Uyumlu Kesirli Navier-Stokes Denkleminin Güçlü Yöntemle Yeni Sayısal Çözümleri

Year 2025, Volume: 14 Issue: 3, 1331 - 1347, 30.09.2025

Aslı Alkan , Hasan Bulut , Tolga Aktürk

https://doi.org/10.17798/bitlisfen.1595852

Abstract

Bu çalışmada, uyumlu kesirli Navier-Stokes denkleminin yeni sayısal çözümleri, uyumlu q-Shehu homotopi analiz dönüşümü yöntemi ile elde edilmiştir. Kesirli Navier-Stokes denklemi, hafıza etkileri ve anormal difüzyon gibi karmaşık akış davranışlarını hesaba katmak için kesirli türevleri içeren klasik Navier-Stokes denklemlerinin bir genişlemesidir. Özellikle Newton olmayan akışkanlar, gözenekli ortamlar ve karmaşık zaman veya mekansal bağımlılıklara sahip diğer sistemlerdeki akışkan dinamiğini tanımlamada faydalıdır. Ayrıca, elde edilen çözümlerin iki ve üç boyutlu grafikleri çizilmiştir. Ayrıca, şemanın doğruluğunu değerlendirmek için bir hata analizi yürütüyoruz. Yaklaşan yöntemin doğruluğunu doğrulamak için hesaplamalı simülasyonlar gerçekleştirilir. Bu makale, sayısal ve grafiksel analizden elde edilen sonuçları sunmaktadır.

Keywords

Navier-Stokes denklemi , Uyumlu q-Shehu homotopi analiz dönüşüm metodu , Shehu dönüşümü

The Novel Numerical Solutions of Conformable Fractional Navier-Stokes Equation with the Robust Method

Abstract

In this study, new numerical solutions of the conformable fractional Navier-Stokes equation are obtained by the conformable q-Shehu homotopy analysis transform method. The fractional Navier-Stokes equation is an extension of the classical Navier-Stokes equations that incorporates fractional derivatives to account for complex flow behaviors such as memory effects and anomalous diffusion. It is particularly useful in describing fluid dynamics in non-Newtonian fluids, porous media, and other systems with intricate time or spatial dependencies. In addition, two and three-dimensional graphs of the obtained solutions were drawn. We also conduct an error analysis to evaluate the accuracy of the scheme. Computational simulations are performed to validate the accuracy of the upcoming method. This paper presents the conclusions derived from the numerical and graphical analysis.

Keywords

Navier-Stokes equation , Conformable q-Shehu homotopy analysis transform method , Shehu transform

Ethical Statement

The study is complied with research and publication ethics.

References

• K. S. Miller, and B. Ross, An introduction to the fractional calculus and fractional differential equations, Wiley, New York, 1993, pp. 363.
• I. Podlubny, Fractional Differential Equations, Academic Press, New York, 1999, pp. 337.
• D. Baleanu, K. Diethelm, E. Scalas, and J. J. Trujillo, Fractional calculus: models and numerical methods, World Scientific, London, 2012, pp. 476.
• Y. Povstenko, Linear fractional diffusion-wave equation for scientists and engineers, Birkhäuser, Switzerland, 2015, pp. 460.
• D. Baleanu, G. C. Wu, and S. D. Zeng, “Chaos analysis and asymptotic stability of generalized Caputo fractional differential equations,” Chaos Soliton Fractals, vol. 102, pp. 99–105, 2017.
• N. H. Sweilam, M. M. Abou Hasan, D. Baleanu, “New studies for general fractional financial models of awareness and trial advertising decisions,” Chaos Soliton Fractals, vol. 104, pp. 772-784, 2017.
• D. Y. Liu, O. Gibaru, W. Perruquetti, T. M. Laleg-Kirati, “Fractional order differentiation by integration and error analysis in noisy environment,” IEEE Trans Automat, vol. 60, pp. 2945–2960, 2015.
• A. Esen, T. A. Sulaiman, H. Bulut, and H. M. Baskonus, “Optical solitons to the space-time fractional (1+1)-dimensional coupled nonlinear Schrödinger equation,” Optik, vol. 167, pp. 150–156, 2018.
• R. Caponetto, G. Dongola, L. Fortuna, and A. Gallo, “New results on the synthesis of FO-PID controllers,” Commun. Nonlinear Sci. Numer. Simul., vol. 15, pp. 997-1007, 2010.
• P. Veeresha, D. G. Prakasha, and H. M. Baskonus, “Novel simulations to the time-fractional Fisher’s equation,” Math. Sci., vol. 13, no. 1, pp. 33-42, 2019.
• R. Khalil, M. Al Horani, A. Yousef, and M. Sababheh, “A new definition of fractional derivative,” J. Comput. Appl. Math., vol. 264, pp. 65-70, 2014.
• T. Abdeljawad, “On conformable fractional calculus,” J. Comput. Appl. Math., vol. 279, pp. 57-66, 2015.
• A. Alkan, T. Aktürk, and H. Bulut, “ The Traveling Wave Solutions of the Conformable Time-Fractional Zoomeron Equation by Using the Modified Exponential Function Method,” Eskişehir Technical University Journal of Science and Technology A-Applied Sciences and Engineering, vol. 25, no. 1, pp. 108-114, 2024.
• T. Aktürk, A. Alkan, H. Bulut, and N. Güllüoğlu, “The Traveling Wave Solutions of Date–Jimbo–Kashiwara–Miwa Equation with Conformable Derivative Dependent on Time Parameter,” Ordu Üniversitesi Bilim ve Teknoloji Dergisi, vol. 14, no. 1, pp. 38-51, 2024.
• A. Kartal, H. Anaç, and A. Olgun, “Numerical solution of conformable time fractional generalized Burgers equation with proportional delay by new methods,” Karadeniz Fen Bilimleri Dergisi, vol. 13, no. 2, pp. 310-335, 2023.
• A. Kartal, H. Anaç, and A. Olgun, “The new numerical solutions of conformable time fractional generalized Burgers equation with proportional delay,” Gümüşhane Üniversitesi Fen Bilimleri Dergisi, vol. 13, no. 4, pp. 927-938, 2023.
• H. Poincaré, “Sur l’équilibre d’une masse fluide animée d’un mouvement de rotation,” Bulletin Astronomique, vol. 2, no.1, pp. 109-118, 1885.
• G. Adomian, “Analytical solution of Navier-Stokes flow of a viscous compressible fluid,” Found. Phys. Lett., vol. 8, pp. 389–400, 1995.
• M. Krasnoschok, V. Pata, S. V. Siryk, and N. Vasylyeva, “A subdiffusive Navier-Stokes-Voigt system,” Phys. D Nonlinear Phenom., vol. 409, pp. 132503, 2020.
• S. Momani, and Z. Odibat, “ Analytical solution of a time-fractional Navier–Stokes equation by Adomian decomposition method,” Applied Mathematics and Computation, vol. 177, no. 2, pp. 488-494, 2006.
• Q. Yu, J. Song, F. Liu, V. Anh, I. Turner, “An approximate solution for the Rayleigh-Stokes problem for a heated generalized second grade fluid with fractional derivative model using the Adomian decomposition method,” J. Algorithms Comput. Technol., vol. 3, pp. 553-571, 2009.
• M. Krasnoschok, V. Pata, S. V. Siryk, N. Vasylyeva, “Equivalent definitions of Caputo derivatives and applications to subdiffusion equations,” Dyn. PDE, vol. 17, pp. 383–402, 2020.
• E. Bazhlekova, B. Jin, R. Lazarov, Z. Zhou, “An analysis of the Rayleigh-Stokes problem for a generalized second-grade fluid,” Numer. Math., vol. 131, pp. 1–31, 2015.
• M. El-Shahed, and A. Salem, “On the generalized Navier-Stokes equations,” Appl. Math. Comput., vol. 156, pp. 287–293, 2004.
• D. Kumar, J. Singh, S. Kumar, “A fractional model of Navier-Stokes equation arising in unsteady flow of a viscous fluid,” J. Assoc. Arab. Univ. Basic Appl. Sci., vol. 17, pp. 14–19, 2015.
• Z. Z. Ganji, D. D. Ganji, A. D. Ganji, and M. Rostamian, “Analytical solution of time-fractional Navier-Stokes equation in polar coordinate by homotopy perturbation method,” Numerical Methods for Partial Differential Equations: An International Journal, vol. 26, pp. 117–124, 2010.
• A. A. Ragab, K. M. Hemida, M. S. Mohamed, and M. A. Abd El Salam, “Solution of time-fractional Navier-Stokes equation by using homotopy analysis method,” Gen. Math. Notes, vol. 13, pp. 13–21, 2012.
• S. Maitama, “Analytical solution of time-fractional Navier-Stokes equation by natural homotopy perturbation method,” Prog. Fract. Differ. Appl., vol. 4, pp. 123–131, 2018.
• G. A. Birajdar, “Numerical solution of time fractional Navier-Stokes equation by discrete Adomian decomposition method,” Nonlinear Eng., vol. 3, pp. 21–26, 2014.
• S. Kumar, S.; Kumar, D.; Abbasbandy, S.; Rashidi, M.M. Analytical solution of fractional Navier-Stokes equation by using modified Laplace decomposition method. Ain Shams Eng. J., 2014, 5, 569–574.
• V. B. L. Chaurasia, and D. Kumar, “Solution of the time-fractional Navier-Stokes equation,” Gen. Math. Notes, vol. 4, pp. 49–59, 2011.
• A. Prakash, P. Veeresha, D. G. Prakasha, and M. Goyal, “A new efficient technique for solving fractional coupled Navier-Stokes equations using q-homotopy analysis transform method,” Pramana, vol. 93, pp. 1–10, 2019.
• S. Mukhtar, R. Shah, and S. Noor, “The numerical investigation of a fractional-order multi-dimensional Model of Navier–Stokes equation via novel techniques,” Symmetry, vol. 14, no. 6, pp. 1102, 2022.
• M. A. El-Tawil, and S. N. Huseen, “The q-homotopy analysis method (q-HAM),” Int. J. Appl. Math. Mech, vol. 8, no. 15, pp. 51-75, 2012.
• S. Liao, On the homotopy analysis method for nonlinear problems,” Applied mathematics and computation, vol. 147, no. 2, pp. 499-513, 2004.
• S. Liao, “Comparison between the homotopy analysis method and homotopy perturbation method,” Applied Mathematics and Computation, vol. 169, no. 2, pp.1186-1194, 2005.
• S. J. Liao, “Homotopy analysis method and its applications in mathematics,” J. Basic Sci. Eng., vol. 5, no. 2, pp. 111–125, 1997.
• S. J. Liao, “Homotopy analysis method: a new analytic method for nonlinear problems,” Appl. Math. Mech., vol. 19, pp. 957–962, 1998.
• J. H. He, “Homotopy perturbation method: a new nonlinear analytical technique,” Applied Mathematics and computation, vol. 135, no. 1, pp. 73-79, 2003.
• J. H. He, “Addendum: new interpretation of homotopy perturbation method,” International journal of modern physics B, vol. 20, no. 18, pp. 2561-2568, 2006.
• S.A Khuri, “A Laplace decomposition algorithm applied to class of nonlinear differential equations,” J Math. Appl., vol. 4, pp. 141 – 155, 2001.
• S. A. Khuri, “A new approach to Bratu’s problem,” Applied mathematics and computation, vol. 147, no. 1, pp. 131-136, 2004.
• H. Anaç, “Conformable Fractional Elzaki Decomposition Method of Conformable Fractional Space-Time Fractional Telegraph Equations,” Ikonion Journal of Mathematics, vol. 4, no. 2, pp. 42-55, 2022.
• A. S. Erol, H. Anaç, and A. Olgun, “Numerical solutions of conformable time-fractional Swift-Hohenberg equation with proportional delay by the novel methods,” Karamanoğlu Mehmetbey Üniversitesi Mühendislik ve Doğa Bilimleri Dergisi, vol. 5, no. 1, pp. 1-24, 2023.
• M. E. Benattıa, and K. Belghaba, “Shehu conformable fractional transform, theories and applications,” Cankaya University Journal of Science and Engineering, vol. 18, no. 1, pp. 24-32, 2021.
• D. Kumar, J. Singh, and D. Baleanu, “A new analysis for fractional model of regularized long-wave equation arising in ion acoustic plasma waves,” Math. Methods Appl. Sci., vol. 40, no. 15, pp. 5642–5653, 2017.
• Á. A. Magreñán, “A new tool to study real dynamics: the convergence plane,” Appl. Math. Comput., vol. 248, pp. 215–224, 2014.
• H. A. Peker, and F. A. Cuha, “Application of Kashuri Fundo transform and homotopy perturbation methods to fractional heat transfer and porous media equations,” Thermal Science, vol. 26, no. 4A, pp. 2877-2884, 2022.
• D. R. Anderson, E. Camrud, and D. J. Ulness, “On the nature of the conformable derivative and its applications to physics,” J. Fract. Calc. Appl, vol. 10, no. 2, pp. 92-135, 2019.
• R. Kumar, R. Dharra, and S. Kumar, “Comparative qualitative analysis and numerical solution of conformable fractional derivative generalized KdV-mKdV equation,” International Journal of System Assurance Engineering and Management, vol. 14, no. 4, pp. 1247-1254, 2023.