Analytical solutions for fractional Navier–Stokes equation using residual power series with $ \mathit{\phi } $-Caputo generalized fractional derivative

Omar Barkat; Awatif Muflih Alqahtani; Omar Barkat; Awatif Muflih Alqahtani

doi:10.3934/math.2025694

AIMS Mathematics

2025, Volume 10, Issue 7: 15476-15496. doi: 10.3934/math.2025694

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Research article

Analytical solutions for fractional Navier–Stokes equation using residual power series with $ \mathit{\phi } $-Caputo generalized fractional derivative

Omar Barkat ^{1,2
,
,},
Awatif Muflih Alqahtani ³

1.
Department of Mathematics, University Center of Barika, Algeria
2.
Laboratory of Science for Mathematics, Computer science and Engineering applications
3.
Department of Mathematics, Shaqra University, Riyadh 11972, Saudi Arabia

Received: 25 May 2025 Revised: 20 June 2025 Accepted: 27 June 2025 Published: 07 July 2025
MSC : 26A33, 40C15, 76D05

In this study, we aimed to derive analytical solutions for a system of nonlinear time-fractional Navier–Stokes equations in Cartesian coordinates by employing the residual power series method. Moreover, we showed that the $ \phi $-Caputo fractional derivative describes these equations in time, enabling the Riemann-Liouville, Hadamard, and Katugampola fractional derivatives to be generalized into a unified form. Additionally, we provide results for certain cases that are given in the literature. Therefore, the solutions obtained for the time-fractional Navier–Stokes equations are presented graphically in the Caputo–Hadamard sense.
Citation: Omar Barkat, Awatif Muflih Alqahtani. Analytical solutions for fractional Navier–Stokes equation using residual power series with $ \mathit{\phi } $-Caputo generalized fractional derivative[J]. AIMS Mathematics, 2025, 10(7): 15476-15496. doi: 10.3934/math.2025694

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Abstract

In this study, we aimed to derive analytical solutions for a system of nonlinear time-fractional Navier–Stokes equations in Cartesian coordinates by employing the residual power series method. Moreover, we showed that the $ \phi $-Caputo fractional derivative describes these equations in time, enabling the Riemann-Liouville, Hadamard, and Katugampola fractional derivatives to be generalized into a unified form. Additionally, we provide results for certain cases that are given in the literature. Therefore, the solutions obtained for the time-fractional Navier–Stokes equations are presented graphically in the Caputo–Hadamard sense.

References

[1]	R. Herrmann, Fractional calculus: an introduction for physicists, Singapore: World Scientific Publishing Company, 2011. https://doi.org/10.1142/8072
[2]	V. S. Kiryakova, Generalized fractional calculus and applications, Pitman research notes in mathematics series, Longman Scientific & Technical, Harlow, 1994.
[3]	K. S. Miller, B. Ross, An introduction to the fractional calculus and fractional differential equations, John Willey & Sons, 1993.
[4]	A. A. Kilbas, H. M. Srivastava, J. J. Trujillo, Theory and applications of fractional differential equations, Vol. 204, Elsevier, 2006.
[5]	I. Podlubny, Fractional differential equations calculus, an introduction to fractional derivatives, fractional differential equations, to methods of their solution and some of their applications, Vol. 198, New, York: Academic, Press, 1999.
[6]	A. M. Alqahtani, H. Mihoubi, Y. Arioua, B. Bouderah, Analytical solutions of time-fractional Navier–Stokes equations employing homotopy Perturbation–Laplace transform method, Fractal Fract., 9 (2024), 23. https://doi.org/10.3390/fractalfract9010023 doi: 10.3390/fractalfract9010023
[7]	J. Zuo, C. Liu, C. Vetro, Normalized solutions to the fractional Schrödinger equation with potential, Mediterr. J. Math., 20 (2023), 216. https://doi.org/10.1007/s00009-023-02422-1 doi: 10.1007/s00009-023-02422-1
[8]	J. Zuo, T. An, G. Ye, Z. Qiao, Nonhomogeneous fractional p-Kirchhoff problems involving a critical nonlinearity, Electron. J. Qual. Theory Differ. Eq., 41 (2019), 1–15. https://doi.org/10.14232/ejqtde.2019.1.41 doi: 10.14232/ejqtde.2019.1.41
[9]	T. Öziş, D. Ağırseven, He's homotopy perturbation method for solving heat-like and wave-like equations with variable coefficients, Phys. Lett. A, 372 (2008), 5944–5950. https://doi.org/10.1016/j.physleta.2008.07.060 doi: 10.1016/j.physleta.2008.07.060
[10]	R. M. Jena, S. Chakraverty, Solving time-fractional Navier–Stokes equations using homotopy perturbation Elzaki transform, SN. Appl. Sci., 1 (2019), 16. https://doi.org/10.1007/s42452-018-0016-9 doi: 10.1007/s42452-018-0016-9
[11]	S. J. Liao, Beyond perturbation: introduction to the homotopy analysis method, 1 Ed., New York: Chapman and Hall/CRC Press, 2003. https://doi.org/10.1201/9780203491164
[12]	G. Bakicierler, S. Alfaqeih, E. Mısırlı, Application of the modified simple equation method for solving two nonlinear time-fractional long water wave equations, Rev. Mex. Fís., 67 (2021), 1–7. https://doi.org/10.31349/revmexfis.67.060701 doi: 10.31349/revmexfis.67.060701
[13]	R. Abazari, B. Soltanalizadeh, Reduced differential transform method and its application on Kawahara equations, Thai J. Math., 11 (2000), 199–216.
[14]	Y. Keskin, G. Oturanc, The Reduced differential transform method: a new approach to fractional partial differential equations, Nonlinear Sci. Lett. A, 1 (2010), 207–217.
[15]	O. A. Arqub, Series solution of fuzzy differential equations under strongly generalized differentiability, J. Adv. Res. Appl. Math., 5 (2013), 31–52.
[16]	K. K. Jaber, R. S. Ahmad, Analytical solution of the time fractional Navier–Stokes equation, Ain Shams Eng. J., 9 (2018), 1917–1927. https://doi.org/10.1016/j.asej.2016.08.021 doi: 10.1016/j.asej.2016.08.021
[17]	A. Arafa, G. Elmahdy, Application of residual power series method to fractional coupled physical equations arising in fluids flow, Int. J. Differ. Equ., 10 (2018), 7692849. https://doi.org/10.1155/2018/7692849 doi: 10.1155/2018/7692849
[18]	T. Ogawa, S. V. Rajopadhye, M. E. Schooner, Energy decay for a weak solution of the Navier–Stokes equation with slowly varying external forces, Energy J. Funct. Anal., 144 (1997), 325–358. https://doi.org/10.1006/jfan.1996.3011 doi: 10.1006/jfan.1996.3011
[19]	E. S. Baranovskii, Exact solutions for non-isothermal flows of second grade fluid between parallel plates, Nanomaterials, 13 (2023), 1409. https://doi.org/10.3390/nano13081409 doi: 10.3390/nano13081409
[20]	E. S. Baranovskii, Analytical unsteady poiseuille flow of a second grade fluid with slip boundary conditions, Polymers, 16 (2024), 179. https://doi.org/10.3390/polym16020179 doi: 10.3390/polym16020179
[21]	M. El-Shahed, A. Salem, On the generalized Navier–Stokes equations, Appl. Math. Comput., 156 (2004), 287–293. https://doi.org/10.1016/j.amc.2003.07.022 doi: 10.1016/j.amc.2003.07.022
[22]	J. Zhang, J. Wang, Numerical analysis for Navier–Stokes equations with time fractional derivatives, Appl. Math. Comput., 336 (2018), 481–489. https://doi.org/10.1016/j.amc.2018.04.036 doi: 10.1016/j.amc.2018.04.036
[23]	W. Sawangtong, P. Dunnimit, B. Wiwatanapataphe, P. Sawangtong, An analytical solution to the time fractional Navier–Stokes equation based on the Katugampola derivative in Caputo sense by the generalized Shehu residual power series approach, Partial Differ. Equ. Appl. Math., 11 (2024), 100890. https://doi.org/10.1016/j.padiff.2024.100890 doi: 10.1016/j.padiff.2024.100890
[24]	T. M. Atanackovic, S. Pilipovic, B. Stankovic, D. Zorica, Fractional calculus with applications in mechanics, vibrations and diffusion processes, John Wiley & Sons, Inc., 2014. https://doi.org/10.1002/9781118577530
[25]	C. G. Eliana, E. C. De Oliveira, Fractional versions of the fundamental theorem of calculus, Appl. Math., 4 (2013), 23–33. https://doi.org/10.4236/am.2013.47A006 doi: 10.4236/am.2013.47A006
[26]	S. R. Bistafa, On the development of the Navier-Stokes equation by Navier, Rev. Bras. Ensino Fís., 40 (2017), e2603. https://doi.org/10.1590/1806-9126-rbef-2017-0239 doi: 10.1590/1806-9126-rbef-2017-0239
[27]	V. B. L. Chaurasia, D. Kumar, Solution of the time-fractional Navier–Stokes equation, Gen. Math. Notes, 4 (2011), 49–59.
[28]	C. Dou, Z. Zhao, Analytical solution to 1D compressible Navier-Stokes equations, J. Func. Spaces, 2021 (2021), 6. https://doi.org/10.1155/2021/6339203 doi: 10.1155/2021/6339203
[29]	H. Mihoubi, A. M. Alqahtani, Y. Arioua, B. Bouderah, T. Tayebi, Homotopy perturbation ρ- Laplace transform approach for numerical simulation of fractional Navier–Stokes equations, Contemp. Math., 6 (2025), 1–29. https://doi.org/10.37256/cm.6320255978 doi: 10.37256/cm.6320255978
[30]	F. Jarad, T. Abdeljawad, D. Baleanu, Caputo-type modification of the Hadamard fractional derivatives, Adv. Differ. Equ., 142 (2012), 8. https://doi.org/10.1186/1687-1847-2012-142 doi: 10.1186/1687-1847-2012-142
[31]	Y. Gambo, F. Jarad, D. Baleanu, T. Abdeljawad, On Caputo modification of the Hadamard fractional derivatives, Adv. Differ. Equ., 2014 (2014), 10. https://doi.org/10.1186/1687-1847-2014-10 doi: 10.1186/1687-1847-2014-10
[32]	A. M. Alqahtani, H. Mihoubi, Analytical solutions for fractional Black-Scholes European option pricing equation by using Homotopy perturbation method with Caputo fractional derivative, Math. Model. Eng. Prob., 12 (2025), 1562–1570. https://doi.org/10.18280/mmep.120510 doi: 10.18280/mmep.120510
[33]	R. Almeida, A Caputo fractional derivative of a function with respect to another function, Commun. Nonlinear Sci. Numer. Simul., 44 (2017), 460–481. https://doi.org/10.1016/j.cnsns.2016.09.006 doi: 10.1016/j.cnsns.2016.09.006
[34]	S. G. Samko, A. A. Kilbas, O. I. Marichev, Fractional integrals and derivatives: theory and applications, Gordon and Breach Science Publishers, 1993.
[35]	A. M. Alqahtani, A. Shukla, Computational analysis of multi-layered Navier–Stokes system by Atangana–Baleanu derivative, Appl. Math. Sci. Eng., 32 (2024), 2290723. https://doi.org/10.1080/27690911.2023.2290723 doi: 10.1080/27690911.2023.2290723
[36]	R. Company, L. Jódar, J. R. Pintos, A numerical method for European Option Pricing with transaction costs nonlinear equation, Math. Comput. Model., 50 (2009), 910–920. https://doi.org/10.1016/j.mcm.2009.05.019 doi: 10.1016/j.mcm.2009.05.019
[37]	R. K. Gazizov, N. H. Ibragimov, Lie symmetry analysis of differential equations in finance, Nonlinear Dyn., 17(1998), 387–407. https://doi.org/10.1023/A:1008304132308 doi: 10.1023/A:1008304132308
[38]	P. Amster, C. G. Averbuj, M. C. Mariani, Stationary solutions for two nonlinear Black–Scholes type equations, Appl. Numer. Math., 47 (2003), 275–280. https://doi.org/10.1016/S0168-9274(03)00070-9 doi: 10.1016/S0168-9274(03)00070-9
[39]	R. M. Jena, S. Chakraverty, Solving time-fractional Navier–Stokes equations using homotopy perturbation Elzaki transform, SN. Appl. Sci., 1 (2019), 16. https://doi.org/10.1007/s42452-018-0016-9 doi: 10.1007/s42452-018-0016-9

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