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An Efficient Collocation Method for Solving the Variable-Order Time-Fractional Partial Differential Equations Arising from the Physical Phenomenon

Authors: Haniye Dehestani, Yadollah Ordokhani


In this work, we present an efficient approach for solving variable-order time-fractional partial differential equations, which are based on Legendre and Laguerre polynomials. First, we introduced the pseudo-operational matrices of integer and variable fractional order of integration by use of some properties of Riemann-Liouville fractional integral. Then, applied together with collocation method and Legendre-Laguerre functions for solving variable-order time-fractional partial differential equations. Also, an estimation of the error is presented. At last, we investigate numerical examples which arise in physics to demonstrate the accuracy of the present method. In comparison results obtained by the present method with the exact solution and the other methods reveals that the method is very effective.

Keywords: Collocation method, fractional partial differential equations, Legendre-Laguerre functions, pseudo-operational matrix of integration.

Digital Object Identifier (DOI):

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[1] R. L. Bagley and P. J. Torvik, Fractional calculus: A different approach to the analysis of viscoelastically damped structures, Aerosp. Am. vol. 21, no. 5, pp. 741−748, 1983.
[2] R. L. Bagley and P. J. Torvik, Fractional calculus in the transient analysis of viscoelastically damped structures, Aerosp. Am. vol. 23, pp. 918−925, 1985.
[3] R. L. Magin, Fractional calculus in bioengineering, Critical Reviews in Biomedical Engineering, vol. 32, pp. 1−104, 2004.
[4] D. A. Robinson, The use of control systems analysis in neurophysiology of eye movements, Annual Review of Neuroscience, vol. 4, pp. 462−503, 1981.
[5] R. T. Baillie, Long memory processes and fractional integration in econometrics, J. Econom. vol. 73, pp. 5−59, 1996.
[6] M. G. Hall and T. R. Barrick, From diffusion-weighted MRI to anomalous diffusion imaging, Magn. Reson. Med. vol. 59, pp. 447−455, 2008.
[7] J. H. He, Nonlinear oscillation with fractional derivative and its applications, in: Proceedings of the International Conference on Vibrating Engineering 98, Dalian, China, 1988.
[8] B. Mandelbrot, Some noises with 1/f spectrum, a bridge between direct current and white noise, IEEE Trans. Inf. Theory, vol. 13, no. 2, pp. 289−298, 1967.
[9] Y. Z. Povstenko, Signaling problem for time-fractional diffusion-wave equation in a half-space in the case of angular symmetry, Nonlinear Dyn. vol. 55, pp. 593−605, 2010.
[10] N. Engheta, On fractional calculus and fractional multipoles in electromagnetism, IEEE Trans. Antennas Propag. vol. 44, no. 4, pp. 554−566, 1996.
[11] K. B. Oldham, Fractional differential equations in electrochemistry, Adv. Eng. Softw. vol. 41, no. 1, pp. 9−12, 2010.
[12] C. Lederman, J. M. Roquejoffre and N. Wolanski, Mathematical justification of a nonlinear integro-differential equation for the propagation of spherical flames, Annali di Matematica Pura ed Applicata, vol. 183, pp. 173−239, 2004.
[13] F. Mainardi, Fractional calculus: some basic problems in continuum and statistical mechanics, in: A. Carpinteri, F. Mainardi (Eds.), Fractals and Fractional Calculus in Continuum Mechanics, Springer Verlag, New York, pp. 291−348, 1997.
[14] Y. A. Rossikhin and M. V. Shitikova, Applications of fractional calculus to dynamic problems of linear and nonlinear hereditary mechanics of solids, Appl. Mech. Rev. vol. 50, no. 1, pp. 15−67, 1997.
[15] J. H. He, Some applications of nonlinear fractional differential equations and their approximations, Bull. Sci. Technol. Soc. vol. 15, no. 2, pp. 86−90, 1999.
[16] P. Kumar and O. P. Agrawal, An approximate method for numerical solution of fractional differential equations, Signal processing, vol. 86, pp. 2602−2610, 2006.
[17] I. T. F. Liu and V. Anh, Numerical solution of the space fractional Fokker-Planck equation, J. Comput. Appl. Math. vol. 166, pp. 209−219, 2004.
[18] E. Keshavarz, Y. Ordokhani and M. Razzaghi, Bernoulli wavelet operational matrix of fractional order integration and its applications in solving the fractional order differential equations, Appl. Math. Model. vol. 38, pp. 6038−6051, 2014.
[19] S. Kazem, S. Abbasbandy and S. Kumar, Fractional-order Legendre functions for solving fractional-order differential equations, Appl. Math. Model. vol. 37, pp. 5498−5510, 2013.
[20] Y. Chen, Y. Sun and L. Liu, Numerical solution of fractional partial differential equations with variable coefficients using generalized fractional-order Legendre functions, Appl. Math. Comput. vol. 244, pp. 847−858, 2014.
[21] L. Wang, Y. Ma and Z. Meng, Haar wavelet method for solving fractional partial differential equations numerically, Appl. Math. Comput. vol. 227, pp. 66−76, 2014.
[22] J. Rena, Z. Z. Sun and W. Dai, New approximations for solving the Caputo-type fractional partial differential equations, Appl. Math. Model. vol. 40, pp. 2625−2636, 2016.
[23] F. Zhou and X. Xu, The third kind Chebyshev wavelets collocation method for solving the time-fractional convection diffusion equations with variable coefficients, Appl. Math. Comput. vol. 280, pp. 11−29, 2016.
[24] U. Saeed and M. Rehman, Haar wavelet Picard method for fractional nonlinear partial differential equations, Appl. Math. Comput. vol. 264, pp. 310−322, 2015.
[25] A. H. Bhrawy and M. A. Zaky, A method based on the Jacobi tau approximation for solving multi-term time-space fractional partial differential equations, J. Comput. Phys. vol. 281, pp. 876−895, 2015.
[26] N. Mollahasani, M. M. Moghadama and K. Afrooz, A new treatment based on hybrid functions to the solution of telegraph equations of fractional order, Appl. Math. Model. vol. 40, pp. 2804−2814, 2016.
[27] P. Rahimkhani, Y. Ordokhani and E. Babolian, Numerical solution of fractional pantograph differential equations by using generalized fractional-order Bernoulli wavelet, J. Comput. Appl. Math. vol. 309, pp. 493−510, 2017.
[28] M. Dehghan, M. Abbaszadeh and A. Mohebbi, An implicit RBF meshless approach for solving the time fractional nonlinear sine-Gordon and Klein-Gordon equations, Eng. Anal. Bound Elem. vol. 50, pp. 412−434, 2015.
[29] S. G. Samko and B. Ross, Integration and differentiation to a variable fractional order, Integral Transforms and Special Functions, vol. 1, no. 4, pp. 277−300, 1993.
[30] S. G. Samko, Variable Order and the Spaces LP. Operator Theory for Complex and Hypercomplex Analysis: Operator Theory for Complex and Hypercomplex Analysis, December 12-17, 1994, Mexico City, Mexico 212, 203, 1998.
[31] Ya. L. Kobelev, L. Ya. Kobelev and Yu. L. Klimontovich, Statistical physics of dynamic systems with variable memory, Doklady Physics. vol. 48, no. 6, Nauka/Interperiodica, 2003.
[32] 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, Eur. Phys. J. Spec. Top. vol. 193, pp. 185−192, 2011.
[33] B. P. Moghaddam and J. A. T. Machado, A stable three-level explicit spline finite difference scheme for a class of nonlinear time variable order fractional partial differential equations, Comput. Math. Appl. vol. 73, no. 6, pp. 1262−1269, 2017.
[34] S. Yaghoobi and B. P. Moghaddam, An efficient cubic spline approximation for variable-order fractional differential equations with time delay, Nonlinear Dyn. vol. 87, pp. 815−826, 2017.
[35] Y. M. Chen, Y. Q. Wei, D.Y. Liu and H. Yu, Numerical solution for a class of nonlinear variable order fractional differential equations with Legendre wavelets, Applied Mathematics Letters, vol. 46, pp. 83−88, 2015.
[36] A. Atangana, On the stability and convergence of the time-fractional variable order telegraph equation, J. Comput. Phys. vol. 293, pp. 104−114, 2015.
[37] X. Li, H. Li and B. Wu, A new numerical method for variable order fractional functional differential equations, Applied Mathematics Letters, vol. 68, pp. 80−86, 2017.
[38] X. Li and B. Wu, A numerical technique for variable fractional functional boundary value problems, Applied Mathematics Letters, vol. 43, pp. 108−113, 2015.
[39] N. H. Sweilam, A. M. Nagy, T. A. Assiri and N. Y. Ali, Numerical simulations for variable-order fractional nonlinear delay differential equations, Journal of Fractional Calculus and Applications, vol. 6, no. 1, pp. 71−82, 2015.
[40] W. Jiang and N. Liu, A numerical method for solving the time variable fractional order mobile-immobile advection-dispersion model, Applied Numerical Mathematics, vol. 119, pp. 18−32, 2017.
[41] H. Zhang, F. Liu, M. S. Phanikumar and M. M. Meerschaert, A novel numerical method for the time variable fractional order mobile-immobile advection-dispersion model, Comput. Math. Appl. vol. 66, pp. 693−701, 2013.
[42] R. Schumer, D. A. Benson, M. M. Meerschaert and B. Baeumer, Fractal mobile/immobile solute transport, Water Resour. Res. vol. 39, no. 10, pp. 1296, 2003.
[43] Y. Zhang, D. A. Benson and D. M. Reeves, Time and space nonlocalities underlying fractional-derivative models: Distinction and literature review of field applications, Adv. Water Resour. vol. 32, pp. 561−581, 2009.
[44] H. Zhang, F. Liu, P. Zhuang, I. Turner and V. Anh, Numerical analysis of a new space-time variable fractional-order advection-dispersion equation, Appl. Math. Comput. vol. 242, pp. 541−550, 2014.
[45] H. Ma and Y. Yang, Jacobi Spectral Collocation Method for the Time Variable-Order Fractional Mobile-Immobile Advection-Dispersion Solute Transport Model, East Asian J. Applied Math. vol. 6, no. 3, pp. 337−352, 2016.
[46] H. Pourbashash, D. Baleanu and M. M. Al-Qurashi, On solving fractional mobile/immobile equation, Advances in Mechanical Engineering, vol. 9, no. 1, pp. 1−12, 2017.
[47] M. A. Abdelkawy, M. A. Zaky, A. H. Bhrawy and D. Baleanu, Numerical simulation of time variable fractional order Mobile-Immobile advection-dispersion model, Rom. Rep. Phys. vol. 67, no. 3, pp. 773−791, 2015.
[48] A. Scott, Nonlinear Science: Emergence and Dynamics of Coherent Structure, vol. 8 of Oxford Texts in Applied and Engineering Mathematics, Oxford University Press, Oxford, UK, 2nd edition, 2003.
[49] T. Dauxois and M. Peyrard, Physics of Solitons, Cambridge University Press, 2006.
[50] N. Laskin and G. Zaslavsky, Nonlinear fractional dynamics on a lattice with long-range interactions, Physica A, vol. 368, pp. 38−54, 2006.
[51] A. Mohebbi and M. Dehghan, High-order solution of one-dimensional sine-Gordon equation using compact finite difference and DIRKN methods, Math. Comput. Model. vol. 51, pp. 537−549, 2010.
[52] A. Akgul and M. Inc, Numerical solution of one-dimensional Sine-Gordon equation using Reproducing Kernel Hilbert Space Method, arXiv:1304.0534v1
[math.NA], 2 Apr 2013.
[53] M. A. Yousif and B. A. Mahmood, Approximate solutions for solving the Klein-Gordon and sine-Gordon equations, Journal of the Association of Arab Universities for Basic and Applied Sciences, vol. 22, pp. 83−90, 2017.
[54] M. Dehghan, M. Abbaszadeh and A. Mohebbi, An implicit RBF meshless approach for solving the time fractional nonlinear sine-Gordon and Klein-Gordon equations, Eng. Anal. Bound. Elem. vol. 50, pp. 412−434, 2015.
[55] Y. Chen, L. Liu, B. Li and Y. Sun, Numerical solution for the variable order linear cable equation with bernstein polynomials, Appl. Math. Comput. vol. 238, pp. 329−341, 2014.
[56] S. Shen, F. Liu, J. Chen, I. Turner and V. Anh, Numerical techniques for the variable order time fractional diffusion equation, Appl. Math. Comput. vol. 218, pp. 10861−10870, 2012.
[57] S. Nemati, P. M. Lima and Y. Ordokhani, Numerical solution of a class of two-dimensional nonlinear Volterra integral equations using Legendre polynomials, J. Comput. Appl. Math. vol. 242, pp. 53−69, 2013.
[58] M. Gulsu, B. Gurbuz, Y. Ozturk and M. Sezer, Lagurre polynomial approach for solving linear delay difference equations, Appl. Math. Comput. vol. 217, pp. 6765−6776, 2011.
[59] K. Wang and Q. Wang, Taylor collocation method and convergence analysis for the Volterra-Fredholm integral equations, J. Comput. Appl. Math. vol. 260, pp. 294−300, 2014.
[60] G. M. Phillips and P. J. Taylor, Theory and Application of Numerical Analysis, Academic Press, New York 1973.
[61] L. Hormander, The analysis of Linear partial Differential operators, Springer, 1 1990.