A spectral approach for time-fractional diffusion and subdiffusion equations in a large interval
Abstract
In this paper, we concentrate on a class of time-fractional diffusion and subdiffusion equations. To solve the mentioned problems, we construct twodimensional Genocchi-fractional Laguerre functions (G-FLFs). Then, the pseudooperational matrices are used to convert the proposed equations to systems of algebraic equations. The properties of pseudo-operational matrices have reflected well in the process of the numerical technique and create an approximate solution with high precision. Finally, several examples are presented to illustrate the accuracy and effectiveness of the technique.
Keyword : Genocchi-fractional Laguerre functions, collocation method, time-fractional diffusion equations, time-fractional subdiffusion equations, pseudo-operational matrix
How to Cite
Dehestani, H., Ordokhani, Y., & Razzaghi, M. (2022). A spectral approach for time-fractional diffusion and subdiffusion equations in a large interval. Mathematical Modelling and Analysis, 27(1), 19–40. https://doi.org/10.3846/mma.2022.13579
This work is licensed under a Creative Commons Attribution 4.0 International License.
References
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M. Cui. Compact finite difference method for the fractional diffusion equation. Journal of Computational Physics, 228(20):7792–7804, 2009. https://doi.org/10.1016/j.jcp.2009.07.021
H. Dehestani, Y. Ordokhani and M. Razzaghi. Fractional-order Legendre– Laguerre functions and their applications in fractional partial differential equations. Applied Mathematics and Computation, 336:433–453, 2018. https://doi.org/10.1016/j.amc.2018.05.017
H. Dehestani, Y. Ordokhani and M. Razzaghi. Hybrid functions for numerical solution of fractional Fredholm-Volterra functional integro-differential equations with proportional delays. International Journal of Numerical Modelling: Electronic Networks, Devices and Fields, 32(5):e2606, 2019. https://doi.org/10.1002/jnm.2606
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H. Dehestani, Y. Ordokhani and M. Razzaghi. The novel operational matrices based on 2D-Genocchi polynomials: solving a general class of variable-order fractional partial integro-differential equations. Computational and Applied Mathematics, 39(4):1–32, 2020. https://doi.org/10.1007/s40314-020-01314-4
M. Dehghan, M. Abbaszadeh and A. Mohebbi. The numerical solution of nonlinear high dimensional generalized Benjamin–Bona–Mahony– Burgers equation via the meshless method of radial basis functions. Computers & Mathematics with Applications, 68(3):212–237, 2014. https://doi.org/10.1016/j.camwa.2014.05.019
M. Dehghan, M. Abbaszadeh and A. Mohebbi. An implicit RBF meshless approach for solving the time fractional nonlinear sine-Gordon and Klein–Gordon equations. Engineering Analysis with Boundary Elements, 50:412–434, 2015. https://doi.org/10.1016/j.enganabound.2014.09.008
M. Dehghan, E.A. Hamedi and H. Khosravian-Arab. A numerical scheme for the solution of a class of fractional variational and optimal control problems using the modified Jacobi polynomials. Journal of Vibration and Control, 22(6):1547– 1559, 2016. https://doi.org/10.1177/1077546314543727
E.H. Doha, M.A. Abdelkawy, A.Z.M. Amin and A. Lopes. A space–time spectral approximation for solving nonlinear variable-order fractional sine and Klein–Gordon differential equations. Computational and Applied Mathematics, 37(5):6212–6229, 2018. https://doi.org/10.1007/s40314-018-0695-2
G. Gao and Z. Sun. A compact finite difference scheme for the fractional subdiffusion equations. Journal of Computational Physics, 230(3):586–595, 2011. https://doi.org/10.1016/j.jcp.2010.10.007
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M.H. Heydari. Wavelets Galerkin method for the fractional subdiffusion equation. Journal of Computational and Nonlinear Dynamics, 11(6), 2016. https://doi.org/10.1115/1.4034391
A. Isah and C. Phang. Operational matrix based on Genocchi polynomials for solution of delay differential equations. Ain Shams Engineering Journal, 9(4):2123–2128, 2018. https://doi.org/10.1016/j.asej.2016.09.015
A. Isah and C. Phang. New operational matrix of derivative for solving non-linear fractional differential equations via Genocchi polynomials. Journal of King Saud University-Science, 31(1):1–7, 2019. https://doi.org/10.1016/j.jksus.2017.02.001
Y. Jiang and J. Ma. High-order finite element methods for time-fractional partial differential equations. Journal of Computational and Applied Mathematics, 235(11):3285–3290, 2011. https://doi.org/10.1016/j.cam.2011.01.011
A. Kumar, A. Bhardwaj and S. Dubey. A local meshless method to approximate the time-fractional telegraph equation. Engineering with Computers, 37:3473– 3488, 2021. https://doi.org/10.1007/s00366-020-01006-x
A. Kumar, A. Bhardwaj and B.V. Rathish Kumar. A meshless local collocation method for time fractional diffusion wave equation. Computers & Mathematics with Applications, 78(6):1851–1861, 2019. https://doi.org/10.1016/j.camwa.2019.03.027
S. Kumar and C. Piret. Numerical solution of space-time fractional PDEs using RBF-QR and Chebyshev polynomials. Applied Numerical Mathematics, 143:300–315, 2019. https://doi.org/10.1016/j.apnum.2019.04.012
C. Li, T. Zhao, W. Deng and Y. Wu. Orthogonal spline collocation methods for the subdiffusion equation. Journal of Computational and Applied Mathematics, 255:517–528, 2014. https://doi.org/10.1016/j.cam.2013.05.022
F. Liu, M. Meerschaert, R¿ McGough, P. Zhuang and Q. Liu. Numerical methods for solving the multi-term time-fractional wave-diffusion equation. Fractional Calculus and Applied Analysis, 16(1):9–25, 2013. https://doi.org/10.2478/s13540-013-0002-2
J.R. Loh, C. Phang and A. Isah. New operational matrix via Genocchi polynomials for solving Fredholm-Volterra fractional integrodifferential equations. Advances in Mathematical Physics, 2017, 2017. https://doi.org/10.1155/2017/3821870
R.L. Magin. Fractional calculus models of complex dynamics in biological tissues. Computers & Mathematics with Applications, 59(5):1586–1593, 2010. https://doi.org/10.1016/j.camwa.2009.08.039
A. Mardani, M.R. Hooshmandasl, M.H. Heydari and C. Cattani. A meshless method for solving the time fractional advection–diffusion equation with variable coefficients. Computers & Mathematics with Applications, 75(1):122–133, 2018. https://doi.org/10.1016/j.camwa.2017.08.038
R. Metzler and J. Klafter. The random walk’s guide to anomalous diffusion: a fractional dynamics approach. Physics reports, 339(1):1–77, 2000. https://doi.org/10.1016/S0370-1573(00)00070-3
N. Mollahasani, M.M. Moghadam and K. Afrooz. A new treatment based on hybrid functions to the solution of telegraph equations of fractional order. Applied Mathematical Modelling, 40(4):2804–2814, 2016. https://doi.org/10.1016/j.apm.2015.08.020
K. Mustapha. An implicit finite-difference time-stepping method for a sub-diffusion equation, with spatial discretization by finite elements. IMA Journal of Numerical Analysis, 31(2):719–739, 2011. https://doi.org/10.1093/imanum/drp057
M.D. Ortigueira and J.A.T. Machado. Fractional signal processing and applications. Signal processing, 83(11):2285–2286, 2003. https://doi.org/10.1016/S0165-1684(03)00181-6
J. Ren, Z.Z. Sun and X. Zhao. Compact difference scheme for the fractional sub-diffusion equation with Neumann boundary conditions. Journal of Computational Physics, 232(1):456–467, 2013. https://doi.org/10.1016/j.jcp.2012.08.026
R. Ren, H. Li, W. Jiang and M.Y. Song. An efficient Chebyshev-tau method for solving the space fractional diffusion equations. Applied Mathematics and Computation, 224:259–267, 2013. https://doi.org/10.1016/j.amc.2013.08.073
S.Y. Reutskiy. A new semi-analytical collocation method for solving multi-term fractional partial differential equations with time variable coefficients. Applied Mathematical Modelling, 45:238–254, 2017. https://doi.org/10.1016/j.apm.2016.12.029
R. Salehi. A meshless point collocation method for 2-D multi-term time fractional diffusion-wave equation. Numerical Algorithms, 74(4):1145–1168, 2017. https://doi.org/10.1007/s11075-016-0190-z
X. Si, C. Wang, Y. Shen and L. Zheng. Numerical method to initial-boundary value problems for fractional partial differential equations with time-space variable coefficients. Applied Mathematical Modelling, 40(7-8):4397–4411, 2016. https://doi.org/10.1016/j.apm.2015.11.039
S. Singh, V.K. Patel and V.K. Singh. Application of wavelet collocation method for hyperbolic partial differential equations via matrices. Applied Mathematics and Computation, 320:407–424, 2018. https://doi.org/10.1016/j.amc.2017.09.043
M. Uddin and S. Haq. RBFs approximation method for time fractional partial differential equations. Communications in Nonlinear Science and Numerical Simulation, 16(11):4208–4214, 2011. https://doi.org/10.1016/j.cnsns.2011.03.021
Z. Wang and S. Vong. Compact difference schemes for the modified anomalous fractional sub-diffusion equation and the fractional diffusionwave equation. Journal of Computational Physics, 277:1–15, 2014. https://doi.org/10.1016/j.jcp.2014.08.012
M. Yaseen, M. Abbas, A.I. Ismail and T. Nazir. A cubic trigonometric B-spline collocation approach for the fractional sub-diffusion equations. Applied Mathematics and Computation, 293:311–319, 2017. https://doi.org/10.1016/j.amc.2016.08.028
F. Zeng and C. Li. A new Crank–Nicolson finite element method for the timefractional subdiffusion equation. Applied Numerical Mathematics, 121:82–95, 2017. https://doi.org/10.1016/j.apnum.2017.06.011
F. Zeng, C. Li, F. Liu and I. Turner. Numerical algorithms for time-fractional subdiffusion equation with second-order accuracy. SIAM Journal on Scientific Computing, 37(1):A55–A78, 2015. https://doi.org/10.1137/14096390X
Y. Zhang, Z. Sun and H. Wu. Error estimates of Crank–Nicolson-type difference schemes for the subdiffusion equation. SIAM Journal on Numerical Analysis, 49(6):2302–2322, 2011. https://doi.org/10.1137/100812707
F. Zhou and X. Xu. The third kind Chebyshev wavelets collocation method for solving the time-fractional convection diffusion equations with variable coefficients. Applied Mathematics and Computation, 280:11–29, 2016. https://doi.org/10.1016/j.amc.2016.01.029
A. Aminataei and S.K. Vanani. Numerical solution of fractional Fokker-Planck equation using the operational collocation method. Applied Mathematics and Computation, 12(1):33–43, 2013.
I. Aziz, S. Islamb and M. Asif. Haar wavelet collocation method for three-dimensional elliptic partial differential equations. Computers & Mathematics with Applications, 73(9):2023–2034, 2017. https://doi.org/10.1016/j.camwa.2017.02.034
A. Bhardwaj and A. Kumar. A meshless method for time fractional nonlinear mixed diffusion and diffusion-wave equation. Applied Numerical Mathematics, 160:146–165, 2021. https://doi.org/10.1016/j.apnum.2020.09.019
A.H. Bhrawy, E.H. Doha, D. Baleanu and S.S. Ezz-Eldien. A spectral tau algorithm based on Jacobi operational matrix for numerical solution of time fractional diffusion-wave equations. Journal of Computational Physics, 293:142–156, 2015. https://doi.org/10.1016/j.jcp.2014.03.039
A.H. Bhrawy and M.A. Zaky. Shifted fractional-order Jacobi orthogonal functions: application to a system of fractional differential equations. Applied Mathematical Modelling, 40(2):832–845, 2016. https://doi.org/10.1016/j.apm.2015.06.012
M. Cui. Compact finite difference method for the fractional diffusion equation. Journal of Computational Physics, 228(20):7792–7804, 2009. https://doi.org/10.1016/j.jcp.2009.07.021
H. Dehestani, Y. Ordokhani and M. Razzaghi. Fractional-order Legendre– Laguerre functions and their applications in fractional partial differential equations. Applied Mathematics and Computation, 336:433–453, 2018. https://doi.org/10.1016/j.amc.2018.05.017
H. Dehestani, Y. Ordokhani and M. Razzaghi. Hybrid functions for numerical solution of fractional Fredholm-Volterra functional integro-differential equations with proportional delays. International Journal of Numerical Modelling: Electronic Networks, Devices and Fields, 32(5):e2606, 2019. https://doi.org/10.1002/jnm.2606
H. Dehestani, Y. Ordokhani and M. Razzaghi. A numerical technique for solving various kinds of fractional partial differential equations via Genocchi hybrid functions. Revista de la Real Academia de Ciencias Exactas, F´ısicas y Naturales. Serie A. Matem´aticas, 113(4):3297–3321, 2019. https://doi.org/10.1007/s13398-019-00694-5
H. Dehestani, Y. Ordokhani and M. Razzaghi. A novel direct method based on the Lucas multiwavelet functions for variable-order fractional reactiondiffusion and subdiffusion equations. Numerical Linear Algebra with Applications, 28(2):e2346, 2020. https://doi.org/10.1002/nla.2346
H. Dehestani, Y. Ordokhani and M. Razzaghi. The novel operational matrices based on 2D-Genocchi polynomials: solving a general class of variable-order fractional partial integro-differential equations. Computational and Applied Mathematics, 39(4):1–32, 2020. https://doi.org/10.1007/s40314-020-01314-4
M. Dehghan, M. Abbaszadeh and A. Mohebbi. The numerical solution of nonlinear high dimensional generalized Benjamin–Bona–Mahony– Burgers equation via the meshless method of radial basis functions. Computers & Mathematics with Applications, 68(3):212–237, 2014. https://doi.org/10.1016/j.camwa.2014.05.019
M. Dehghan, M. Abbaszadeh and A. Mohebbi. An implicit RBF meshless approach for solving the time fractional nonlinear sine-Gordon and Klein–Gordon equations. Engineering Analysis with Boundary Elements, 50:412–434, 2015. https://doi.org/10.1016/j.enganabound.2014.09.008
M. Dehghan, E.A. Hamedi and H. Khosravian-Arab. A numerical scheme for the solution of a class of fractional variational and optimal control problems using the modified Jacobi polynomials. Journal of Vibration and Control, 22(6):1547– 1559, 2016. https://doi.org/10.1177/1077546314543727
E.H. Doha, M.A. Abdelkawy, A.Z.M. Amin and A. Lopes. A space–time spectral approximation for solving nonlinear variable-order fractional sine and Klein–Gordon differential equations. Computational and Applied Mathematics, 37(5):6212–6229, 2018. https://doi.org/10.1007/s40314-018-0695-2
G. Gao and Z. Sun. A compact finite difference scheme for the fractional subdiffusion equations. Journal of Computational Physics, 230(3):586–595, 2011. https://doi.org/10.1016/j.jcp.2010.10.007
R. Ghaffari and F. Ghoreishi. Reduced spline method based on a proper orthogonal decomposition technique for fractional subdiffusion equations. Applied Numerical Mathematics, 137:62–79, 2019. https://doi.org/10.1016/j.apnum.2018.11.014
A.K. Golmankhaneh, A.K. Golmankhaneh and D. Baleanu. On nonlinear fractional Klein–Gordon equation. Signal Processing, 91(3):446–451, 2011. https://doi.org/10.1016/j.sigpro.2010.04.016
M.H. Heydari. Wavelets Galerkin method for the fractional subdiffusion equation. Journal of Computational and Nonlinear Dynamics, 11(6), 2016. https://doi.org/10.1115/1.4034391
A. Isah and C. Phang. Operational matrix based on Genocchi polynomials for solution of delay differential equations. Ain Shams Engineering Journal, 9(4):2123–2128, 2018. https://doi.org/10.1016/j.asej.2016.09.015
A. Isah and C. Phang. New operational matrix of derivative for solving non-linear fractional differential equations via Genocchi polynomials. Journal of King Saud University-Science, 31(1):1–7, 2019. https://doi.org/10.1016/j.jksus.2017.02.001
Y. Jiang and J. Ma. High-order finite element methods for time-fractional partial differential equations. Journal of Computational and Applied Mathematics, 235(11):3285–3290, 2011. https://doi.org/10.1016/j.cam.2011.01.011
A. Kumar, A. Bhardwaj and S. Dubey. A local meshless method to approximate the time-fractional telegraph equation. Engineering with Computers, 37:3473– 3488, 2021. https://doi.org/10.1007/s00366-020-01006-x
A. Kumar, A. Bhardwaj and B.V. Rathish Kumar. A meshless local collocation method for time fractional diffusion wave equation. Computers & Mathematics with Applications, 78(6):1851–1861, 2019. https://doi.org/10.1016/j.camwa.2019.03.027
S. Kumar and C. Piret. Numerical solution of space-time fractional PDEs using RBF-QR and Chebyshev polynomials. Applied Numerical Mathematics, 143:300–315, 2019. https://doi.org/10.1016/j.apnum.2019.04.012
C. Li, T. Zhao, W. Deng and Y. Wu. Orthogonal spline collocation methods for the subdiffusion equation. Journal of Computational and Applied Mathematics, 255:517–528, 2014. https://doi.org/10.1016/j.cam.2013.05.022
F. Liu, M. Meerschaert, R¿ McGough, P. Zhuang and Q. Liu. Numerical methods for solving the multi-term time-fractional wave-diffusion equation. Fractional Calculus and Applied Analysis, 16(1):9–25, 2013. https://doi.org/10.2478/s13540-013-0002-2
J.R. Loh, C. Phang and A. Isah. New operational matrix via Genocchi polynomials for solving Fredholm-Volterra fractional integrodifferential equations. Advances in Mathematical Physics, 2017, 2017. https://doi.org/10.1155/2017/3821870
R.L. Magin. Fractional calculus models of complex dynamics in biological tissues. Computers & Mathematics with Applications, 59(5):1586–1593, 2010. https://doi.org/10.1016/j.camwa.2009.08.039
A. Mardani, M.R. Hooshmandasl, M.H. Heydari and C. Cattani. A meshless method for solving the time fractional advection–diffusion equation with variable coefficients. Computers & Mathematics with Applications, 75(1):122–133, 2018. https://doi.org/10.1016/j.camwa.2017.08.038
R. Metzler and J. Klafter. The random walk’s guide to anomalous diffusion: a fractional dynamics approach. Physics reports, 339(1):1–77, 2000. https://doi.org/10.1016/S0370-1573(00)00070-3
N. Mollahasani, M.M. Moghadam and K. Afrooz. A new treatment based on hybrid functions to the solution of telegraph equations of fractional order. Applied Mathematical Modelling, 40(4):2804–2814, 2016. https://doi.org/10.1016/j.apm.2015.08.020
K. Mustapha. An implicit finite-difference time-stepping method for a sub-diffusion equation, with spatial discretization by finite elements. IMA Journal of Numerical Analysis, 31(2):719–739, 2011. https://doi.org/10.1093/imanum/drp057
M.D. Ortigueira and J.A.T. Machado. Fractional signal processing and applications. Signal processing, 83(11):2285–2286, 2003. https://doi.org/10.1016/S0165-1684(03)00181-6
J. Ren, Z.Z. Sun and X. Zhao. Compact difference scheme for the fractional sub-diffusion equation with Neumann boundary conditions. Journal of Computational Physics, 232(1):456–467, 2013. https://doi.org/10.1016/j.jcp.2012.08.026
R. Ren, H. Li, W. Jiang and M.Y. Song. An efficient Chebyshev-tau method for solving the space fractional diffusion equations. Applied Mathematics and Computation, 224:259–267, 2013. https://doi.org/10.1016/j.amc.2013.08.073
S.Y. Reutskiy. A new semi-analytical collocation method for solving multi-term fractional partial differential equations with time variable coefficients. Applied Mathematical Modelling, 45:238–254, 2017. https://doi.org/10.1016/j.apm.2016.12.029
R. Salehi. A meshless point collocation method for 2-D multi-term time fractional diffusion-wave equation. Numerical Algorithms, 74(4):1145–1168, 2017. https://doi.org/10.1007/s11075-016-0190-z
X. Si, C. Wang, Y. Shen and L. Zheng. Numerical method to initial-boundary value problems for fractional partial differential equations with time-space variable coefficients. Applied Mathematical Modelling, 40(7-8):4397–4411, 2016. https://doi.org/10.1016/j.apm.2015.11.039
S. Singh, V.K. Patel and V.K. Singh. Application of wavelet collocation method for hyperbolic partial differential equations via matrices. Applied Mathematics and Computation, 320:407–424, 2018. https://doi.org/10.1016/j.amc.2017.09.043
M. Uddin and S. Haq. RBFs approximation method for time fractional partial differential equations. Communications in Nonlinear Science and Numerical Simulation, 16(11):4208–4214, 2011. https://doi.org/10.1016/j.cnsns.2011.03.021
Z. Wang and S. Vong. Compact difference schemes for the modified anomalous fractional sub-diffusion equation and the fractional diffusionwave equation. Journal of Computational Physics, 277:1–15, 2014. https://doi.org/10.1016/j.jcp.2014.08.012
M. Yaseen, M. Abbas, A.I. Ismail and T. Nazir. A cubic trigonometric B-spline collocation approach for the fractional sub-diffusion equations. Applied Mathematics and Computation, 293:311–319, 2017. https://doi.org/10.1016/j.amc.2016.08.028
F. Zeng and C. Li. A new Crank–Nicolson finite element method for the timefractional subdiffusion equation. Applied Numerical Mathematics, 121:82–95, 2017. https://doi.org/10.1016/j.apnum.2017.06.011
F. Zeng, C. Li, F. Liu and I. Turner. Numerical algorithms for time-fractional subdiffusion equation with second-order accuracy. SIAM Journal on Scientific Computing, 37(1):A55–A78, 2015. https://doi.org/10.1137/14096390X
Y. Zhang, Z. Sun and H. Wu. Error estimates of Crank–Nicolson-type difference schemes for the subdiffusion equation. SIAM Journal on Numerical Analysis, 49(6):2302–2322, 2011. https://doi.org/10.1137/100812707
F. Zhou and X. Xu. The third kind Chebyshev wavelets collocation method for solving the time-fractional convection diffusion equations with variable coefficients. Applied Mathematics and Computation, 280:11–29, 2016. https://doi.org/10.1016/j.amc.2016.01.029