An Efficient Collocation Method for Solving the Variable-Order Time-Fractional Partial Differential Equations Arising from the Physical Phenomenon
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.
[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.
[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.
@article{"International Journal of Engineering, Mathematical and Physical Sciences:78235", author = "Haniye Dehestani and Yadollah Ordokhani", title = "An Efficient Collocation Method for Solving the Variable-Order Time-Fractional Partial Differential Equations Arising from the Physical Phenomenon", abstract = "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.", volume = "12", number = "12", pages = "245-12", }
