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Neural Computing and Applications

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05.08.2017 | Original Article

The construction of operational matrix of fractional integration for solving fractional differential and integro-differential equations

verfasst von: Farzane Yousefi, Azim Rivaz, Wen Chen

Erschienen in: Neural Computing and Applications | Ausgabe 6/2019

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Abstract

This study intends to present a general formulation for the hybrid Jacobi and block pulse operational matrix of fractional integral operator in order to solve fractional differential and integro-differential equations. First, we define hybrid Jacobi polynomials and block pulse functions as an orthogonal basis for function approximation. Then, we construct the operational matrix of fractional integration for these hybrid functions. With the combined features of these hybrid functions and their operational matrix of fractional integration, the governing equations that take the form of fractional differential and integro-differential equations are reduced to a system of algebraic equations. Illustrative examples are given to demonstrate the validity and reliability of the present technique.

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Bagley RL, Torvik PJ (1986) On the fractional calculus model of viscoelastic behavior. J Rheol 30(1):133–155. doi:10.1122/1.549887 CrossRefMATH

Calderon AJ, Vinagre BM, Feliu V (2006) Fractional order control strategies for power electronic buck converters. Signal Process 86(10):2803–2819. doi:10.1016/j.sigpro.2006.02.022 CrossRefMATH

Grigorenko I, Grigorenko E (2003) Chaotic dynamics of the fractional Lorenz system. Phys Rev Lett. doi:10.1103/PhysRevLett.91.034101

Kumar S, Kumar D, Singh J (2016) Fractional modelling arising in unidirectional propagation of long waves in dispersive media. Adv Nonlinear Anal 5(4):383–394. doi:10.1515/anona-2013-0033 MathSciNetMATH

Al Jarbouh A (2012) Rheological behaviour modelling of viscoelastic materials by using fractional model. Energy Procedia 19:143–157. doi:10.1016/j.egypro.2012.05.194 CrossRef

Mainardi F (2012) Fractional calculus: some basic problems in continuum and statistical mechanics. arXiv preprint arXiv:1201.0863. In: Carpinteri A, Mainardi F (eds) (1997) Fractals and fractional calculus in continuum mechanics. Springer Verlag, Wien and New York, pp 291–348

Tavazoei MS, Haeri M (2008) Chaos control via a simple fractional-order controller. Phys Lett A 372(6):798–807. doi:10.1016/j.physleta.2007.08.040 CrossRefMATH

Aghajani A, Jalilian Y, Trujillo J (2012) On the existence of solutions of fractional integro-differential equations. Fract Calc Appl Anal 15(1):44–69. doi:10.2478/s13540-012-0005-4 MathSciNetCrossRefMATH

Deng J, Ma L (2010) Existence and uniqueness of solutions of initial value problems for nonlinear fractional differential equations. Appl Math Lett 23(6):676–680. doi:10.1016/j.aml.2010.02.007 MathSciNetCrossRefMATH

10.

Kumar S, Kumar A, Baleanu D (2016) Two analytical methods for time-fractional nonlinear coupled Boussinesq–Burgers equations arise in propagation of shallow water waves. Nonlinear Dyn 85(2):699–715. doi:10.1007/s11071-016-2716-2 MathSciNetCrossRefMATH

11.

Inc M (2008) The approximate and exact solutions of the space-and time-fractional Burgers equations with initial conditions by variational iteration method. J Math Anal Appl 345(1):476–484. doi:10.1016/j.jmaa.2008.04.007 MathSciNetCrossRefMATH

12.

Ray SS (2009) Analytical solution for the space fractional diffusion equation by two-step Adomian decomposition method. Commun Nonlinear Sci Numer Simul 14(4):1295–1306. doi:10.1016/j.cnsns.2008.01.010 MathSciNetCrossRefMATH

13.

Heydari MH, Hooshmandasl MR, Mohammadi F, Cattani C (2014) Wavelets method for solving systems of nonlinear singular fractional Volterra integro-differential equations. Commun Nonlinear Sci Numer Simul 19(1):37–48. doi:10.1016/j.cnsns.2013.04.026 MathSciNetCrossRefMATH

14.

Hosseini VR, Shivanian E, Chen W (2016) Local radial point interpolation (MLRPI) method for solving time fractional diffusion-wave equation with damping. J Comput Phys 312:307–332. doi:10.1016/j.jcp.2016.02.030 MathSciNetCrossRefMATH

15.

Hosseini VR, Shivanian E, Chen W (2015) Local integration of 2-D fractional telegraph equation via local radial point interpolant approximation. EPJ Plus 130(2):1–21. doi:10.1140/epjp/i2015-15033-5

16.

Hosseini VR, Chen W, Avazzadeh Z (2014) Numerical solution of fractional telegraph equation by using radial basis functions. Eng Anal Bound Elem 38:31–39. doi:10.1016/j.enganabound.2013.10.009 MathSciNetCrossRefMATH

17.

Kazem S (2013) An integral operational matrix based on Jacobi polynomials for solving fractional-order differential equations. Appl Math Model 37(3):1126–1136. doi:10.1016/j.apm.2012.03.033 MathSciNetCrossRefMATH

18.

Lia C, Kumarb A, Kumarb S, Yangc XJ (2016) On the approximate solution of nonlinear time-fractional KdV equation via modified homotopy analysis Laplace transform method. J Nonlinear Sci Appl 9:5463–5470MathSciNetCrossRef

19.

Yin XB, Kumar S, Kumar D (2015) A modified homotopy analysis method for solution of fractional wave equations. Adv Mech Eng 7(12):1–8. doi:10.1177/1687814015620330 CrossRef

20.

Atabakzadeh MH, Akrami MH, Erjaee GH (2013) Chebyshev operational matrix method for solving multi-order fractional ordinary differential equations. Appl Math Model 37(20):8903–8911. doi:10.1016/j.apm.2013.04.019 MathSciNetCrossRefMATH

21.

Kilicman A, Al Zhour ZAA (2007) Kronecker operational matrices for fractional calculus and some applications. Appl Math Comput 187(1):250–265. doi:10.1016/j.amc.2006.08.122 MathSciNetMATH

22.

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(24):6038–6051. doi:10.1016/j.apm.2014.04.064 MathSciNetCrossRefMATH

23.

Zhu L, Fan Q (2013) Numerical solution of nonlinear fractional-order Volterra integro-differential equations by SCW. Commun Nonlinear Sci Numer Simul 18(5):1203–1213. doi:10.1016/j.cnsns.2012.09.024 MathSciNetCrossRefMATH

24.

Saeedi H, Moghadam MM, Mollahasani N, Chuev GN (2011) A CAS wavelet method for solving nonlinear Fredholm integro-differential equations of fractional order. Commun Nonlinear Sci Numer Simul 16(3):1154–1163. doi:10.1016/j.cnsns.2010.05.036 MathSciNetCrossRefMATH

25.

Saadatmandi A, Dehghan M (2010) A new operational matrix for solving fractional-order differential equations. Comput Math Appl 59(3):1326–1336. doi:10.1016/j.camwa.2009.07.006 MathSciNetCrossRefMATH

26.

Yi M, Huang J (2014) Wavelet operational matrix method for solving fractional differential equations with variable coefficients. Appl Math Comput 230:383–394. doi:10.1016/j.amc.2013.06.102 MathSciNetMATH

27.

Podlubny I (1999) Fractional differential equations. Academic Press, San DiegoMATH

28.

Canuto C, Hussaini MY, Quarteroni AM, Thomas JrA (2012) Spectral methods in fluid dynamics. Springer, BerlinMATH

29.

Jiang Z, Schoufelberger W, Thoma M, Wyner A (1992) Block pulse functions and their applications in control systems. Springer, New YorkCrossRef

30.

Ervin VJ, Roop JP (2006) Variational formulation for the stationary fractional advection dispersion equation. Numer Methods Partial Differ Equ 22(3):558–576. doi:10.1002/num.20112 MathSciNetCrossRefMATH

Titel: The construction of operational matrix of fractional integration for solving fractional differential and integro-differential equations
verfasst von: Farzane Yousefi
Azim Rivaz
Wen Chen
Publikationsdatum: 05.08.2017
Verlag: Springer London
Erschienen in: Neural Computing and Applications / Ausgabe 6/2019
Print ISSN: 0941-0643
Elektronische ISSN: 1433-3058
DOI: https://doi.org/10.1007/s00521-017-3163-9

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