Spectral tau technique via Lucas polynomials for the time-fractional diffusion equation

Waleed Mohamed Abd-Elhameed; Abdullah F. Abu Sunayh; Mohammed H. Alharbi; Ahmed Gamal Atta; Waleed Mohamed Abd-Elhameed; Abdullah F. Abu Sunayh; Mohammed H. Alharbi; Ahmed Gamal Atta

doi:10.3934/math.20241646

AIMS Mathematics

2024, Volume 9, Issue 12: 34567-34587. doi: 10.3934/math.20241646

Previous Article Next Article

Research article Special Issues

Spectral tau technique via Lucas polynomials for the time-fractional diffusion equation

1.
Department of Mathematics, Faculty of Science, Cairo University, Giza 12613, Egypt
2.
Department of Mathematics and Statistics, College of Science, University of Jeddah, Jeddah, Saudi Arabia
3.
Department of Mathematics, Faculty of Education, Ain Shams University, Roxy 11341, Cairo, Egypt

Received: 08 September 2024 Revised: 21 November 2024 Accepted: 03 December 2024 Published: 10 December 2024
MSC : 35R11, 65N35, 40A05

Here, we provide a new method to solve the time-fractional diffusion equation (TFDE) following the spectral tau approach. Our proposed numerical solution is expressed in terms of a double Lucas expansion. The discretization of the technique is based on several formulas about Lucas polynomials, such as those for explicit integer and fractional derivatives, products, and certain definite integrals of these polynomials. These formulas aid in transforming the TFDE and its conditions into a matrix system that can be treated through a suitable numerical procedure. We conduct a study on the convergence analysis of the double Lucas expansion. In addition, we provide a few examples to ensure that the proposed numerical approach is applicable and efficient.
- time-fractional diffusion equation,
- Lucas polynomials,
- spectral methods,
- error bound
Citation: Waleed Mohamed Abd-Elhameed, Abdullah F. Abu Sunayh, Mohammed H. Alharbi, Ahmed Gamal Atta. Spectral tau technique via Lucas polynomials for the time-fractional diffusion equation[J]. AIMS Mathematics, 2024, 9(12): 34567-34587. doi: 10.3934/math.20241646

Related Papers:

Abstract

Here, we provide a new method to solve the time-fractional diffusion equation (TFDE) following the spectral tau approach. Our proposed numerical solution is expressed in terms of a double Lucas expansion. The discretization of the technique is based on several formulas about Lucas polynomials, such as those for explicit integer and fractional derivatives, products, and certain definite integrals of these polynomials. These formulas aid in transforming the TFDE and its conditions into a matrix system that can be treated through a suitable numerical procedure. We conduct a study on the convergence analysis of the double Lucas expansion. In addition, we provide a few examples to ensure that the proposed numerical approach is applicable and efficient.

References

[1]	A. A. Kilbas, H. M. Srivastava, J. J. Trujillo, Theory and applications of fractional differential equations, Elsevier, 2006.
[2]	S. G. Samko, Fractional integrals and derivatives: theory and applications, Gordon and Breach Science Publishers, 1993.
[3]	I. Podlubny, Fractional differential equations: an introduction to fractional derivatives, fractional differential equations, to methods of their solution and some of their applications, Elsevier, 1998.
[4]	D. Albogami, D. Maturi, H. Alshehri, Adomian decomposition method for solving fractional time-Klein-Gordon equations using Maple, Appl. Math., 14 (2023), 411–418. https://doi.org/10.4236/am.2023.146024 doi: 10.4236/am.2023.146024
[5]	K. Sadri, K. Hosseini, E. Hinçal, D. Baleanu, S. Salahshour, A pseudo-operational collocation method for variable-order time-space fractional KdV–Burgers–Kuramoto equation, Math. Methods Appl. Sci., 46 (2023), 8759–8778. https://doi.org/10.1002/mma.9015 doi: 10.1002/mma.9015
[6]	S. M. Sivalingam, V. Govindaraj, A novel numerical approach for time-varying impulsive fractional differential equations using theory of functional connections and neural network, Expert Syst. Appl., 238 (2024), 121750. https://doi.org/10.1016/j.eswa.2023.121750 doi: 10.1016/j.eswa.2023.121750
[7]	W. M. Abd-Elhameed, H. M. Ahmed, Spectral solutions for the time-fractional heat differential equation through a novel unified sequence of Chebyshev polynomials, AIMS Math., 9 (2024), 2137–2166. https://doi.org/10.3934/math.2024107 doi: 10.3934/math.2024107
[8]	M. Izadi, Ş. Yüzbaşı W. Adel, A new Chelyshkov matrix method to solve linear and nonlinear fractional delay differential equations with error analysis, Math. Sci., 17 (2023), 267–284. https://doi.org/10.1007/s40096-022-00468-y doi: 10.1007/s40096-022-00468-y
[9]	H. Alrabaiah, I. Ahmad, R. Amin, K. Shah, A numerical method for fractional variable order pantograph differential equations based on Haar wavelet, Eng. Comput., 38 (2022), 2655–2668. https://doi.org/10.1007/s00366-020-01227-0 doi: 10.1007/s00366-020-01227-0
[10]	Kamran, S. Ahmad, K. Shah, T. Abdeljawad, B. Abdalla, On the approximation of fractal-fractional differential equations using numerical inverse Laplace transform methods, Comput. Model. Eng. Sci., 135 (2023), 3. https://doi.org/10.32604/cmes.2023.023705 doi: 10.32604/cmes.2023.023705
[11]	L. Qing, X. Li, Meshless analysis of fractional diffusion-wave equations by generalized finite difference method, Appl. Math. Lett., 157 (2024), 109204. https://doi.org/10.1016/j.camwa.2024.08.008 doi: 10.1016/j.camwa.2024.08.008
[12]	L. Qing, X. Li, Analysis of a meshless generalized finite difference method for the time-fractional diffusion-wave equation, Comput. Math. Appl., 172 (2024), 134–151. https://doi.org/10.1016/j.camwa.2024.08.008 doi: 10.1016/j.camwa.2024.08.008
[13]	T. Koshy, Fibonacci and Lucas numbers with applications, John Wiley & Sons, 2011.
[14]	I. Ali, S. Haq, S. F. Aldosary, K. S. Nisar, F. Ahmad, Numerical solution of one-and two-dimensional time-fractional Burgers equation via Lucas polynomials coupled with finite difference method, Alex. Eng. J., 61 (2022), 6077–6087. https://doi.org/10.1016/j.aej.2021.11.032 doi: 10.1016/j.aej.2021.11.032
[15]	I. Ali, S. Haq, K. S. Nisar, S. U. Arifeen, Numerical study of 1D and 2D advection-diffusion-reaction equations using Lucas and Fibonacci polynomials, Arab. J. Math., 10 (2021), 513–526. https://doi.org/10.1007/s40065-021-00330-4 doi: 10.1007/s40065-021-00330-4
[16]	M. N. Sahlan, H. Afshari, Lucas polynomials based spectral methods for solving the fractional order electrohydrodynamics flow model, Commun. Nonlinear Sci. Numer. Simul., 107 (2022), 106108. https://doi.org/10.1016/j.cnsns.2021.106108 doi: 10.1016/j.cnsns.2021.106108
[17]	P. K. Singh, S. S. Ray, An efficient numerical method based on Lucas polynomials to solve multi-dimensional stochastic Itô-Volterra integral equations, Math. Comput. Simul., 203 (2023), 826–845. https://doi.org/10.1016/j.matcom.2022.06.029 doi: 10.1016/j.matcom.2022.06.029
[18]	A. M. S. Mahdy, D. Sh. Mohamed, Approximate solution of Cauchy integral equations by using Lucas polynomials, Comput. Appl. Math., 41 (2022), 403. https://doi.org/10.1007/s40314-022-02116-6 doi: 10.1007/s40314-022-02116-6
[19]	B. P. Moghaddam, A. Dabiri, A. M. Lopes, J. A. T. Machado, Numerical solution of mixed-type fractional functional differential equations using modified Lucas polynomials, Comput. Appl. Math., 38 (2019), 1–12. https://doi.org/10.1007/s40314-019-0813-9 doi: 10.1007/s40314-019-0813-9
[20]	O. Oruç, A new algorithm based on Lucas polynomials for approximate solution of 1D and 2D nonlinear generalized Benjamin–Bona–Mahony–Burgers equation, Comput. Math. Appl., 74 (2017), 3042–3057. https://doi.org/10.1016/j.camwa.2017.07.046 doi: 10.1016/j.camwa.2017.07.046
[21]	M. Çetin, M. Sezer, H. Kocayiğit, An efficient method based on Lucas polynomials for solving high-order linear boundary value problems, Gazi Univ. J. Sci., 28 (2015), 483–496.
[22]	P. Roul, V. Goura, R. Cavoretto, A numerical technique based on B-spline for a class of time-fractional diffusion equation, Numer. Methods Partial Differ. Equ., 39 (2023), 45–64. https://doi.org/10.1002/num.22790 doi: 10.1002/num.22790
[23]	Y. H. Youssri, A. G. Atta, Petrov-Galerkin Lucas polynomials procedure for the time-fractional diffusion equation, Contemp. Math., 4 (2023), 230–248. https://doi.org/10.37256/cm.4220232420 doi: 10.37256/cm.4220232420
[24]	P. Lyu, S. Vong, A fast linearized numerical method for nonlinear time-fractional diffusion equations, Numer. Algorithms, 87 (2021), 381–408. https://doi.org/10.1007/s11075-020-00971-0 doi: 10.1007/s11075-020-00971-0
[25]	X. M. Gu, H. W. Sun, Y. L. Zhao, X. Zheng, An implicit difference scheme for time-fractional diffusion equations with a time-invariant type variable order, Appl. Math. Lett., 120 (2021), 107270. https://doi.org/10.1016/j.aml.2021.107270 doi: 10.1016/j.aml.2021.107270
[26]	A. Khibiev, A. Alikhanov, C. Huang, A second-order difference scheme for generalized time-fractional diffusion equation with smooth solutions, Comput. Methods Appl. Math., 24 (2024), 101–117. https://doi.org/10.1515/cmam-2022-0089 doi: 10.1515/cmam-2022-0089
[27]	J. L. Zhang, Z. W. Fang, H. W. Sun, Exponential-sum-approximation technique for variable-order time-fractional diffusion equations, J. Appl. Math. Comput., 68 (2022), 323–347. https://doi.org/10.1007/s12190-021-01528-7 doi: 10.1007/s12190-021-01528-7
[28]	C. Canuto, M. Y. Hussaini, A. Quarteroni, T. A. Zang, Spectral methods in fluid dynamics, Springer-Verlag, 1988.
[29]	J. Hesthaven, S. Gottlieb, D. Gottlieb, Spectral methods for time-dependent problems, Cambridge University Press, 2007.
[30]	J. P. Boyd, Chebyshev and Fourier spectral methods, Courier Corporation, 2001.
[31]	W. M. Abd-Elhameed, M. M. Alsyuti, Numerical treatment of multi-term fractional differential equations via new kind of generalized Chebyshev polynomials, Fractal Fract., 7 (2023), 74. https://doi.org/10.3390/fractalfract7010074 doi: 10.3390/fractalfract7010074
[32]	M. M. Alsuyuti, E. H. Doha, S. S. Ezz-Eldien, I. K. Youssef, Spectral Galerkin schemes for a class of multi-order fractional pantograph equations, J. Comput. Appl. Math., 384 (2021), 113157. https://doi.org/10.1016/j.cam.2020.113157 doi: 10.1016/j.cam.2020.113157
[33]	R. M. Hafez, M. A. Zaky, M. A. Abdelkawy, Jacobi spectral Galerkin method for distributed-order fractional Rayleigh–Stokes problem for a generalized second grade fluid, Front. Phys., 7 (2020), 240. https://doi.org/10.3389/fphy.2019.00240 doi: 10.3389/fphy.2019.00240
[34]	W. M. Abd-Elhameed, A. M. Al-Sady, O. M. Alqubori, A. G. Atta, Numerical treatment of the fractional Rayleigh-Stokes problem using some orthogonal combinations of Chebyshev polynomials, AIMS Math., 9 (2024), 25457–25481. https://doi.org/10.3934/math.20241243 doi: 10.3934/math.20241243
[35]	A. A. El-Sayed, S. Boulaaras, N. H. Sweilam, Numerical solution of the fractional-order logistic equation via the first-kind Dickson polynomials and spectral tau method, Math. Methods Appl. Sci., 46 (2023), 8004–8017. https://doi.org/10.1002/mma.7345 doi: 10.1002/mma.7345
[36]	H. Hou, X. Li, A meshless superconvergent stabilized collocation method for linear and nonlinear elliptic problems with accuracy analysis, Appl. Math. Comput., 477 (2024), 128801. https://doi.org/10.1016/j.amc.2024.128801 doi: 10.1016/j.amc.2024.128801
[37]	A. G. Atta, Two spectral Gegenbauer methods for solving linear and nonlinear time fractional cable problems, Int. J. Mod. Phys. C, 35 (2024), 2450070. https://doi.org/10.1142/S0129183124500700 doi: 10.1142/S0129183124500700
[38]	M. M. Khader, M. Adel, Numerical and theoretical treatment based on the compact finite difference and spectral collocation algorithms of the space fractional-order Fisher's equation, Int. J. Mod. Phys. C, 31 (2020), 2050122. https://doi.org/10.1142/S0129183120501223 doi: 10.1142/S0129183120501223
[39]	A. Z. Amin, M. A. Abdelkawy, I. Hashim, A space-time spectral approximation for solving nonlinear variable-order fractional convection-diffusion equations with nonsmooth solutions, Int. J. Mod. Phys. C, 34 (2023), 2350041. https://doi.org/10.1142/S0129183123500419 doi: 10.1142/S0129183123500419
[40]	H. M. Ahmed, New generalized Jacobi polynomial Galerkin operational matrices of derivatives: An algorithm for solving boundary value problems, Fractal Fract., 8 (2024), 199. https://doi.org/10.3390/fractalfract8040199 doi: 10.3390/fractalfract8040199
[41]	İ. Avcı Spectral collocation with generalized Laguerre operational matrix for numerical solutions of fractional electrical circuit models, Math. Model. Numer. Simul. Appl., 4 (2024), 110–132. https://doi.org/10.53391/mmnsa.1428035 doi: 10.53391/mmnsa.1428035
[42]	H. Singh, Chebyshev spectral method for solving a class of local and nonlocal elliptic boundary value problems, Int. J. Nonlinear Sci. Numer. Simul., 24 (2023), 899–915. https://doi.org/10.1515/ijnsns-2020-0235 doi: 10.1515/ijnsns-2020-0235
[43]	M. Abdelhakem, A. Ahmed, D. Baleanu, M. El-Kady, Monic Chebyshev pseudospectral differentiation matrices for higher-order IVPs and BVP: Applications to certain types of real-life problems, Comput. Appl. Math., 41 (2022), 253. https://doi.org/10.1007/s40314-022-01940-0 doi: 10.1007/s40314-022-01940-0
[44]	Priyanka, S. Sahani, S. Arora, An efficient fourth order Hermite spline collocation method for time fractional diffusion equation describing anomalous diffusion in two space variables, Comput. Appl. Math., 43 (2024), 193. https://doi.org/10.1007/s40314-024-02708-4 doi: 10.1007/s40314-024-02708-4
[45]	W. M. Abd-Elhameed, A. Napoli, Some novel formulas of Lucas polynomials via different approaches, Symmetry, 15 (2023), 185. https://doi.org/10.3390/sym15010185 doi: 10.3390/sym15010185
[46]	W. M. Abd-Elhameed, N. A. Zeyada, New formulas including convolution, connection and radicals formulas of k-Fibonacci and k-Lucas polynomials, Indian J. Pure Appl. Math., 53 (2022), 1006–1016. https://doi.org/10.1007/s13226-021-00214-5 doi: 10.1007/s13226-021-00214-5
[47]	W. M. Abd-Elhameed, A. K. Amin, Novel identities of Bernoulli polynomials involving closed forms for some definite integrals, Symmetry, 14 (2022), 2284. https://doi.org/10.3390/sym14112284 doi: 10.3390/sym14112284
[48]	W. M. Abd-Elhameed, Y. H. Youssri, Generalized Lucas polynomial sequence approach for fractional differential equations, Nonlinear Dynam., 89 (2017), 1341–1355. https://doi.org/10.1007/s11071-017-3519-9 doi: 10.1007/s11071-017-3519-9
[49]	Y. H. Youssri, W. M. Abd-Elhameed, A. G. Atta, Spectral Galerkin treatment of linear one-dimensional telegraph type problem via the generalized Lucas polynomials, Arab. J. Math., 11 (2022), 601–615. https://doi.org/10.1007/s40065-022-00374-0 doi: 10.1007/s40065-022-00374-0
[50]	G. J. O. Jameson, The incomplete gamma functions, Math. Gaz., 100 (2016), 298–306. https://doi.org/10.1017/mag.2016.67 doi: 10.1017/mag.2016.67
[51]	F. Zeng, C. Li, A new Crank–Nicolson finite element method for the time-fractional subdiffusion equation, Appl. Numer. Math., 121 (2017), 82–95. https://doi.org/10.1016/j.apnum.2017.06.011 doi: 10.1016/j.apnum.2017.06.011

Reader Comments

Your name:*

Email:*
© 2024 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)