Numerical solutions of space-fractional diffusion equations via the exponential decay kernel

Manal Alqhtani; Khaled M. Saad; Manal Alqhtani; Khaled M. Saad

doi:10.3934/math.2022364

AIMS Mathematics

2022, Volume 7, Issue 4: 6535-6549. doi: 10.3934/math.2022364

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Numerical solutions of space-fractional diffusion equations via the exponential decay kernel

Manal Alqhtani ,
Khaled M. Saad ^,

Department of Mathematics, College of Sciences and Arts, Najran University, Najran, Saudi Arabia

Received: 04 November 2021 Revised: 15 December 2021 Accepted: 04 January 2022 Published: 21 January 2022
MSC : 41A50, 65L12, 65N12, 65N35

The main object of this paper is to investigate the spectral collocation method for three new models of space fractional Fisher equations based on the exponential decay kernel, for which properties of Chebyshev polynomials are utilized to reduce these models to a set of differential equations. We then numerically solve these differential equations using finite differences, with the resulting algebraic equations solved using Newton 's method. The accuracy of the numerical solution is verified by computing the residual error function. Additionally, the numerical results are compared with other results obtained using the power law kernel and the Mittag-Leffler kernel. The advantage of the present work stems from the use of spectral methods, which have high accuracy and exponential convergence for problems with smooth solutions. The numerical solutions based on Chebyshev polynomials are in remarkably good agreement with numerical solutions obtained using the power law and the Mittag-Leffler kernels. Mathematica was used to obtain the numerical solutions.
- power law kernel,
- exponential decay kernel,
- Mittag-Leffler kernel,
- Chebyshev polynomials approximation,
- finite difference method,
- space fractional Fisher equations
Citation: Manal Alqhtani, Khaled M. Saad. Numerical solutions of space-fractional diffusion equations via the exponential decay kernel[J]. AIMS Mathematics, 2022, 7(4): 6535-6549. doi: 10.3934/math.2022364

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Abstract

The main object of this paper is to investigate the spectral collocation method for three new models of space fractional Fisher equations based on the exponential decay kernel, for which properties of Chebyshev polynomials are utilized to reduce these models to a set of differential equations. We then numerically solve these differential equations using finite differences, with the resulting algebraic equations solved using Newton 's method. The accuracy of the numerical solution is verified by computing the residual error function. Additionally, the numerical results are compared with other results obtained using the power law kernel and the Mittag-Leffler kernel. The advantage of the present work stems from the use of spectral methods, which have high accuracy and exponential convergence for problems with smooth solutions. The numerical solutions based on Chebyshev polynomials are in remarkably good agreement with numerical solutions obtained using the power law and the Mittag-Leffler kernels. Mathematica was used to obtain the numerical solutions.

References

[1]	A. A. Kilbas, H. M. Srivastava, J. J. Trujillo, Theory and Applications of Fractional Differential Equations, NorthHolland Mathematical Studies, Vol. 204, Elsevier (North-Holland) Science Publishers, Amsterdam, London and New York, 2006.
[2]	I. Podlubny, An Introduction to Fractional Derivatives, Fractional Differential Equations, to Methods of Their Solution and Some of Their Applications, Mathematics in Science and Engineering, Fractional Differential Equations, Vol. 198, Academic Press, New York, London, Sydney, Tokyo and Toronto, 1999.
[3]	O. Nikan, Z. Avazzadeh, J. A. Tenreiro Machadoc, Numerical study of the nonlinear anomalous reaction-subdiffusion process arising in the electroanalytical chemistry, J. Comput. Sci., 53 (2021), 101394. https://doi.org/10.1016/j.jocs.2021.101394 doi: 10.1016/j.jocs.2021.101394
[4]	Q. Rubbab, M. Nazeer, F. Ahmad, Y. M. Chu, M. Ijaz Khan, S. Kadry, Numerical simulation of advection-diffusion equation with caputo–fabrizio time fractional derivative in cylindrical domains: Applications of pseudo-spectral collocation method, Alexandria Eng. J., 60 (2021), 1731–1738. https://doi.org/10.1016/j.aej.2020.11.022 doi: 10.1016/j.aej.2020.11.022
[5]	O. Nikan, Z. Avazzadeh, J. A. Tenreiro Machadoc, A local stabilized approach for approximating the modified time-fractional diffusion problem arising in heat and mass transfer, J. Adv. Res., 32 (2021), 45–60. https://doi.org/10.1016/j.jare.2021.03.002 doi: 10.1016/j.jare.2021.03.002
[6]	Y. M. Chu, N. A. Shah, H. Ahmad, J. D. Chung, S. M. Khaled, A comparative Study of Semi-Analytical Methods for Solving Fractional-Order Cauchy Reaction Diffusion Equations, Fractals, 29, (2021), 2150143. https://doi.org/10.1142/S0218348X21501437
[7]	O. Nikan, Z. Avazzadeh, An improved localized radial basis-pseudospectral method for solving fractional reaction-subdiffusion problem, Res. Phys., 23 (2021), 104048. https://doi.org/10.1016/j.rinp.2021.104048 doi: 10.1016/j.rinp.2021.104048
[8]	M. Fardi, J. Alidousti, A Legendre spectral-finite difference method for Caputo-Fabrizio time-fractional distributed-order diffusion equation, Math. Sci., (2021). https://doi.org/10.1007/s40096-021-00430-4
[9]	M. M. Alsuyuti, E. H. Doha, S. S. Ezz-Eldien, I. K. Youssefe 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
[10]	H. M. Srivastava, A. K. N. Alomari, K. M. Saad, W. M. Hamanah, Some Dynamical Models Involving Fractional-Order Derivatives with the Mittag-Leffler Type Kernels and Their Applications Based upon the Legendre Spectral Collocation Method, Fractal Fract., 5 (2021), 131. https://doi.org/10.3390/fractalfract5030131 doi: 10.3390/fractalfract5030131
[11]	A. Atangana, J. f. Gómez-Aguilar, Numerical approximation of Riemann-Liouville definition of fractional derivative: from Riemann-Liouville to Atangana-Baleanu, Numer. Methods Partial Differ. Equ., 34 (2018), 1502–1523. https://doi.org/10.1002/num.22195 doi: 10.1002/num.22195
[12]	K. Hosseini, K. Sadri, M. Mirzazadeh, A. Ahmadian, Y. M. Chu, S. Salahshour, Reliable methods to look for analytical and numerical solutions of a nonlinear differential equation arising in heat transfer with the conformable derivative, Math. Methods Appl. Sci., (2021). https://doi.org/10.1002/mma.7582
[13]	O. Nikan, Z. Avazzadeh, J. A. Tenreiro Machado, Numerical approximation of the nonlinear time-fractional telegraph equation arising in neutron transport, Commun. Nonlinear Sci. Numer. Simul., 99 (2021), 105755. https://doi.org/10.1016/j.cnsns.2021.105755 doi: 10.1016/j.cnsns.2021.105755
[14]	T. U. Khan, M. A. Khan, Y. M. Chu, A new generalized Hilfer-type fractional derivative with applications to space-time diffusion equation, Res. Phys., 22 (2021), 103953. https://doi.org/10.1016/j.rinp.2021.103953 doi: 10.1016/j.rinp.2021.103953
[15]	C. Çelik, M. Duman, Crank-Nicolson method for the fractional diffusion equation with the Riesz fractional derivative, J. Comput. Phys., 231 (2012), 1743–1750. https://doi.org/10.1016/j.jcp.2011.11.008 doi: 10.1016/j.jcp.2011.11.008
[16]	M. Caputo, M. Fabrizio, A new definition of fractional derivative without singular kernel, Progr. Fract. Differ. Appl., 1 (2015), 73–85.
[17]	N. Al-Salti, E. Karimov, K. Sadarangani, On a differential equation with Caputo-Fabrizio fractional derivative of order $1 < \beta\leq2$ and application to mass-spring-damper system, Progr. Fract. Differ. Appl., 2 (2015), 257–263. https://doi.org/10.18576/pfda/020403 doi: 10.18576/pfda/020403
[18]	J. E. Macías-Díaz, A structure-preserving method for a class of nonlinear dissipative wave equations with Riesz space-fractional derivatives, J. Comput. Phys., 351 (2017), 40–58. https://doi.org/10.1016/j.jcp.2017.09.028 doi: 10.1016/j.jcp.2017.09.028
[19]	M. M. Khader, K. M. Saad, Numerical studies of the fractional Korteweg-de Vries, Korteweg–de Vries–Burgers and Burgers equations, Proc. Natl. Acad. Sci., India, Sect. A: Phys., 91 (2021), 67–77. https://doi.org/10.1007/s40010-020-00656-2 doi: 10.1007/s40010-020-00656-2
[20]	H. C. Yaslan, Numerical solution of fractional Riccati differential equation via shifted Chebyshev polynomials of the third kind, J. Eng. Technol. Appl. Sci., 2 (2017), 1–11. https://doi.org/10.30931/jetas.304377 doi: 10.30931/jetas.304377
[21]	J. R. Loh, A. Isah, C. Phang, Y. T. Toh, On the new properties of Caputo-Fabrizio operator and its application in deriving shifted Legendre operational matrix, Appl. Numer. Math., 132 (2018), 138–153. https://doi.org/10.1016/j.apnum.2018.05.016 doi: 10.1016/j.apnum.2018.05.016
[22]	T. Akram, M. Abbas, A. Ali, A. Iqbal, D. Baleanu, A Numerical Approach of a Time Fractional Reaction-Diffusion Model with a Non-Singular Kernel, Symmetry, 12 (2020), 1653. https://doi.org/10.3390/sym12101653 doi: 10.3390/sym12101653
[23]	S. Kumar, J. F. Gómez-Aguilar, Numerical Solution of Caputo-Fabrizio Time Fractional Distributed Order Reaction-diffusion Equation via Quasi Wavelet based Numerical Method, J. Appl. Comput. Mech., 6 (2020), 848–861.
[24]	K. M. Saad, New fractional derivative with non-singular kernel for deriving Legendre spectral collocation method, Alexandria Eng. J., 59 (2020), 1909–1917. https://doi.org/10.1016/j.aej.2019.11.017 doi: 10.1016/j.aej.2019.11.017
[25]	S. Rashid, K. T. Kubra, A. Rauf, Y. M. Chu, Y. S. Hamed, New numerical approach for time-fractional partial differential equations arising in physical system involving natural decomposition method, Phys. Scr., 96, (2021), 105204. https://doi.org/10.1088/1402-4896/ac0bce
[26]	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. https://doi.org/10.1016/j.chaos.2020.110174 doi: 10.1016/j.chaos.2020.110174
[27]	K. M. Saad, M. M. Khader, J. F. Gómez-Aguilar, D. Baleanu, Numerical solutions of the fractional Fishers type equations with Atangana-Baleanu fractional derivative by using spectral collocation methods, Chaos, 29 (2019), 023116. https://doi.org/10.1063/1.5086771 doi: 10.1063/1.5086771
[28]	M. M. Khader, K. M. Saad, Z. Hammouch, D. Baleanu, A spectral collocation method for solving fractional KdV and KdV-Burgers equations with non-singular kernel derivatives, Appl. Numer. Math., 161 (2021), 137–146. https://doi.org/10.1016/j.apnum.2020.10.024 doi: 10.1016/j.apnum.2020.10.024
[29]	K. M. Saad, Comparative study on Fractional Isothermal Chemical Model, Alexandria Eng. J., 60 (2021), 3265–3274. https://doi.org/10.1016/j.aej.2021.01.037 doi: 10.1016/j.aej.2021.01.037
[30]	M. A. Snyder, Chebyshev Methods in Numerical Approximation, Prentice-Hall, Inc. Englewood Cliffs, 1966.
[31]	A. H. Bhrawy, M. A. Alghamdi, Approximate solutions of Fishers type equations with variable coefficients, Abstr. Appl. Anal., 1 (2013), 1–16. https://doi.org/10.1155/2013/176730 doi: 10.1155/2013/176730
[32]	C. Tadjeran, M. M. Meerschaert, A second-order accurate numerical method for the two dimensional fractional diffusion equation, J. Comput. Phys., 220 (2007), 813–823. https://doi.org/10.1016/j.jcp.2006.05.030 doi: 10.1016/j.jcp.2006.05.030
[33]	N. J. Zabusky, M. D. Kruskal, Interaction of solitons in a collisionless plasma and the recurrence of initial states, Phys. Rev. Lett., 15 (1965), 240–243. https://doi.org/10.1103/PhysRevLett.15.240 doi: 10.1103/PhysRevLett.15.240

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