**Commenced**in January 2007

**Frequency:**Monthly

**Edition:**International

**Paper Count:**30135

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

**Authors:**
Haniye Dehestani,
Yadollah Ordokhani

**Abstract:**

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

**Digital Object Identifier (DOI):**
doi.org/10.5281/zenodo.2022075

**References:**

[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.