Meshfree numerical approach for some time-space dependent order partial differential equations in porous media

Abdul Samad; Imran Siddique; Zareen A. Khan; Abdul Samad; Imran Siddique; Zareen A. Khan

doi:10.3934/math.2023665

AIMS Mathematics

2023, Volume 8, Issue 6: 13162-13180. doi: 10.3934/math.2023665

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Meshfree numerical approach for some time-space dependent order partial differential equations in porous media

1.
School of Mathematics, Northwest University, Xi'an 710127, China
2.
Department of Mathematics, University of Management and Technology, Lahore 54770, Pakistan
3.
Department of Mathematical Sciences, College of Sciences, Princess Nourah bint Abdulrahman University, P.O. Box 84428, Riyadh 11671, Saudi Arabia

Received: 07 January 2023 Revised: 12 March 2023 Accepted: 17 March 2023 Published: 03 April 2023
MSC : 35G31, 35G35, 65D12

In this article, the meshfree radial basis function method based on the Gaussian function is proposed for some time-space dependent fractional order partial differential equation (PDE) models. These PDE models have significant applications in chemical engineering and physical science. Some main advantages of the proposed method are that it is easy to implement, and the output response is quick and highly accurate, especially in the higher dimension. In this method, the time-dependent derivative terms are treated by Caputo fractional derivative while space-dependent derivative terms are treated by Riesz, Riemann-Liouville, and Grünwald-Letnikov derivatives. The proposed method is tested on some numerical examples and the accuracy is analyzed by $ \|L\|_\infty $.
- advection-dispersion PDEs,
- Caputo derivatives,
- Grünwald-Letnikov derivatives,
- Riesz fractional derivatives,
- Riemann-Liouville fractional derivatives
Citation: Abdul Samad, Imran Siddique, Zareen A. Khan. Meshfree numerical approach for some time-space dependent order partial differential equations in porous media[J]. AIMS Mathematics, 2023, 8(6): 13162-13180. doi: 10.3934/math.2023665

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Abstract

In this article, the meshfree radial basis function method based on the Gaussian function is proposed for some time-space dependent fractional order partial differential equation (PDE) models. These PDE models have significant applications in chemical engineering and physical science. Some main advantages of the proposed method are that it is easy to implement, and the output response is quick and highly accurate, especially in the higher dimension. In this method, the time-dependent derivative terms are treated by Caputo fractional derivative while space-dependent derivative terms are treated by Riesz, Riemann-Liouville, and Grünwald-Letnikov derivatives. The proposed method is tested on some numerical examples and the accuracy is analyzed by $ \|L\|_\infty $.

References

[1]	M. M. Meerschaert, D. A. Benson, B. Bäumer, Multidimensional advection and fractional dispersion, Phys. Rev. E, 59 (1999), 5026. https://doi.org/10.1103/physreve.59.5026 doi: 10.1103/physreve.59.5026
[2]	J. Bear, Dynamics of fluids in porous media, New York: American Elsevier Publishing Company, 1972.
[3]	R. Schumer, D. A. Benson, M. M. Meerschaert, B. Baeumer, Fractal mobile/immobile solute transport, Water Resour. Res., 39 (2003), 1296. https://doi.org/10.1029/2003WR002141 doi: 10.1029/2003WR002141
[4]	K. H. Coats, B. D. Smith, Dead-end pore volume and dispersion in porous media, SPE J., 4 (1964), 73–84. https://doi.org/10.2118/647-PA doi: 10.2118/647-PA
[5]	F. Bauget, M. Fourar, Non-fickian dispersion in a single fracture, J. Contam. Hydrol., 100 (2008), 137–148. https://doi.org/10.1016/j.jconhyd.2008.06.005 doi: 10.1016/j.jconhyd.2008.06.005
[6]	B. Berkowitz, Charaterizing flow and transport in fractured geological media: a review, Adv. Water Resour., 25 (2002), 861–884. https://doi.org/10.1016/S0309-1708(02)00042-8 doi: 10.1016/S0309-1708(02)00042-8
[7]	H. Scher, M. Lax, Stochastic transport in a disordered solid. I. theory, Phys. Rev. B, 7 (1973), 4491. https://doi.org/10.1103/PhysRevB.7.4491 doi: 10.1103/PhysRevB.7.4491
[8]	I. Podulbny, Fractional differential equations, Academic Press, 1998.
[9]	K. S. Miller, B. Ross, An introduction to the fractional calculus and fractional differential equations, New York: Wiley, 1993.
[10]	D. Baleanu, J. A. T. Machado, A. C. J. Luo, Fractional dynamics and control, New York: Springer, 2012. https://doi.org/10.1007/978-1-4614-0457-6
[11]	V. E. Tarasov, Fractional dynamics: applications of fractional calculus to dynamics of particles, fields and media, Berlin, Heidelberg: Springer, 2010.
[12]	R. Hilfer, Applications of fractional calculus in physics, Singapore: World Scientific, 2000. https://doi.org/10.1142/3779
[13]	M. A. Abdelkawy, M. A. Zaky, A. H. Bhrawy, D. Baleanu, Numerical simulation of time variable fractional order mobile-immobile advection-dispersion model, Rom. Rep. Phys., 67 (2015), 773–791.
[14]	B. Y. Wang, J. Y. Zhang, G. W. Yan, Numerical simulation of the fractional dispersion advection equations based on the lattice Boltzmann model, Math. Probl. Eng., 2020 (2020), 2570252. https://doi.org/10.1155/2020/2570252 doi: 10.1155/2020/2570252
[15]	I. I. Gorial, A reliable algorithm for multi-dimensional mobile/immobile advection-dispersion equation with variable order fractional, Indian J. Sci. Technol., 11 (2018), 1–9. https://doi.org/10.17485/ijst/2018/v11i30/127486 doi: 10.17485/ijst/2018/v11i30/127486
[16]	B. Yu, X. Y. Jiang, H. T. Qi, Numerical method for the estimation of the fractional parameters in the fractional mobile/immobile advection-diffusion model, Int. J. Comput. Math., 95 (2018), 1131–1150. https://doi.org/10.1080/00207160.2017.1378811 doi: 10.1080/00207160.2017.1378811
[17]	H. Pourbashash, D. Baleanu, M. M. A. Qurashi, On solving fractional mobile/immobile equation, Adv. Mech. Eng., 9 (2017), 1–12.
[18]	P. D. Lax, R. D. Richtmyer, Survey of the stability of linear finite difference equations, Commun. Pure Appl. Math., 9 (1956), 267–293. https://doi.org/10.1002/cpa.3160090206 doi: 10.1002/cpa.3160090206
[19]	O. Nikan, Z. Avazzadeh, J. A. T. Machado, Numerical approach for modeling fractional heat conduction in porous medium with the generalized Cattaneo model, Appl. Math. Model., 100 (2021), 107–124. https://doi.org/10.1016/j.apm.2021.07.025 doi: 10.1016/j.apm.2021.07.025
[20]	O. Nikan, Z. Avazzadeh, Numerical simulation of fractional evolution model arising in viscoelastic mechanics, Appl. Numer. Math., 169 (2021), 303–320. https://doi.org/10.1016/j.apnum.2021.07.008 doi: 10.1016/j.apnum.2021.07.008
[21]	X. T. Liu, H. G. Sun, Y. Zhang, Z. J. Fu, A scale-dependent finite difference approximation for time fractional differential equation, Comput. Mech., 63 (2019), 429–442. https://doi.org/10.1007/s00466-018-1601-x doi: 10.1007/s00466-018-1601-x
[22]	Z. C. Tang, Z. J. Fu, H. G. Sun, X. T. Liu, An efficient localized collocation solver for anomalous diffusion on surfaces, Fract. Calc. Appl. Anal., 24 (2021), 865–894. https://doi.org/10.1515/fca-2021-0037 doi: 10.1515/fca-2021-0037
[23]	Z. J. Fu, L. W. Yang, Q. Xi, C. S. Liu, A boundary collocation method for anomalous heat conduction analysis in functionally graded materials, Comput. Math. Appl., 88 (2021), 91–109. https://doi.org/10.1016/j.camwa.2020.02.023 doi: 10.1016/j.camwa.2020.02.023
[24]	H. Xu, S. J. Liao, X. C. You, Analysis of nonlinear fractional partial differential equations with the homotopy analysis method, Commun. Nonlinear Sci. Numer. Simul., 14 (2009), 1152–1156. https://doi.org/10.1016/j.cnsns.2008.04.008 doi: 10.1016/j.cnsns.2008.04.008
[25]	A. A. Ragab, K. M. Hemida, M. S. Mohamed, M. A. A. E. Salam, Solution of time-fractional Navier-Stokes equation by using homotopy analysis method, Gen. Math. Notes, 13 (2012), 13–21.
[26]	S. Chen, F. Liu, P. Zhuang, V. Anh, Finite difference approximations for the fractional Fokker-Planck equation, Appl. Math. Model., 33 (2009), 256–273. https://doi.org/10.1016/j.apm.2007.11.005 doi: 10.1016/j.apm.2007.11.005
[27]	Y. M. Lin, C. J. Xu, Finite difference/spectral approximations for the time-fractional diffusion equation, J. Comput. Phys., 225 (2007), 1533–1552. https://doi.org/10.1016/j.jcp.2007.02.001 doi: 10.1016/j.jcp.2007.02.001
[28]	C. P. Li, Y. H. Wang, Numerical algorithm based on adomian decomposition for fractional differential equations, Comput. Math. Appl., 57 (2009), 1672–1681. https://doi.org/10.1016/j.camwa.2009.03.079 doi: 10.1016/j.camwa.2009.03.079
[29]	H. Fatoorehchi, R. Rach, H. Sakhaeinia, Explicit Frost-Kalkwarf type equations for calculation of vapour pressure of liquids from triple to critical point by the adomian decomposition method, Can. J. Chem. Eng., 95 (2017), 2199–2208. https://doi.org/10.1002/cjce.22853 doi: 10.1002/cjce.22853
[30]	A. Samad, J. Muhammad, Meshfree collocation method for higher order KdV equations, J. Appl. Comput. Mech., 7 (2021), 422–431. https://doi.org/10.22055/JACM.2020.34874.2493 doi: 10.22055/JACM.2020.34874.2493
[31]	P. Thounthong, M. N. Khan, I. Hussain, I. Ahmad, P. Kumam, Symmetric radial basis function method for simulation of elliptic partial differential equations, Mathematics, 6 (2018), 327. https://doi.org/10.3390/math6120327 doi: 10.3390/math6120327
[32]	G. E. Fasshauer, Meshfree approximation methods with Matlab, Word Scientific, 2007. https://doi.org/10.1142/6437
[33]	V. R. Hosseini, W. Chen, Z. Avazzadeh, Numerical solution of fractional telegraph equation by using radial basis functions, Eng. Anal. Bound. Elem., 38 (2014), 31–39. https://doi.org/10.1016/j.enganabound.2013.10.009 doi: 10.1016/j.enganabound.2013.10.009
[34]	A. Samad, I. Siddique, F. Jarad, Meshfree numerical integration for some challenging multi-term fractional order PDEs, AIMS Math., 7 (2022), 14249–14269. https://doi.org/10.3934/math.2022785 doi: 10.3934/math.2022785
[35]	F. Z. Wang, K. H. Zheng, I. Ahmad, H. Ahmad, Gaussian radial basis functions method for linear and nonlinear convection-diffusion models in physical phenomena, Open Phys., 19 (2021), 69–76. https://doi.org/10.1515/phys-2021-0011 doi: 10.1515/phys-2021-0011
[36]	F. Z. Wang, I. Ahmad, H. Ahmad, M. D. Alsulami, K. S. Alimgeer, C. Cesarano, et al., Meshless method based on RBFs for solving three-dimensional multi-term time fractional PDEs arising in engineering phenomenons, J. King Saud Univ. Sci., 33 (2021), 101604. https://doi.org/10.1016/j.jksus.2021.101604 doi: 10.1016/j.jksus.2021.101604
[37]	M. N. Khan, I. Ahmad, H. Ahmad, A radial basis function collocation method for space-dependent inverse heat problems, J. Appl. Comput. Mech., 6 (2020), 1187–1199.
[38]	A. Ali, S. Islam, S. Haq, A computational meshfree technique for the numerical solution of the two dimensional coupled Burgers' equations, Int. J. Comput. Methods Eng. Sci. Mech., 10 (2009), 406–422. https://doi.org/10.1080/15502280903108016 doi: 10.1080/15502280903108016
[39]	C. F. M. Coimbra, Mechanica with variable-order differential operators, Ann. Phys., 12 (2003), 692–703.
[40]	H. Jiang, F. Liu, I. Turner, K. Burrage, Analytical solutions for the multi-term time-space Caputo-Riesz fractional advection-diffusion equations on a finite domain, J. Math. Anal. Appl., 389 (2012), 1117–1127. https://doi.org/10.1016/j.jmaa.2011.12.055 doi: 10.1016/j.jmaa.2011.12.055

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