Numerical treatment of the fractional Rayleigh-Stokes problem using some orthogonal combinations of Chebyshev polynomials

Waleed Mohamed Abd-Elhameed; Ahad M. Al-Sady; Omar Mazen Alqubori; Ahmed Gamal Atta; Waleed Mohamed Abd-Elhameed; Ahad M. Al-Sady; Omar Mazen Alqubori; Ahmed Gamal Atta

doi:10.3934/math.20241243

AIMS Mathematics

2024, Volume 9, Issue 9: 25457-25481. doi: 10.3934/math.20241243

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Numerical treatment of the fractional Rayleigh-Stokes problem using some orthogonal combinations of Chebyshev polynomials

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

Received: 27 May 2024 Revised: 05 August 2024 Accepted: 19 August 2024 Published: 30 August 2024
MSC : 26A33, 33C45, 65Mxx, 65M70

This work aims to provide a new Galerkin algorithm for solving the fractional Rayleigh-Stokes equation (FRSE). We select the basis functions for the Galerkin technique to be appropriate orthogonal combinations of the second kind of Chebyshev polynomials (CPs). By implementing the Galerkin approach, the FRSE, with its governing conditions, is converted into a matrix system whose entries can be obtained explicitly. This system can be obtained by expressing the derivatives of the basis functions in terms of the second-kind CPs and after computing some definite integrals based on some properties of CPs of the second kind. A thorough investigation is carried out for the convergence analysis. We demonstrate that the approach is applicable and accurate by providing some numerical examples.
- Chebyshev polynomials,
- recurrence relation,
- spectral methods,
- fractional differential equations,
- convergence analysis
Citation: Waleed Mohamed Abd-Elhameed, Ahad M. Al-Sady, Omar Mazen Alqubori, Ahmed Gamal Atta. Numerical treatment of the fractional Rayleigh-Stokes problem using some orthogonal combinations of Chebyshev polynomials[J]. AIMS Mathematics, 2024, 9(9): 25457-25481. doi: 10.3934/math.20241243

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Abstract

This work aims to provide a new Galerkin algorithm for solving the fractional Rayleigh-Stokes equation (FRSE). We select the basis functions for the Galerkin technique to be appropriate orthogonal combinations of the second kind of Chebyshev polynomials (CPs). By implementing the Galerkin approach, the FRSE, with its governing conditions, is converted into a matrix system whose entries can be obtained explicitly. This system can be obtained by expressing the derivatives of the basis functions in terms of the second-kind CPs and after computing some definite integrals based on some properties of CPs of the second kind. A thorough investigation is carried out for the convergence analysis. We demonstrate that the approach is applicable and accurate by providing some numerical examples.

References

[1]	H. M. Ahmed, Numerical solutions for singular Lane-Emden equations using shifted Chebyshev polynomials of the first kind, Contemp. Math., 4 (2023), 132–149. https://doi.org/10.37256/cm.4120232254 doi: 10.37256/cm.4120232254
[2]	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
[3]	A. H. Bhrawy, M. A. Abdelkawy, F. Mallawi, An accurate Chebyshev pseudospectral scheme for multi-dimensional parabolic problems with time delays, Bound. Value Probl., 2015 (2015), 1–20. https://doi.org/10.1186/s13661-015-0364-y doi: 10.1186/s13661-015-0364-y
[4]	H. M. Ahmed, W. M. Abd-Elhameed, Spectral solutions of specific singular differential equations using a unified spectral Galerkin-collocation algorithm, J. Nonlinear Math. Phys., 31 (2024), 42. https://doi.org/10.1007/s44198-024-00194-0 doi: 10.1007/s44198-024-00194-0
[5]	Y. Xu, An integral formula for generalized Gegenbauer polynomials and Jacobi polynomials, Adv. Appl. Math., 29 (2002), 328–343. https://doi.org/10.1016/s0196-8858(02)00017-9 doi: 10.1016/s0196-8858(02)00017-9
[6]	A. Draux, M. Sadik, B. Moalla, Markov-Bernstein inequalities for generalized Gegenbauer weight, Appl. Numer. Math., 61 (2011), 1301–1321. https://doi.org/10.1016/j.apnum.2011.09.003 doi: 10.1016/j.apnum.2011.09.003
[7]	A. G. Atta, W. M. Abd-Elhameed, G. M. Moatimid, Y. H. Youssri, Advanced shifted sixth-kind Chebyshev tau approach for solving linear one-dimensional hyperbolic telegraph type problem, Math. Sci., 17 (2023), 415–429. https://doi.org/10.1007/s40096-022-00460-6 doi: 10.1007/s40096-022-00460-6
[8]	A. Eid, M. M. Khader, A. M. Megahed, Sixth-kind Chebyshev polynomials technique to numerically treat the dissipative viscoelastic fluid flow in the rheology of Cattaneo-Christov model, Open Phys., 22 (2024), 20240001. https://doi.org/10.1515/phys-2024-0001 doi: 10.1515/phys-2024-0001
[9]	M. Obeid, M. A. Abd El Salam, J. A. Younis, Operational matrix-based technique treating mixed type fractional differential equations via shifted fifth-kind Chebyshev polynomials, Appl. Math. Sci. Eng., 31 (2023), 2187388. https://doi.org/10.1080/27690911.2023.2187388 doi: 10.1080/27690911.2023.2187388
[10]	K. Sadri, H. Aminikhah, A new efficient algorithm based on fifth-kind Chebyshev polynomials for solving multi-term variable-order time-fractional diffusion-wave equation, Int. J. Comput. Math., 99 (2022), 966–992. https://doi.org/10.1080/00207160.2021.1940977 doi: 10.1080/00207160.2021.1940977
[11]	W. M. Abd-Elhameed, Y. H. Youssri, A. K. Amin, A. G. Atta, Eighth-kind Chebyshev polynomials collocation algorithm for the nonlinear time-fractional generalized Kawahara equation, Fractal Fract., 7 (2023), 1–23. https://doi.org/10.3390/fractalfract7090652 doi: 10.3390/fractalfract7090652
[12]	H. M. Ahmed, R. M. Hafez, W. M. Abd-Elhameed, A computational strategy for nonlinear time-fractional generalized Kawahara equation using new eighth-kind Chebyshev operational matrices, Phys. Scr., 99 (2024), 045250. https://doi.org/10.1088/1402-4896/ad3482 doi: 10.1088/1402-4896/ad3482
[13]	R. L. Magin, Fractional calculus in bioengineering, Part 1, Crit. Rev. Biomed. Eng., 32 (2004), 1–104. https://doi.org/10.1615/CritRevBiomedEng.v32.i1.10 doi: 10.1615/CritRevBiomedEng.v32.i1.10
[14]	V. E. Tarasov, Fractional dynamics: applications of fractional calculus to dynamics of particles, fields and media, Berlin, Heidelberg: Springer, 2010. https://doi.org/10.1007/978-3-642-14003-7
[15]	D. Baleanu, Z. B. Guvenc, J. A. T. Machado, New trends in nanotechnology and fractional calculus applications, Dordrecht: Springer, 2010. https://doi.org/10.1007/978-90-481-3293-5
[16]	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
[17]	A. Kanwal, S. Boulaaras, R. Shafqat, B. Taufeeq, M. ur Rahman, Explicit scheme for solving variable-order time-fractional initial boundary value problems, Sci. Rep., 14 (2024), 5396. https://doi.org/10.1038/s41598-024-55943-4 doi: 10.1038/s41598-024-55943-4
[18]	L. Y. Qing, X. L. 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.aml.2024.109204 doi: 10.1016/j.aml.2024.109204
[19]	A. Z. Amin, A. M. Lopes, I. Hashim, A space-time spectral collocation method for solving the variable-order fractional Fokker-Planck equation, J. Appl. Anal. Comput., 13 (2023), 969–985. https://doi.org/10.11948/20220254 doi: 10.11948/20220254
[20]	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), 2743–2765. https://doi.org/10.32604/cmes.2023.023705 doi: 10.32604/cmes.2023.023705
[21]	A. Burqan, R. Saadeh, A. Qazza, S. Momani, ARA-residual power series method for solving partial fractional differential equations, Alexandria Eng. J., 62 (2023), 47–62. https://doi.org/10.1016/j.aej.2022.07.022 doi: 10.1016/j.aej.2022.07.022
[22]	S. N. Hajiseyedazizi, M. E. Samei, J. Alzabut, Y. M. Chu, On multi-step methods for singular fractional q-integro-differential equations, Open Math., 19 (2021), 1378–1405. https://doi.org/10.1515/math-2021-0093 doi: 10.1515/math-2021-0093
[23]	R. Amin, K. Shah, M. Asif, I. Khan, F. Ullah, An efficient algorithm for numerical solution of fractional integro-differential equations via Haar wavelet, J. Comput. Appl. Math., 381 (2021), 113028. https://doi.org/10.1016/j.cam.2020.113028 doi: 10.1016/j.cam.2020.113028
[24]	H. M. Ahmed, New generalized Jacobi Galerkin operational matrices of derivatives: an algorithm for solving multi-term variable-order time-fractional diffusion-wave equations, Fractal Fract., 8 (2024), 1–26. https://doi.org/10.3390/fractalfract8010068 doi: 10.3390/fractalfract8010068
[25]	H. M. Ahmed, Enhanced shifted Jacobi operational matrices of derivatives: spectral algorithm for solving multiterm variable-order fractional differential equations, Bound. Value Probl., 2023 (2023), 108. https://doi.org/10.1186/s13661-023-01796-1 doi: 10.1186/s13661-023-01796-1
[26]	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
[27]	Y. F. Wei, Y. Guo, Y. Li, A new numerical method for solving semilinear fractional differential equation, J. Appl. Math. Comput., 68 (2022), 1289–1311. https://doi.org/10.1007/s12190-021-01566-1 doi: 10.1007/s12190-021-01566-1
[28]	M. A. Zaky, I. G. Ameen, M. Babatin, A. Akgül, M. Hammad, A. Lopes, Non-polynomial collocation spectral scheme for systems of nonlinear Caputo-Hadamard differential equations, Fractal Fract., 8 (2024), 1–16. https://doi.org/10.3390/fractalfract8050262 doi: 10.3390/fractalfract8050262
[29]	M. H. Alharbi, A. F. Abu Sunayh, A. G. Atta, W. M. Abd-Elhameed, Novel approach by shifted Fibonacci polynomials for solving the fractional Burgers equation, Fractal Fract., 8 (2024), 1–22. https://doi.org/10.3390/fractalfract8070427 doi: 10.3390/fractalfract8070427
[30]	M. A. Abdelkawy, A. M. Lopes, M. M. Babatin, Shifted fractional Jacobi collocation method for solving fractional functional differential equations of variable order, Chaos Solitons Fract., 134 (2020), 109721. https://doi.org/10.1016/j.chaos.2020.109721 doi: 10.1016/j.chaos.2020.109721
[31]	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
[32]	W. M. Abd-Elhameed, M. M. Alsuyuti, Numerical treatment of multi-term fractional differential equations via new kind of generalized Chebyshev polynomials, Fractal Fract., 7 (2023), 1–22. https://doi.org/10.3390/fractalfract7010074 doi: 10.3390/fractalfract7010074
[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]	S. M. Sivalingam, P. Kumar, V. Govindaraj, A neural networks-based numerical method for the generalized Caputo-type fractional differential equations, Math. Comput. Simul., 213 (2023), 302–323. https://doi.org/10.1016/j.matcom.2023.06.012 doi: 10.1016/j.matcom.2023.06.012
[35]	N. H. Tuan, N. D. Phuong, T. N. Thach, New well-posedness results for stochastic delay Rayleigh-Stokes equations, Discrete Contin. Dyn. Syst. Ser. B, 28 (2023), 347–358. https://doi.org/10.3934/dcdsb.2022079 doi: 10.3934/dcdsb.2022079
[36]	J. N. Wang, Y. Zhou, A. Alsaedi, B. Ahmad, Well-posedness and regularity of fractional Rayleigh-Stokes problems, Z. Angew. Math. Phys., 73 (2022), 161. https://doi.org/10.1007/s00033-022-01808-7 doi: 10.1007/s00033-022-01808-7
[37]	L. Peng, Y. Zhou, The well-posedness results of solutions in Besov-Morrey spaces for fractional Rayleigh-Stokes equations, Qual. Theory Dyn. Syst., 23 (2024), 43. https://doi.org/10.1007/s12346-023-00897-7 doi: 10.1007/s12346-023-00897-7
[38]	Z. Guan, X. D. Wang, J. Ouyang, An improved finite difference/finite element method for the fractional Rayleigh-Stokes problem with a nonlinear source term, J. Appl. Math. Comput., 65 (2021), 451–479. https://doi.org/10.1007/s12190-020-01399-4 doi: 10.1007/s12190-020-01399-4
[39]	Ö. Oruç, An accurate computational method for two-dimensional (2D) fractional Rayleigh-Stokes problem for a heated generalized second grade fluid via linear barycentric interpolation method, Comput. Math. Appl., 118 (2022), 120–131. https://doi.org/ 10.1016/j.camwa.2022.05.012 doi: 10.1016/j.camwa.2022.05.012
[40]	Y. L. Zhang, Y. H. Zhou, J. M. Wu, Quadratic finite volume element schemes over triangular meshes for a nonlinear time-fractional Rayleigh-Stokes problem, Comput. Model. Eng. Sci., 127 (2021), 487–514. https://doi.org/10.32604/cmes.2021.014950 doi: 10.32604/cmes.2021.014950
[41]	M. Saffarian, A. Mohebbi, High order numerical method for the simulation of Rayleigh-Stokes problem for a heated generalized second grade fluid with fractional derivative on regular and irregular domains, Eng. Comput., 39 (2023), 2851–2868. https://doi.org/10.1007/s00366-022-01647-0 doi: 10.1007/s00366-022-01647-0
[42]	F. Salehi, H. Saeedi, M. M. Moghadam, Discrete Hahn polynomials for numerical solution of two-dimensional variable-order fractional Rayleigh-Stokes problem, Comput. Appl. Math., 37 (2018), 5274–5292. https://doi.org/10.1007/s40314-018-0631-5 doi: 10.1007/s40314-018-0631-5
[43]	O. Nikan, A. Golbabai, J. A. T. Machado, T. Nikazad, Numerical solution of the fractional Rayleigh-Stokes model arising in a heated generalized second-grade fluid, Eng. Comput., 37 (2021), 1751–1764. https://doi.org/10.1007/s00366-019-00913-y doi: 10.1007/s00366-019-00913-y
[44]	J. Shen, T. Tang, L. L. Wang, Spectral methods: algorithms, analysis and applications, Berlin, Heidelberg: Springer, 2011. https://doi.org/10.1007/978-3-540-71041-7
[45]	B. Shizgal, Spectral methods in chemistry and physics, Dordrecht: Springer, 2015. https://doi.org/10.1007/978-94-017-9454-1
[46]	C. Canuto, M. Y. Hussaini, A. Quarteroni, T. A. Zang, Spectral methods in fluid dynamics, Berlin, Heidelberg: Springer, 1988. https://doi.org/10.1007/978-3-642-84108-8
[47]	M. M. Alsuyuti, E. H. Doha, S. S. Ezz-Eldien, Galerkin operational approach for multi-dimensions fractional differential equations, Commun. Nonlinear Sci. Numer. Simul., 114 (2022), 106608. https://doi.org/10.1016/j.cnsns.2022.106608 doi: 10.1016/j.cnsns.2022.106608
[48]	M. M. Alsuyuti, E. H. Doha, S. S. Ezz-Eldien, B. I. Bayoumi, D. Baleanu, Modified Galerkin algorithm for solving multitype fractional differential equations, Math. Methods Appl. Sci., 42 (2019), 1389–1412. https://doi.org/10.1002/mma.5431 doi: 10.1002/mma.5431
[49]	E. M. Abdelghany, W. M. Abd-Elhameed, G. M. Moatimid, Y. H. Youssri, A. G. Atta, A tau approach for solving time-fractional heat equation based on the shifted sixth-kind Chebyshev polynomials, Symmetry, 15 (2023), 1–17. https://doi.org/10.3390/sym15030594 doi: 10.3390/sym15030594
[50]	E. Magdy, W. M. Abd-Elhameed, Y. H. Youssri, G. M. Moatimid, A. G. Atta, A potent collocation approach based on shifted Gegenbauer polynomials for nonlinear time fractional Burgers' equations, Contemp. Math., 4 (2023), 647–665. https://doi.org/10.37256/cm.4420233302 doi: 10.37256/cm.4420233302
[51]	W. M. Abd-Elhameed, M. S. Al-Harbi, A. G. Atta, New convolved Fibonacci collocation procedure for the Fitzhugh-Nagumo non-linear equation, Nonlinear Eng., 13 (2024), 20220332. https://doi.org/10.1515/nleng-2022-0332 doi: 10.1515/nleng-2022-0332
[52]	J. Shen, Efficient spectral-Galerkin method I. Direct solvers of second- and fourth-order equations using Legendre polynomials, SIAM J. Sci. Comput., 15 (1994), 1489–1505. https://doi.org/10.1137/0915089 doi: 10.1137/0915089
[53]	J. Shen, Efficient spectral-Galerkin method Ⅱ. Direct solvers of second- and fourth-order equations using Chebyshev polynomials, SIAM J. Sci. Comput., 16 (1995), 74–87. https://doi.org/10.1137/0916006 doi: 10.1137/0916006
[54]	E. H. Doha, W. M. Abd-Elhameed, A. H. Bhrawy, New spectral-Galerkin algorithms for direct solution of high even-order differential equations using symmetric generalized Jacobi polynomials, Collect. Math., 64 (2013), 373–394. https://doi.org/10.1007/s13348-012-0067-y doi: 10.1007/s13348-012-0067-y
[55]	I. Podlubny, Fractional differential equations, Elsevier, 1998.
[56]	W. M. Abd-Elhameed, Y. H. Youssri, Explicit shifted second-kind Chebyshev spectral treatment for fractional Riccati differential equation, Comput. Model. Eng. Sci., 121 (2019), 1029–1049. https://doi.org/10.32604/cmes.2019.08378 doi: 10.32604/cmes.2019.08378
[57]	A. G. Atta, Y. H. Youssri, Shifted second-kind Chebyshev spectral collocation-based technique for time-fractional KdV-Burgers' equation, Iran. J. Math. Chem., 14 (2023), 207–224. https://doi.org/ 10.22052/ijmc.2023.252824.1710 doi: 10.22052/ijmc.2023.252824.1710
[58]	H. Mesgarani, Y. E. Aghdam, M. Khoshkhahtinat, B. Farnam, Analysis of the numerical scheme of the one-dimensional fractional Rayleigh-Stokes model arising in a heated generalized problem, AIP Adv., 13 (2023), 085024. https://doi.org/10.1063/5.0156586 doi: 10.1063/5.0156586
[59]	E. W. Weisstein, Regularized hypergeometric function. Available from: https://mathworld.wolfram.com/RegularizedHypergeometricFunction.html.
[60]	X. D. Zhao, L. L. Wang, Z. Q. Xie, Sharp error bounds for Jacobi expansions and Gegenbauer-Gauss quadrature of analytic functions, SIAM J. Numer. Anal., 51 (2013), 1443–1469. https://doi.org/10.1137/12089421X doi: 10.1137/12089421X
[61]	M. A. Zaky, An improved tau method for the multi-dimensional fractional Rayleigh-Stokes problem for a heated generalized second grade fluid, Comput. Math. Appl., 75 (2018), 2243–2258. https://doi.org/10.1016/j.camwa.2017.12.004 doi: 10.1016/j.camwa.2017.12.004

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