New recursive approximations for variable-order fractional operators with applications
Abstract
To broaden the range of applicability of variable-order fractional differential models, reliable numerical approaches are needed to solve the model equation.In this paper, we develop Laguerre spectral collocation methods for solving variable-order fractional initial value problems on the half line. Specifically, we derive three-term recurrence relations to efficiently calculate the variable-order fractional integrals and derivatives of the modified generalized Laguerre polynomials, which lead to the corresponding fractional differentiation matrices that will be used to construct the collocation methods. Comparison with other existing methods shows the superior accuracy of the proposed spectral collocation methods.
Keyword : spectral collocation methods, modified generalized Laguerre polynomials, variable order fractional integrals and derivatives, Bagley-Torvik equation
How to Cite
Zaky, M. A., Doha, E. H., Taha, T. M., & Baleanu, D. (2018). New recursive approximations for variable-order fractional operators with applications. Mathematical Modelling and Analysis, 23(2), 227-239. https://doi.org/10.3846/mma.2018.015
This work is licensed under a Creative Commons Attribution 4.0 International License.
References
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[21] Ch.-Ch. Tseng. Design of variable and adaptive fractional order FIR differentiators. Signal Processing, 86(10):2554–2566, 2006. https://doi.org/10.1016/j.sigpro.2006.02.004
[22] S. Wei, W. Chen and J. Zhang. Time-fractional derivative model for chloride ions sub-diffusion in reinforced concrete. European Journal of Environmental and Civil Engineering, 21(3):319–331, 2017. https://doi.org/10.1080/19648189.2015.1116467
[23] M.A. Zaky. A Legendre spectral quadrature tau method for the multi-term time-fractional diffusion equations. Computational and Applied Mathematics, pp. 1–14, 2017. https://doi.org/10.1007/s40314-017-0530-1
[24] M.A. Zaky, S.S. Ezz-Eldien, E.H. Doha, J.A. Tenreiro Machado and A.H. Bhrawy. An efficient operational matrix technique for multidimensional variable-order time fractional diffusion equations. Journal of Computational and Nonlinear Dynamics, 11(6):061002, 2016. https://doi.org/10.1115/1.4033723
[25] M. Zayernouri and G.E. Karniadakis. Fractional spectral collocation methods for linear and nonlinear variable order FPDEs. Journal of Computational Physics, 293:312–338, 2015. https://doi.org/10.1016/j.jcp.2014.12.001
[2] T. Atanackovic and S. Pilipovic. Hamilton’s principle with variable order fractional derivatives. Fractional Calculus and Applied Analysis, 14(1):94–109, 2011. https://doi.org/10.2478/s13540-011-0007-7
[3] A. Atangana. On the stability and convergence of the time-fractional variable order telegraph equation. Journal of Computational Physics, 293:104–114, 2015. https://doi.org/10.1016/j.jcp.2014.12.043
[4] D. Baleanu, A.H. Bhrawy and T.M. Taha. A modified generalized Laguerre spectral method for fractional differential equations on the half line. Abstract and Applied Analysis, 2013:1–12, 2013. https://doi.org/10.1155/2013/413529
[5] A.H. Bhrawy and M.A. Zaky. Numerical simulation for two-dimensional variable-order fractional nonlinear cable equation. Nonlinear Dynamics, 80(1):101–116, 2015. https://doi.org/10.1007/s11071-014-1854-7
[6] A.H. Bhrawy and M.A. Zaky. An improved collocation method for multi-dimensional space–time variable-order fractional Schrödinger equations. Applied Numerical Mathematics, 111:197–218, 2017. https://doi.org/10.1016/j.apnum.2016.09.009
[7] J. Cao and Y. Qiu. A high order numerical scheme for variable order fractional ordinary differential equation. Applied Mathematics Letters, 61:88–94, 2016. https://doi.org/10.1016/j.aml.2016.05.012
[8] C.F.M. Coimbra. Mechanics with variable-order differential operators. Annalen der Physik, 12(11-12):692–703, 2003. https://doi.org/10.1002/andp.200310032
[9] G.R.J Cooper and D.R. Cowan. Filtering using variable order vertical derivatives. Computers & Geosciences, 30(5):455–459, 2004. https://doi.org/10.1016/j.cageo.2004.03.001
[10] Z.-J. Fu, W. Chen and L. Ling. Method of approximate particular solutions for constant- and variable-order fractional diffusion models. Engineering Analysis with Boundary Elements, 57:37–46, 2015. https://doi.org/10.1016/j.enganabound.2014.09.003
[11] D. Ingman and J. Suzdalnitsky. Application of differential operator with servo-order function in model of viscoelastic deformation process. Journal of Engineering Mechanics, 131(7):763–767, 2005. https://doi.org/10.1061/(ASCE)0733-9399(2005)131:7(763)
[12] C.F. Lorenzo and T.T. Hartley. Variable order and distributed order fractional operators. Nonlinear Dynamics, 29(1):57–98, 2002. https://doi.org/10.1023/A:1016586905654
[13] B.P. Moghaddam and J.A.T. Machado. Extended algorithms for approximating variable order fractional derivatives with applications. Journal of Scientific Computing, 71(3):1351–1374, 2017. https://doi.org/10.1007/s10915-016-0343-1
[14] B.P. Moghaddam, J.A.T. Machado and H. Behforooz. An integro quadratic spline approach for a class of variable-order fractional initial value problems. Chaos, Solitons & Fractals, 102:354–360, 2017. https://doi.org/10.1016/j.chaos.2017.03.065
[15] H. Pedro, M. Kobayashi, J. Pereira and C. Coimbra. Variable order modeling of diffusive-convective effects on the oscillatory flow past a sphere. Journal of Vibration and Control, 14(9-10):1659–1672, 2008. https://doi.org/10.1177/1077546307087397
[16] L. Ramirez and C.F.M. Coimbra. A variable order constitutive relation for viscoelasticity. Annalen der Physik, 16(7-8):543–552, 2007. https://doi.org/10.1002/andp.200710246
[17] S. Samko and B. Ross. Integration and differentiation to a variable fractional orde. Integral Transforms and Special Functions, 1(4):277–300, 1993. https://doi.org/10.1080/10652469308819027
[18] C.M. Soon, C.F.M. Coimbra and M. Kobayashi. The variable viscoelasticity oscillator. Annalen der Physik, 14(6):378–389, 2005. https://doi.org/10.1002/andp.200410140
[19] H.G. Sun., W. Chen, H. Wei and Y.Q. Chen. A comparative study of constant-order and variable-order fractional models in characterizing memory property of systems. The European Physical Journal Special Topics, 193(1):185–192, 2011. https://doi.org/10.1140/epjst/e2011-01390-6
[20] A. Tayebi, Y. Shekari and M.H. Heydari. A meshless method for solving two-dimensional variable-order time fractional advection–diffusion equation. Journal of Computational Physics, 340:655–669, 2017. https://doi.org/10.1016/j.jcp.2017.03.061
[21] Ch.-Ch. Tseng. Design of variable and adaptive fractional order FIR differentiators. Signal Processing, 86(10):2554–2566, 2006. https://doi.org/10.1016/j.sigpro.2006.02.004
[22] S. Wei, W. Chen and J. Zhang. Time-fractional derivative model for chloride ions sub-diffusion in reinforced concrete. European Journal of Environmental and Civil Engineering, 21(3):319–331, 2017. https://doi.org/10.1080/19648189.2015.1116467
[23] M.A. Zaky. A Legendre spectral quadrature tau method for the multi-term time-fractional diffusion equations. Computational and Applied Mathematics, pp. 1–14, 2017. https://doi.org/10.1007/s40314-017-0530-1
[24] M.A. Zaky, S.S. Ezz-Eldien, E.H. Doha, J.A. Tenreiro Machado and A.H. Bhrawy. An efficient operational matrix technique for multidimensional variable-order time fractional diffusion equations. Journal of Computational and Nonlinear Dynamics, 11(6):061002, 2016. https://doi.org/10.1115/1.4033723
[25] M. Zayernouri and G.E. Karniadakis. Fractional spectral collocation methods for linear and nonlinear variable order FPDEs. Journal of Computational Physics, 293:312–338, 2015. https://doi.org/10.1016/j.jcp.2014.12.001