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