A novel generalized symmetric spectral Galerkin numerical approach for solving fractional differential equations with singular kernel

Mohamed Obeid; Mohamed A. Abd El Salam; Mohamed S. Mohamed; Mohamed Obeid; Mohamed A. Abd El Salam; Mohamed S. Mohamed

doi:10.3934/math.2023855

AIMS Mathematics

2023, Volume 8, Issue 7: 16724-16747. doi: 10.3934/math.2023855

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A novel generalized symmetric spectral Galerkin numerical approach for solving fractional differential equations with singular kernel

1.
Department of Mathematics, Faculty of Science, Al-Azhar University, Nasr City 11884, Cairo, Egypt
2.
Basic Science Department, October High Institute for Engineering and Technology, Giza, Egypt
3.
Department of Mathematics, College of Science, Taif University, P.O. Box 11099, Taif, 21944, Saudi Arabia

Received: 18 February 2023 Revised: 13 April 2023 Accepted: 27 April 2023 Published: 12 May 2023
MSC : 35G05, 65N35

Polynomial based numerical techniques usually provide the best choice for approximating the solution of fractional differential equations (FDEs). The choice of the basis at which the solution is expanded might affect the results significantly. However, there is no general approach to determine which basis will perform better with a particular problem. The aim of this paper is to develop a novel generalized symmetric orthogonal basis which has not been discussed in the context of numerical analysis before to establish a general numerical treatment for the FDEs with a singular kernel. The operational matrix with four free parameters was derived for the left-sided Caputo fractional operator in order to transform the FDEs into the corresponding algebraic system with the aid of spectral Galerkin method. Several families of the existing polynomials can be obtained as a special case from the new basis beside other new families generated according to the value of the free parameters. Consequently, the operational matrix in terms of these families was derived as a special case from the generalized one up to a coefficient diagonal matrix. Furthermore, different properties relevant to the new generalized basis were derived and the error associated with function approximation by the new basis was performed based on the generalized Taylor's formula.
- symmetric orthogonal polynomials,
- operational matrix,
- spectral Galerkin method,
- error analysis,
- fractional differential equations
Citation: Mohamed Obeid, Mohamed A. Abd El Salam, Mohamed S. Mohamed. A novel generalized symmetric spectral Galerkin numerical approach for solving fractional differential equations with singular kernel[J]. AIMS Mathematics, 2023, 8(7): 16724-16747. doi: 10.3934/math.2023855

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Abstract

Polynomial based numerical techniques usually provide the best choice for approximating the solution of fractional differential equations (FDEs). The choice of the basis at which the solution is expanded might affect the results significantly. However, there is no general approach to determine which basis will perform better with a particular problem. The aim of this paper is to develop a novel generalized symmetric orthogonal basis which has not been discussed in the context of numerical analysis before to establish a general numerical treatment for the FDEs with a singular kernel. The operational matrix with four free parameters was derived for the left-sided Caputo fractional operator in order to transform the FDEs into the corresponding algebraic system with the aid of spectral Galerkin method. Several families of the existing polynomials can be obtained as a special case from the new basis beside other new families generated according to the value of the free parameters. Consequently, the operational matrix in terms of these families was derived as a special case from the generalized one up to a coefficient diagonal matrix. Furthermore, different properties relevant to the new generalized basis were derived and the error associated with function approximation by the new basis was performed based on the generalized Taylor's formula.

References

[1]	N. Sene, Analytical solutions and numerical schemes of certain generalized fractional diffusion models, Eur. Phys. J. Plus, 134 (2019), 199. https://doi.org/10.1140/epjp/i2019-12531-4 doi: 10.1140/epjp/i2019-12531-4
[2]	A. Atangana, S. Qureshi, Modeling attractors of chaotic dynamical systems with fractal-fractional operators, Chaos, Solitons Fract., 123 (2019), 320–337. https://doi.org/10.1016/j.chaos.2019.04.020 doi: 10.1016/j.chaos.2019.04.020
[3]	R. P. Yadav, R. Verma, A numerical simulation of fractional order mathematical modeling of COVID-19 disease in case of Wuhan China, Chaos, Solitons Fract., 140 (2020), 110124. https://doi.org/10.1016/j.chaos.2020.110124 doi: 10.1016/j.chaos.2020.110124
[4]	C. M. Chen, F. Liu, I. Turner, V. Anh, A Fourier method for the fractional diffusion equation describing sub-diffusion, J. Comput. Phys., 227 (2007), 886–897. https://doi.org/10.1016/j.jcp.2007.05.012 doi: 10.1016/j.jcp.2007.05.012
[5]	C. M. Chen, F. Liu, K. Burrage, Finite difference methods and a Fourier analysis for the fractional reaction-subdiffusion equation, Appl. Math. Comput., 198 (2008), 754–769. https://doi.org/10.1016/j.amc.2007.09.020 doi: 10.1016/j.amc.2007.09.020
[6]	H. Gul, S. Ali, K. Shah, S. Muhammad, T. Sitthiwirattham, S. Chasreechai, Application of asymptotic homotopy perturbation method to fractional order partial differential equation, Symmetry, 13 (2021), 2215. https://doi.org/10.3390/sym13112215 doi: 10.3390/sym13112215
[7]	M. S. Arshad, D. Baleanu, M. B. Riaz, M. Abbas, A novel 2-stage fractional Runge–kutta method for a time-fractional logistic growth model, Discrete Dyn. Nat. Soc., 2020 (2020), 1–8. https://doi.org/10.1155/2020/1020472 doi: 10.1155/2020/1020472
[8]	O. González-Gaxiola, A. Biswas, W-shaped optical solitons of Chen–Lee–Liu equation by Laplace–Adomian decomposition method, Opt. Quant. Electron., 50 (2018), 1–11. https://doi.org/10.1007/s11082-018-1583-0 doi: 10.1007/s11082-018-1583-0
[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]	S. Ibrahim, A. Isah, Solving system of fractional order differential equations using Legendre operational matrix of derivatives, Eurasian J. Sci. Eng., 7 (2021), 25–37. https://doi.org/10.23918/eajse.v7i1p25 doi: 10.23918/eajse.v7i1p25
[11]	A. Zamiri, A. Borhanifar, A. Ghannadiasl, Laguerre collocation method for solving Lane–Emden type equations, Comput. Methods Differ. Equ., 9 (2021), 1176–1197. https://dx.doi.org/10.22034/cmde.2020.35895.1621 doi: 10.22034/cmde.2020.35895.1621
[12]	A. Yari, Numerical solution for fractional optimal control problems by Hermite polynomials, J. Vib. Control, 27 (2021), 698–716. https://doi.org/10.1177/1077546320933129 doi: 10.1177/1077546320933129
[13]	M. H. Derakhshan, Numerical solution of a coupled system of fractional order integro differential equations by an efficient numerical method based on the second kind Chebyshev polynomials, Math. Anal. Contemp. Appl., 3 (2021), 25–40. https://doi.org/10.30495/maca.2021.1938222.1025 doi: 10.30495/maca.2021.1938222.1025
[14]	S. N. Tural-Polat, A. T. Dincel, Numerical solution method for multi-term variable order fractional differential equations by shifted Chebyshev polynomials of the third kind, Alex. Eng. J., 61 (2022), 5145–5153. https://doi.org/10.1016/j.aej.2021.10.036 doi: 10.1016/j.aej.2021.10.036
[15]	C. Cesarano, S. Pinelas, P. E. Ricci, The third and fourth kind pseudo-Chebyshev polynomials of half-integer degree, Symmetry, 11 (2019), 274. https://doi.org/10.3390/sym11020274 doi: 10.3390/sym11020274
[16]	A. Secer, S. Altun, A new operational matrix of fractional derivatives to solve systems of fractional differential equations via legendre wavelets, Mathematics, 6 (2018), 238. https://doi.org/10.3390/math6110238 doi: 10.3390/math6110238
[17]	C. Baishya, P. Veeresha, Laguerre polynomial-based operational matrix of integration for solving fractional differential equations with non-singular kernel, Proc. R. Soc. A, 477 (2021), 20210438. https://doi.org/10.1098/rspa.2021.0438 doi: 10.1098/rspa.2021.0438
[18]	W. M. Abd-Elhameed, Y. H. Youssri, Fifth-kind orthonormal Chebyshev polynomial solutions for fractional differential equations, Comput. Appl. Math., 37 (2018), 2897–2921. https://doi.org/10.1007/s40314-017-0488-z doi: 10.1007/s40314-017-0488-z
[19]	K. Sadri, H. Aminikhah, Chebyshev polynomials of sixth kind for solving nonlinear fractional PDEs with proportional delay and its convergence analysis, J. Funct. Spaces, 2022 (2022), 1–20. https://doi.org/10.1155/2022/9512048 doi: 10.1155/2022/9512048
[20]	M. Abdelhakem, D. Mahmoud, D. Baleanu, M. El-kady, Shifted ultraspherical pseudo-Galerkin method for approximating the solutions of some types of ordinary fractional problems, Adv. Differ. Equ., 2021 (2021), 1–18. https://doi.org/10.1186/s13662-021-03247-6 doi: 10.1186/s13662-021-03247-6
[21]	F. A. Shah, R. Abass, Solution of fractional oscillator equations using ultraspherical wavelets, Int. J. Geom. Methods Mod. Phys., 16 (2019), 1950075. https://doi.org/10.1142/S0219887819500750 doi: 10.1142/S0219887819500750
[22]	D. Fortunato, N. Hale, A. Townsend, The ultraspherical spectral element method, J. Comput. Phys., 436 (2021), 110087. https://doi.org/10.1016/j.jcp.2020.110087 doi: 10.1016/j.jcp.2020.110087
[23]	A. A. El-Sayed, D. Baleanu, P. Agarwal, A novel Jacobi operational matrix for numerical solution of multi-term variable-order fractional differential equations, J. Taibah Univ. Sci., 14 (2020), 963–974. https://doi.org/10.1080/16583655.2020.1792681 doi: 10.1080/16583655.2020.1792681
[24]	K. Maleknejad, J. Rashidinia, T. Eftekhari, A new and efficient numerical method based on shifted fractional‐order Jacobi operational matrices for solving some classes of two‐dimensional nonlinear fractional integral equations, Numer. Methods Partial Differ. Equ., 37 (2021), 2687–2713. https://doi.org/10.1002/num.22762 doi: 10.1002/num.22762
[25]	J. Rashidinia, T. Eftekhari, K. Maleknejad, Numerical solutions of two-dimensional nonlinear fractional Volterra and Fredholm integral equations using shifted Jacobi operational matrices via collocation method, J. King Saud Univ.-Sci., 33 (2021), 101244. https://doi.org/10.1016/j.jksus.2020.101244 doi: 10.1016/j.jksus.2020.101244
[26]	M. Masjed-Jamei, A generalization of classical symmetric orthogonal functions using a symmetric generalization of Sturm–Liouville problems, Integr. Transf. Spec. Funct., 18 (2007), 871–883. https://doi.org/10.1080/10652460701510949 doi: 10.1080/10652460701510949
[27]	S. G. Samko, A. A. Kilbas, O. I. Marichev, Fractional integrals and derivatives, Vol. 1, Yverdon-les-Bains, Switzerland: Gordon and breach science publishers, Yverdon, 1993.
[28]	Z. M. Odibat, S. Momani, An algorithm for the numerical solution of differential equations of fractional order, J. Appl. Math. Inform., 26 (2008), 15–27.
[29]	Z. M. Odibat, N. T. Shawagfeh, Generalized Taylor's formula, Appl. Math. Comput., 186 (2007), 286–293. https://doi.org/10.1016/j.amc.2006.07.102 doi: 10.1016/j.amc.2006.07.102
[30]	A. El-Ajou, O. A. Arqub, M. Al-Smadi, A general form of the generalized Taylor's formula with some applications, Appl. Math. Comput., 256 (2015), 851–859. https://doi.org/10.1016/j.amc.2015.01.034 doi: 10.1016/j.amc.2015.01.034
[31]	A. E. Choque-Rivero, I. Area, A Favard type theorem for Hurwitz polynomials, Discrete Cont. Dyn. Syst.-Ser. B, 25 (2020), 529–544. https://doi/10.3934/dcdsb.2019252 doi: 10.3934/dcdsb.2019252
[32]	H. M. Srivastava, J. Choi, Zeta and q-Zeta functions and associated series and integrals, El-sevier, 2011.
[33]	A. W. Naylor, G. R. Sell, Linear operator theory in engineering and science, Springer Science and Business Media, 1982.
[34]	H. M. Srivastava, F. A. Shah, R. Abass, An application of the Gegenbauer wavelet method for the numerical solution of the fractional Bagley–Torvik equation, Russ. J. Math. Phys., 26 (2019), 77–93. https://doi.org/10.1134/S1061920819010096 doi: 10.1134/S1061920819010096
[35]	S. K. Damarla, M. Kundu, Numerical solution of multi-order fractional differential equations using generalized triangular function operational matrices, Appl. Math. Comput., 263 (2015), 189–203. https://doi.org/10.1016/j.amc.2015.04.051 doi: 10.1016/j.amc.2015.04.051
[36]	W. M. Abd-Elhameed, Y. H. Youssri, Sixth-kind Chebyshev spectral approach for solving fractional differential equations, Int. J. Nonlinear Sci. Numer. Simul., 20 (2019), 191–203. https://doi.org/10.1515/ijnsns-2018-0118 doi: 10.1515/ijnsns-2018-0118

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