz M (1973) Handbook of mathematical functions. National Bureau of Standards, Applied Mathematics Series 55
Bagley RL, Torvik PJ (1985) Fractional calculus in the transient analysis of viscoelastically damped structures. AIAA J 23:918–925
Baillie RT (1996) Long memory processes and fractional integration in econometrics. J Econom 73:5–59
Bhrawy A, Alhamed Y, Baleanu D, Al-Zahrani A (2014) New spectral techniques for systems of fractional differential equations using fractional-order generalized laguerre orthogonal functions. Fract Calc Appl Anal 17:1137–1157
Bhrawy A, Tharwat M, Yildirim A (2013) A new formula for fractional integrals of Chebyshev polynomials: application for solving multi-term fractional differential equations. Appl Math Model 37:4245–4252
Carpinteri A, Mainardi F (2014) Fractals and fractional calculus in continuum mechanics, vol. 378. Springer
Deng W, Du S, Wu Y (2013) High order finite difference WENO schemes for fractional differential equations. Appl Math Lett 26:362–366
Ertürk VS, Odibat ZM, Momani S (2011) An approximate solution of a fractional order differential equation model of human T-cell lymphotropic virus I (HTLV-I) infection of CD4+ T-cells. Comput Math Appl 62:996–1002
Gupta S, Kumar D, Singh J (2015) Numerical study for systems of fractional differential equations via Laplace transform. J Egypt Math Soc 23:256–262
Hall MG, Barrick TR (2008) From diffusion-weighted MRI to anomalous diffusion imaging. Magn Reson Med 59:447–455
He J (1999) Some applications of nonlinear fractional differential equations and their approximations. Bull Sci Technol 15:86–90
Heydari M, Hooshmandasl MR, Mohammadi F (2014) Legendre wavelets method for solving fractional partial differential equations with Dirichlet boundary conditions. Appl Math Comput 234:267–276
Kazem S, Abbasbandy S, Kumar S (2013) Fractional-order Legendre functions for solving fractional-order differential equations. Appl Math Model 37:5498–5510
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 38:6038–6051
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
Khalil H, Khan RA (2014) A new method based on Legendre polynomials for solutions of the fractional two-dimensional heat conduction equation. Comput Math Appl 67:1938–1953
Li M, Zhao W (2013) Solving Abel’s type integral equation with Mikusinski’s operator of fractional order. Adv Math Phys
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 216:2276–2285
Machado JT, Kiryakova V, Mainardi F (2011) Recent history of fractional calculus. Commun Nonlinear Sci Numer Simul 16:1140–1153
Mandelbrot B (1967) Some noises with I/F spectrum, a bridge between direct current and white noise. IEEE Trans Inf Theory 13:289–298
Mashayekhi S, Razzaghi M (2016a) Numerical solution of distributed order fractional differential equations by hybrid functions. J Comput Phys 315:169–181
Mashayekhi S, Razzaghi M (2016b) Numerical solution of the fractional Bagley–Torvik equation by using hybrid functions approximation. Math Method Appl Sci 39:353–365
Meerschaert MM, Tadjeran C (2006) Finite difference approximations for two-sided space-fractional partial differential equations. Appl Numer Math 56:80–90
Miller KS, Ross B (1993) An introduction to the fractional calculus and fractional differential equations. Wiley, London
Mohammadi F, Cattani C (2018) A generalized fractional-order Legendre wavelet tau method for solving fractional differential equations. J Comput Appl Math 339:306–316
Nemati S, Torres DF (2020) A new spectral method based on two classes of hat functions for solving systems of fractional differential equations and an application to respiratory syncytial virus infection. Soft Comput, 1–13
Odibat ZM, Shawagfeh NT (2007) Generalized Taylors formula. Appl Math Comput 186:286–293
Oldham KB (2010) Fractional differential equations in electrochemistry. Adv Eng Softw 41:9–12
Parand K, Nikarya M (2014) Application of bessel functions for solving differential and integro-differential equations of the fractional order. Appl Math Model 38:4137–4147
Podlubny I (1998) Fractional differential equations: an introduction to fractional derivatives, fractional differential equations, to methods of their solution and some of their applications. Elsevier, Amsterdam
Povstenko Y (2010) Signaling problem for time-fractional diffusion-wave equation in a half-space in the case of angular symmetry. Nonlinear Dyn 59:593–605
Rahimkhani P, Ordokhani Y, Babolian E (2016) Fractional-order Bernoulli wavelets and their applications. Appl Math Model 40:8087–8107
Rahimkhani P, Ordokhani Y, Babolian E (2017) Numerical solution of fractional pantograph differential equations by using generalized fractional-order Bernoulli wavelet. J Comput Appl Math 309:493–510
Rosa S, Torres DF (2018a) Optimal control of a fractional order epidemic model with application to human respiratory syncytial virus infection. Chaos Soliton Fract 117:142–149
Rosa S, Torres DF (2018b) Parameter estimation, sensitivity analysis and optimal control of a periodic epidemic model with application to HRSV in Florida. Stat Optim Inf Comput 6:139–49
Rossikhin YA, Shitikova MV (1997) Applications of fractional calculus to dynamic problems of linear and nonlinear hereditary mechanics of solids. Appl Mech Rev 50:15–67
Saadatmandi A (2014) Bernstein operational matrix of fractional derivatives and its applications. Appl Math Model 38:1365–1372
Saeedi H, Moghadam MM, Mollahasani N, Chuev G (2011) A CAS wavelet method for solving nonlinear fredholm integro-differential equations of fractional order. Commun Nonlinear Sci Num Simul 16:1154–1163
Shen S, Liu F, Anh VV (2019) The analytical solution and numerical solutions for a two-dimensional multi-term time fractional diffusion and diffusion-wave equation. J Comput Appl Math 345:515–534
Yi M, Huang J, Wei J (2013) Block pulse operational matrix method for solving fractional partial differential equation. Appl Math Comput 221:121–131
Yüzbaşı Ş (2013) Numerical solutions of fractional Riccati type differential equations by means of the bernstein polynomials. Appl Math Comput 219:6328–6343
Zhu L, Fan Q (2012) Solving fractional nonlinear Fredholm integro-differential equations by the second kind Chebyshev wavelet. Commun Nonlinear Sci Num Simul 17:2333–2341
Zurigat M, Momani S, Odibat Z, Alawneh A (2010) The homotopy analysis method for handling systems of fractional differential equations. Appl Math Model 34:24–35
Acknowledgements
The authors wish to express their sincere thanks to the anonymous referee for valuable suggestions that improved the final version of the manuscript.
Author information
Authors and Affiliations
Corresponding author
Ethics declarations
Conflict of interest
The authors declare no potential conflict of interests.
Ethical approval
This article does not contain any studies with human participants or animals performed by any of the authors.
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
Vo, T.N., Razzaghi, M. & Toan, P.T. Fractional-order generalized Taylor wavelet method for systems of nonlinear fractional differential equations with application to human respiratory syncytial virus infection. Soft Comput 26, 165–173 (2022). https://doi.org/10.1007/s00500-021-06436-3
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00500-021-06436-3