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Abstrakty
In this paper we discuss different definitions of variable-order derivatives of high order and we propose accurate and robust algorithms for their approximate calculation. The proposed algorithms are based on finite difference approximations and B-spline interpolation. We compare the performance of the algorithms by experimental convergence order. Numerical examples are presented demonstrating the efficiency and accuracy of the proposed algorithms.
Wydawca
Czasopismo
Rocznik
Tom
Strony
293--311
Opis fizyczny
Bibliogr. 42 poz., tab., wykr.
Twórcy
autor
- Department of Mathematics, Lahijan Branch, Islamic Azad University, Lahijan, Iran
autor
- Department of Electrical Engineering, Institute of Engineering, Rua Dr. António Bernardino de Almeida, 431, Porto 4249-015, Portugal
Bibliografia
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- [5] Machado JAT. Numerical calculation of the left and right fractional derivatives. Journal of Computational Physics. 2015; 293: 96-103. doi: 10.1016/j.jcp.2014.05.029.
- [6] Bhrawy AH, Doha EH, Machado JAT, Ezz-Eldien SS. An efficient numerical scheme for solving multi-dimensional fractional optimal control problems with a quadratic performance index. Asian Journal of Control. 2015; 17 (6): 2389-2402. doi: 10.1002/asjc.1109.
- [7] Yang XJ, Baleanu D, Srivastava HM. Local fractional similarity solution for the diffusion equation defined on Cantor sets. Applied Mathematics Letters. 2015; 47: 54-60. doi: 10.1016/j.am1.2015.02.024.
- [8] Yang XJ, Machado JAT, Srivastava HM. A new numerical technique for solving the local fractional diffusion equation: two-dimensional extended differential transform approach. Applied Mathematics and Computation. 2016; 274: 143-151. doi: 10.1016/j.amc.2015.10.072.
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- [10] Zayernouri M, Karniadakis GE. Fractional spectral collocation methods for linear and nonlinear variable order FPDEs. Journal of Computational Physics. 2015; 293: 312-338. doi: 10.1016/j.jcp.2014.12.001.
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- [12] Ross B, Samko S. Fractional integration operator of variable order in the holder spaces H λ(x). International Journal of Mathematics and Mathematical Sciences. 1995; 18 (4): 777-788. doi: 10.1155/s0161171295001001.
- [13] Coimbra CFM. Mechanics with variable-order differential operators. Ann Phys. 2003; 12 (1112): 692-703. doi: 10.1002/andp.200310032.
- [14] Almeida A, Samko S. Fractional and hypersingular operators in variable exponent spaces on metric measure spaces. Mediterranean Journal of Mathematics. 2009; 6 (2): 215-232. doi: 10.1007/s00009-009-0006-7.
- [15] Lorenzo CF, Hartley TT. Variable order and distributed order fractional operators. Nonlinear dynamics. 2002; 29 (l-4): 57-98. doi: 10.1023/A:1016586905654.
- [16] Moghaddam BP, Yaghoobi S, Machado JAT. An extended predictor-corrector algorithm for variable-order fractional delay differential equations. Journal of Computational and Nonlinear Dynamics. 2016; 11 (6): 061001. doi: 10.1115/1.4032574.
- [17] Diaz G, Coimbra CFM. Nonlinear dynamics and control of a variable order oscillator with application to the van der Pol equation. Nonlinear Dynamics. 2009; 56 (l-2): 145-157. doi: 10.1007/s11071-008-9385-8.
- [18] Pedro HTC, Kobayashi MH, Pereira JMC, Coimbra CFM. Variable order modeling of diffusive-convective effects on the oscillatory flow past a sphere. Journal of Vibration and Control. 2008; 14 (9-10): 1659-1672. doi: 10.1177/1077546307087397.
- [19] Ramirez LES, Coimbra CFM. On the variable order dynamics of the nonlinear wake caused by a sedimenting particle. Physica D: Nonlinear Phenomena. 2011; 240 (13): 1111-1118. doi: 10.1016/j.physd.2011.04.001.
- [20] Son HG, Chen W, Wei H, Chen YQ. A comparative study of constant-order and variable-order fractional models in characterizing memory property of systems. The European Physical Journal Special Topics. 2011; 193 (1): 185-192. doi: 10.1140/epjst/e2011-01390-6.
- [21] Orosco J, Coimbra CFM. On the control and stability of variable-order mechanical systems. Nonlinear Dynamics. 2016; 86 (1): 695-710. doi: 10.1007/s11071-016-2916-9.
- [22] Atanackovic T, Janev M, Pilipovic S, Zorica D. An expansion formula for fractional derivatives of variable order. Open Physics. 2013; 11 (10). doi: 10.2478/s11534-013-0243-z.
- [23] Tavares D, Almeida R, Torres DFM. Caputo derivatives of fractional variable order: numerical approximations. Communications in Nonlinear Science and Numerical Simulation. 2016; 35: 69-87. doi: 10.1016/j.cnsns.2015.10.027.
- [24] Valério D, Vinagre G, Domingues J, Costa JS. Variable-order fractional derivatives and their numerical approximations I-real orders. Caparica; 2009.
- [25] Valério D, Costa JS. Variable-order fractional derivatives and their numerical approximations II complex orders. Symposium on Fractional Signals and Systems; 2009.
- [26] Sierociuk D, Malesza W, Macias M. Derivation, interpretation, and analog modelling of fractional variable order derivative definition. Applied Mathematical Modelling. 2015; 39 (13): 3876-3888. doi: 10.1016/j.apm.2014.12.009.
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- [28] Sun H, Chen W, Sheng H, Chen YQ. On mean square displacement behaviors of anomalous diffusions with variable and random orders. Physics Letters A. 2010; 374 (7): 906-910. doi: 10.1016/j.physleta.2009.12.021.
- [29] Sun HG, Chen W, Chen YQ. Variable-order fractional differential operators in anomalous diffusion modeling. Physica A: Statistical Mechanics and its Applications. 2009; 388 (21): 4586-4592. doi: 10.1016/physa.2009.07.024.
- [30] Ramirez LES, Coimbra CFM. On the selection and meaning of variable order operators for dynamic modeling. International Journal of Differential Equations. 2010; 2010: 1-16. doi: 10.1155/2010/846107.
- [31] Sun Z, Wu X. A fully discrete difference scheme for a diffusion-wave system. Applied Numerical Mathematics. 2006; 56 (2): 193-209. doi: 10.1016/j.apnum.2005.03.003.
- [32] Lin Y, Xu C. Finite difference/spectral approximations for the time-fractional diffusion equation. Journal of Computational Physics. 2007; 225 (2): 1533-1552. doi: 10.1016/j.jcp.2007.02.001.
- [33] Moghaddam BP, Mostaghim ZS. A novel matrix approach to fractional finite difference for solving models based on nonlinear fractional delay differential equations. Ain Shams Engineering Journal. 2014; 5 (2): 585-594. doi: 10.1016/j.asej.2013.11.007.
- [34] Moghaddam BP, Mostaghim ZS. Modified finite difference method for solving fractional delay differential equations. B Soc Paran Mat. 2016; 35 (2): 49. doi: 10.5269/bspm.v35i2.25081.
- [35] Zhuang P, Liu F, Anh V, Turner I. Numerical Methods for the Variable-Order Fractional Advection-Diffusion Equation with a Nonlinear Source Term. SIAM Journal on Numerical Analysis. 2009; 47 (3): 1760-1781. doi: 10.1137/080730597.
- [36] Shen S, Liu F, Chen J, Turner I, Anh V. Numerical techniques for the variable order time fractional diffusion equation. Applied Mathematics and Computation. 2012; 218 (22): 10861-10870. doi: 10.1016/j.amc.2012.04.047.
- [37] Chen CM, Liu F, Burrage K, Chen Y. Numerical methods of the variable-order Rayleigh-Stokes problem for a heated generalized second grade fluid with fractional derivative. IMA Journal of Applied Mathematics. 2012; 78 (5): 924-944. doi: 10.1093/imamat/hxr079.
- [38] Shen S, Liu F, Anh V, Turner I, Chen J. A characteristic difference method for the variable-order fractional advection-diffusion equation. Journal of Applied Mathematics and Computing. 2013; 42 (1-2): 371-386. doi: 10.1007/s12190-012-0642-0.
- [39] Zhao X, Sun ZZ, Karniadakis GE. Second-order approximations for variable order fractional derivatives: Algorithms and applications. Journal of Computational Physics. 2015; 293: 184-200. doi: 10.1016/j.jcp.2014.08.015.
- [40] Moghaddam BP, Machado JAT. A stable three-level explicit spline finite difference scheme for a class of nonlinear time variable order fractional partial differential equations. Computers & Mathematics with Applications. 2016; doi: 10.1016/j.camwa.2016.07.010.
- [41] Yaghoobi S, Moghaddam BP, Ivaz K. An efficient cubic spline approximation for variable-order fractional differential equations with time delay. Nonlinear Dynamics. 2016; doi: 10.1007/s11071-016-3079-4.
- [42] Bartels S. Numerical Approximation of Partial Differential Equations. Springer; 2016.
Typ dokumentu
Bibliografia
Identyfikator YADDA
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