Spectral collocation approach with shifted Chebyshev sixth-kind series approximation for generalized space fractional partial differential equations

K. Ali Khalid; Aiman Mukheimer; A. Younis Jihad; Mohamed A. Abd El Salam; Hassen Aydi; K. Ali Khalid; Aiman Mukheimer; A. Younis Jihad; Mohamed A. Abd El Salam; Hassen Aydi

doi:10.3934/math.2022482

AIMS Mathematics

2022, Volume 7, Issue 5: 8622-8644. doi: 10.3934/math.2022482

Previous Article Next Article

Research article

Spectral collocation approach with shifted Chebyshev sixth-kind series approximation for generalized space fractional partial differential equations

1.
Department of Mathematics, Faculty of Science, Al-Azhar University, Nasr-City, 11884, Cairo, Egypt
2.
Department of Mathematics and Sciences, Prince Sultan University, Riyadh 11586, Saudi Arabia
3.
Department of Mathematics, Aden University, Aden, 6014, Yemen
4.
Basic Science Department, October High Institute for Engineering and Technology, 6th october city, Giza, Egypt
5.
Institut Supérieur d'Informatique et des Techniques de Communication, Université de Sousse, H. Sousse, 4000, Tunisia
6.
China Medical University Hospital, China Medical University, Taichung 40402, Taiwan
7.
Department of Mathematics and Applied Mathematics, Sefako Makgatho Health Sciences University, Ga-Rankuwa, South Africa

Received: 14 January 2022 Revised: 20 February 2022 Accepted: 22 February 2022 Published: 02 March 2022
MSC : 35G05, 65N35

In this paper, we propose a numerical scheme to solve generalized space fractional partial differential equations (GFPDEs). Besides, the proposed GFPDEs represent a great generalization of a significant type of FPDEs and their applications, which contain many previous reports as a special case. Moreover, the proposed scheme uses shifted Chebyshev sixth-kind (SCSK) polynomials with spectral collocation approach. The fractional differential derivatives are expressed in terms of the Caputo's definition. Furthermore, the Chebyshev collocation method together with the finite difference method is used to reduce these types of differential equations to a system of algebraic equations which can be solved numerically. In addition, the classical fourth-order Runge-Kotta method is also used to treat the differential system with the collocation method which obtains a great accuracy. Numerical approximations performed by the proposed method are presented and compared with the results obtained by other numerical methods. The introduced numerical experiments are fractional-order mathematical physics models, as advection-dispersion equation (FADE) and diffusion equation (FDE). The results reveal that our method is a simple and effective numerical method.
Citation: K. Ali Khalid, Aiman Mukheimer, A. Younis Jihad, Mohamed A. Abd El Salam, Hassen Aydi. Spectral collocation approach with shifted Chebyshev sixth-kind series approximation for generalized space fractional partial differential equations[J]. AIMS Mathematics, 2022, 7(5): 8622-8644. doi: 10.3934/math.2022482

Related Papers:

Abstract

In this paper, we propose a numerical scheme to solve generalized space fractional partial differential equations (GFPDEs). Besides, the proposed GFPDEs represent a great generalization of a significant type of FPDEs and their applications, which contain many previous reports as a special case. Moreover, the proposed scheme uses shifted Chebyshev sixth-kind (SCSK) polynomials with spectral collocation approach. The fractional differential derivatives are expressed in terms of the Caputo's definition. Furthermore, the Chebyshev collocation method together with the finite difference method is used to reduce these types of differential equations to a system of algebraic equations which can be solved numerically. In addition, the classical fourth-order Runge-Kotta method is also used to treat the differential system with the collocation method which obtains a great accuracy. Numerical approximations performed by the proposed method are presented and compared with the results obtained by other numerical methods. The introduced numerical experiments are fractional-order mathematical physics models, as advection-dispersion equation (FADE) and diffusion equation (FDE). The results reveal that our method is a simple and effective numerical method.

References

[1]	R. L. Magin, Fractional calculus models of complex dynamics in biological tissues, Comput. Math. Appl., 59 (2010), 1586–1593. http://dx.doi.org/10.1016/j.camwa.2009.08.039 doi: 10.1016/j.camwa.2009.08.039
[2]	D. Kumar, D. Baleanu, Fractional calculus and its applications in physics. Front. Phys., 7 (2019), 81. http://dx.doi.org/10.1016/j.camwa.2009.08.039
[3]	H. Sun, Y. Zhang, D. Baleanu, W. Chen, Y. Chen, A new collection of real world applications of fractional calculus in science and engineering, Commun. Nonlinear Sci., 64 (2018), 213–231. http://dx.doi.org/10.1016/j.cnsns.2018.04.019 doi: 10.1016/j.cnsns.2018.04.019
[4]	S. K. Vanani, A. Aminataei, On the numerical solution of fractional partial differential equations, Mathematical and Computational Applications, 17 (2012), 140–151. http://dx.doi.org/10.3390/mca17020140 doi: 10.3390/mca17020140
[5]	F. Yin, J. Song, Y. Wu, L. Zhang, Numerical solution of the fractional partial differential equations by the two-dimensional fractional-order Legendre functions, Abstr. Appl. Anal., 2013 (2013), 562140. http://dx.doi.org/10.1155/2013/562140 doi: 10.1155/2013/562140
[6]	C. Han, Y. L. Wang, & Z. Y. Li, A high-precision numerical approach to solving space fractional Gray-Scott model, Appl. Math. Lett., 125 (2022), 107759. https://doi.org/10.1016/j.aml.2021.107759 doi: 10.1016/j.aml.2021.107759
[7]	P. Roul, A high accuracy numerical method and its convergence for time-fractional Black-Scholes equation governing European options, Appl. Numer. Math., 151 (2020), 472–493. https://doi.org/10.1016/j.apnum.2019.11.004 doi: 10.1016/j.apnum.2019.11.004
[8]	P. Roul, V. P. Goura, A compact finite difference scheme for fractional black-scholes option pricing model, Appl. Numer. Math., 166 (2021), 40–60. https://doi.org/10.1016/j.apnum.2021.03.017 doi: 10.1016/j.apnum.2021.03.017
[9]	P. Roul, V. P. Goura, A high-order B-spline collocation scheme for solving a nonhomogeneous time–fractional diffusion equation, Math. Method. Appl. Sci., 44 (2021), 546–567. https://doi.org/10.1002/mma.6760 doi: 10.1002/mma.6760
[10]	P. Roul, V. P. Goura, A high order numerical scheme for solving a class of non–homogeneous time-fractional reaction diffusion equation, Numer. Meth. Part. D. E., 37 (2021), 1506–1534. https://doi.org/10.1002/num.22594 doi: 10.1002/num.22594
[11]	P. Roul, V. P. Goura, R. Cavoretto, A numerical technique based on B–spline for a class of time–fractional diffusion equation, Numer. Meth. Part. D. E., 2021 (2021), 1–20. https://doi.org/10.1002/num.22790, 2021
[12]	P. Roul, V. Rohil, G. Espinosa-Paredes, K. Obaidurrahman, An efficient numerical method for fractional neutron diffusion equation in the presence of different types of reactivities, Ann. Nucl. Energy, 152 (2021), 108038. https://doi.org/10.1016/j.anucene.2020.108038 doi: 10.1016/j.anucene.2020.108038
[13]	P. Roul, V. Rohil, G. Espinosa-Paredes, V. P. Goura, R. S. Gedam, K. Obaidurrahman, Design and analysis of a numerical method for fractional neutron diffusion equation with delayed neutrons, Appl. Numer. Math., 157 (2020), 634–653. https://doi.org/10.1016/j.apnum.2020.07.007 doi: 10.1016/j.apnum.2020.07.007
[14]	C. C. Ji, Z. Z. Sun, A high-order compact finite difference scheme for the fractional sub-diffusion equation, J. Sci. Comput., 64 (2015), 959–985. https://doi.org/10.1007/s10915-014-9956-4 doi: 10.1007/s10915-014-9956-4
[15]	A. Ahmadian, F. Ismail, S. Salahshour, D. Baleanu, F. Ghaemi, Uncertain viscoelastic models with fractional order: a new spectral tau method to study the numerical simulations of the solution, Commun. Nonlinear Sci., 53 (2017), 44–64. http://dx.doi.org/10.1016/j.cnsns.2017.03.012 doi: 10.1016/j.cnsns.2017.03.012
[16]	K. R. Raslan, K. K. Ali, M. A. Abd El Salam, E. M. Mohamed, Spectral Tau method for solving general fractional order differential equations with linear functional argument, Journal of the Egyptian Mathematical Society, 27 (2019), 1–16. https://doi.org/10.1186/s42787-019-0039-4 doi: 10.1186/s42787-019-0039-4
[17]	H. M. Srivastava, K. M. Saad, M. M. Khader, An efficient spectral collocation method for the dynamic simulation of the fractional epidemiological model of the Ebola virus, Chaos, Solitons & Fractals, 140 (2020), 110174. http://dx.doi.org/10.1016/j.chaos.2020.110174 doi: 10.1016/j.chaos.2020.110174
[18]	M. A. Abd El Salam, M. A. Ramadan, M. A. Nassar, P. Agarwal, Y. M. Chu, Matrix computational collocation approach based on rational Chebyshev functions for nonlinear differential equations, Adv. Differ. Equ-NY, 2021 (2021), 1–17. http://dx.doi.org/10.1186/s13662-021-03481-y doi: 10.1186/s13662-021-03481-y
[19]	K. K. Ali, E. M. Mohamed, K. S. Nisar, M. M. Khashan, M. Zakarya, A collocation approach for multiterm variable-order fractional delay-differential equations using shifted Chebyshev polynomials, Alex. Eng. J., 61 (2022), 3511–3526. http://dx.doi.org/10.1016/j.aej.2021.08.067 doi: 10.1016/j.aej.2021.08.067
[20]	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. http://dx.doi.org/10.1016/j.cam.2020.113157 doi: 10.1016/j.cam.2020.113157
[21]	D. D. Dan, W. Zhang, Y. L. Wang, T. T. Ban, Using piecewise reproducing kernel method and Legendre polynomial for solving a class of the time variable fractional order advection-reaction-diffusion equation, Therm. Sci., 25 (2021), 1261–1268. https://doi.org/10.2298/TSCI200302021D doi: 10.2298/TSCI200302021D
[22]	C. Han, Y. L. Wang, Z. Y. Li, Numerical Solutions of Space Fractional Variable-Coefficient Kdv?Modified Kdv Equation by Fourier Spectral Method, Fractals, 29 (2021), 2150246. https://doi.org/10.1142/S0218348X21502467 doi: 10.1142/S0218348X21502467
[23]	M. Masjed-Jamei, Some new classes of orthogonal polynomials and special functions: a symmetric generalization of Sturm-Liouville problems and its consequences, Department of Mathematics, University of Kassel, Ph.D. thesis, 2006.
[24]	W. M. Abd-Elhameed, Y. H. Youssri, Fifth-kind orthonormal Chebyshev polynomial solutions for fractional differential equations, Comput. Appl. Math., 37 (2018), 2897–2921. http://dx.doi.org/10.1007/s40314-017-0488-z doi: 10.1007/s40314-017-0488-z
[25]	W. M. Abd-Elhameed, Y. H. Youssri, Sixth-kind Chebyshev spectral approach for solving fractional differential equations, Int. J. Nonlin. Sci. Num., 20 (2019), 191–203. http://dx.doi.org/10.1515/ijnsns-2018-0118 doi: 10.1515/ijnsns-2018-0118
[26]	W. M. Abd-Elhameed, Y. H. Youssri, New formulas of the high-order derivatives of fifth–kind Chebyshev polynomials: Spectral solution of the convection-diffusion equation, Numer. Method. Part. D. E., 2021 (2021), 1–17. http://dx.doi.org/10.1002/num.22756 doi: 10.1002/num.22756
[27]	K. Sadri, K. Hosseini, D. Baleanu, A. Ahmadian, S. Salahshour, Bivariate Chebyshev polynomials of the fifth kind for variable-order time-fractional partial integro-differential equations with weakly singular kernel, Adv. Differ. Equ-NY, 2021 (2021), 1–26. http://dx.doi.org/10.1186/s13662-021-03507-5 doi: 10.1186/s13662-021-03507-5
[28]	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., (just-accepted) (2021), 1–27. http://dx.doi.org/10.1080/00207160.2021.1940977
[29]	A. G. Atta, W. M. Abd-Elhameed, G. M. Moatimid, Y. H. Youssri, Shifted fifth-kind Chebyshev Galerkin treatment for linear hyperbolic first-order partial differential equations, Appl. Numer. Math., 167 (2021), 237–256. http://dx.doi.org/10.1016/j.apnum.2021.05.010 doi: 10.1016/j.apnum.2021.05.010
[30]	W. M. Abd-Elhameed, Y. H. Youssri, Neoteric formulas of the monic orthogonal Chebyshev polynomials of the sixth-kind involving moments and linearization formulas, Adv. Differ. Equ-NY, 2021 (2021), 1–19. http://dx.doi.org/10.1186/s13662-021-03244-9 doi: 10.1186/s13662-021-03244-9
[31]	R. Hilfer, P. L. Butzer, U. Westphal, An introduction to fractional calculus. Appl. Fract. Calc. Phys., World Scientific (2010), 1–85.
[32]	Y. Luchko, J. Trujillo, Caputo-type modification of the Erdélyi-Kober fractional derivative, Fract. Calc. Appl. Anal., 10 (2007), 249–267.
[33]	C. Li, D. Qian, Y. Chen, On Riemann-Liouville and caputo derivatives, Discrete Dyn. Nat. Soc., 2011 (2011). https://doi.org/10.1155/2011/562494
[34]	K. K. Ali, M. A. Abd El Salam, E. M. Mohamed, B. Samet, S. Kumar, M. S. Osman, Numerical solution for generalized nonlinear fractional integro-differential equations with linear functional arguments using Chebyshev series, Adv. Differ. Equ-NY, 2020 (2020), 1–23. http://dx.doi.org/10.1186/s13662-020-02951-z doi: 10.1186/s13662-020-02951-z
[35]	R. M. Ganji, H. Jafari, D. Baleanu, A new approach for solving multi variable orders differential equations with Mittag-Leffler kernel, Chaos, Solitons & Fractals, 130 (2020), 109405. http://dx.doi.org/10.1016/j.chaos.2019.109405 doi: 10.1016/j.chaos.2019.109405
[36]	N. H. Sweilam, A. M. Nagy, A. A. El-Sayed, Second kind shifted Chebyshev polynomials for solving space fractional order diffusion equation, Chaos, Solitons & Fractals, 73 (2015), 141–147. http://dx.doi.org/10.1016/j.chaos.2015.01.010 doi: 10.1016/j.chaos.2015.01.010
[37]	N. H. Sweilam, A. M. Nagy, A. A. El-Sayed, On the numerical solution of space fractional order diffusion equation via shifted Chebyshev polynomials of the third kind, J. King Saud Univ. Sci., 28 (2016), 41–47. http://dx.doi.org/10.1016/j.jksus.2015.05.002 doi: 10.1016/j.jksus.2015.05.002
[38]	M. M. Khader, On the numerical solutions for the fractional diffusion equation, Commun. Nonlinear Sci., 16 (2011), 2535–2542. http://dx.doi.org/10.1016/j.cnsns.2010.09.007 doi: 10.1016/j.cnsns.2010.09.007
[39]	M. A. Ramadan, M. A. Abd El Salam, Spectral collocation method for solving continuous population models for single and interacting species by means of exponential Chebyshev approximation, Int. J. Biomath., 11 (2018), 1850109. http://dx.doi.org/10.1142/S1793524518501097 doi: 10.1142/S1793524518501097
[40]	P. Agarwal, A. A. El-Sayed, Vieta-Lucas polynomials for solving a fractional-order mathematical physics model, Adv. Differ. Equ-NY, 2020 (2020), 1–18. http://dx.doi.org/10.1186/s13662-020-03085-y doi: 10.1186/s13662-020-03085-y
[41]	M. M. Khader, N. H. Sweilam, Approximate solutions for the fractional advection-dispersion equation using Legendre pseudo-spectral method, Comput. Appl. Math., 33 (2014), 739–750. http://dx.doi.org/10.1007/s40314-013-0091-x doi: 10.1007/s40314-013-0091-x
[42]	V. Saw, S. Kumar, Fourth kind shifted Chebyshev polynomials for solving space fractional order advection-dispersion equation based on collocation method and finite difference approximation, International Journal of Applied and Computational Mathematics, 4 (2018), 1–17. http://dx.doi.org/10.1007/s40819-018-0517-7 doi: 10.1007/s40819-018-0517-7
[43]	V. Saw, S. Kumar, Second kind Chebyshev polynomials for solving space fractional advection-dispersion equation using collocation method, Iran. J. Sci. Technol. Trans. Sci., 43 (2019), 1027–1037. http://dx.doi.org/10.1007/s40995-018-0480-5 doi: 10.1007/s40995-018-0480-5

Reader Comments

Your name:*

Email:*
© 2022 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)