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Keywords = fractional Navier–Stokes equations

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25 pages, 8764 KiB  
Article
A Comprehensive Study on the Applications of NTIM and OAFM in Analyzing Fractional Navier–Stokes Equations
by Siddiq Ur Rehman, Rashid Nawaz, Faisal Zia and Nick Fewster-Young
Axioms 2025, 14(7), 521; https://doi.org/10.3390/axioms14070521 - 7 Jul 2025
Viewed by 218
Abstract
This article introduces two enhanced techniques: the Natural Transform Iterative Method (NTIM) and the Optimal Auxiliary Function Method (OAFM). These approaches provide a close approximation for solving fractional-order Navier–Stokes equations, which are widely employed in domains such as biology, ecology, and applied sciences. [...] Read more.
This article introduces two enhanced techniques: the Natural Transform Iterative Method (NTIM) and the Optimal Auxiliary Function Method (OAFM). These approaches provide a close approximation for solving fractional-order Navier–Stokes equations, which are widely employed in domains such as biology, ecology, and applied sciences. By comparing the solutions derived from these methods to exact solutions, it is clear that they provide accurate and efficient outcomes. These findings highlight the straightforward yet effective use of these methodologies in modeling engineering systems. Navier–Stokes equations have numerous practical uses, including analyzing fluid flow in pipelines and channels, predicting weather patterns, and constructing aircraft and vehicles. Full article
(This article belongs to the Special Issue Nonlinear Fractional Differential Equations: Theory and Applications)
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32 pages, 3446 KiB  
Article
Finite Element Method for Time-Fractional Navier–Stokes Equations with Nonlinear Damping
by Shahid Hussain, Xinlong Feng, Arafat Hussain and Ahmed Bakhet
Fractal Fract. 2025, 9(7), 445; https://doi.org/10.3390/fractalfract9070445 - 4 Jul 2025
Viewed by 425
Abstract
We propose a hybrid numerical framework for solving time-fractional Navier–Stokes equations with nonlinear damping. The method combines the finite difference L1 scheme for time discretization of the Caputo derivative (0<α<1) with mixed finite element methods (P1b–P1 and [...] Read more.
We propose a hybrid numerical framework for solving time-fractional Navier–Stokes equations with nonlinear damping. The method combines the finite difference L1 scheme for time discretization of the Caputo derivative (0<α<1) with mixed finite element methods (P1b–P1 and P2P1) for spatial discretization of velocity and pressure. This approach addresses the key challenges of fractional models, including nonlocality and memory effects, while maintaining stability in the presence of the nonlinear damping term γ|u|r2u, for r2. We prove unconditional stability for both semi-discrete and fully discrete schemes and derive optimal error estimates for the velocity and pressure components. Numerical experiments validate the theoretical results. Convergence tests using exact solutions, along with benchmark problems such as backward-facing channel flow and lid-driven cavity flow, confirm the accuracy and reliability of the method. The computed velocity contours and streamlines show close agreement with analytical expectations. This scheme is particularly effective for capturing anomalous diffusion in Newtonian and turbulent flows, and it offers a strong foundation for future extensions to viscoelastic and biological fluid models. Full article
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59 pages, 1417 KiB  
Article
Symmetrized Neural Network Operators in Fractional Calculus: Caputo Derivatives, Asymptotic Analysis, and the Voronovskaya–Santos–Sales Theorem
by Rômulo Damasclin Chaves dos Santos, Jorge Henrique de Oliveira Sales and Gislan Silveira Santos
Axioms 2025, 14(7), 510; https://doi.org/10.3390/axioms14070510 - 30 Jun 2025
Viewed by 277
Abstract
This work presents a comprehensive mathematical framework for symmetrized neural network operators operating under the paradigm of fractional calculus. By introducing a perturbed hyperbolic tangent activation, we construct a family of localized, symmetric, and positive kernel-like densities, which form the analytical backbone for [...] Read more.
This work presents a comprehensive mathematical framework for symmetrized neural network operators operating under the paradigm of fractional calculus. By introducing a perturbed hyperbolic tangent activation, we construct a family of localized, symmetric, and positive kernel-like densities, which form the analytical backbone for three classes of multivariate operators: quasi-interpolation, Kantorovich-type, and quadrature-type. A central theoretical contribution is the derivation of the Voronovskaya–Santos–Sales Theorem, which extends classical asymptotic expansions to the fractional domain, providing rigorous error bounds and normalized remainder terms governed by Caputo derivatives. The operators exhibit key properties such as partition of unity, exponential decay, and scaling invariance, which are essential for stable and accurate approximations in high-dimensional settings and systems governed by nonlocal dynamics. The theoretical framework is thoroughly validated through applications in signal processing and fractional fluid dynamics, including the formulation of nonlocal viscous models and fractional Navier–Stokes equations with memory effects. Numerical experiments demonstrate a relative error reduction of up to 92.5% when compared to classical quasi-interpolation operators, with observed convergence rates reaching On1.5 under Caputo derivatives, using parameters λ=3.5, q=1.8, and n=100. This synergy between neural operator theory, asymptotic analysis, and fractional calculus not only advances the theoretical landscape of function approximation but also provides practical computational tools for addressing complex physical systems characterized by long-range interactions and anomalous diffusion. Full article
(This article belongs to the Special Issue Advances in Fuzzy Logic and Computational Intelligence)
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13 pages, 2314 KiB  
Article
A Novel Approach to Solving Fractional Navier–Stokes Equations Using the Elzaki Transform
by Tarig M. Elzaki and Eltaib M. Abd Elmohmoud
Fractal Fract. 2025, 9(6), 396; https://doi.org/10.3390/fractalfract9060396 - 19 Jun 2025
Viewed by 530
Abstract
A clear method is provided to explain a new approach for solving systems of fractional Navier–Stokes equations (SFNSEs) using initial conditions (ICs) that rely on the Elzaki transform (ET). A few steps show the technique’s validity and utility for handling SFNSE solutions. For [...] Read more.
A clear method is provided to explain a new approach for solving systems of fractional Navier–Stokes equations (SFNSEs) using initial conditions (ICs) that rely on the Elzaki transform (ET). A few steps show the technique’s validity and utility for handling SFNSE solutions. For fractional derivatives, the Caputo sense is used. This method does not need discretization or limiting assumptions and may be used to solve both linear and nonlinear SFNSEs. By eliminating round-off mistakes, the technique reduces the need for numerical calculations. Using examples, the new technique’s accuracy and efficacy are illustrated. Full article
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11 pages, 288 KiB  
Article
Uniform Analyticity and Time Decay of Solutions to the 3D Fractional Rotating Magnetohydrodynamics System in Critical Sobolev Spaces
by Muhammad Zainul Abidin and Abid Khan
Fractal Fract. 2025, 9(6), 360; https://doi.org/10.3390/fractalfract9060360 - 29 May 2025
Viewed by 375
Abstract
In this paper, we investigated a three-dimensional incompressible fractional rotating magnetohydrodynamic (FrMHD) system by reformulating the Cauchy problem into its equivalent mild formulation and working in critical homogeneous Sobolev spaces. For this, we first established the existence and uniqueness of a global mild [...] Read more.
In this paper, we investigated a three-dimensional incompressible fractional rotating magnetohydrodynamic (FrMHD) system by reformulating the Cauchy problem into its equivalent mild formulation and working in critical homogeneous Sobolev spaces. For this, we first established the existence and uniqueness of a global mild solution for small and divergence-free initial data. Moreover, our approach is based on proving sharp bilinear convolution estimates in critical Sobolev norms, which in turn guarantee the uniform analyticity of both the velocity and magnetic fields with respect to time. Furthermore, leveraging the decay properties of the associated fractional heat semigroup and a bootstrap argument, we derived algebraic decay rates and established the long-time dissipative behavior of FrMHD solutions. These results extended the existing literature on fractional Navier–Stokes equations by fully incorporating magnetic coupling and Coriolis effects within a unified fractional-dissipation framework. Full article
22 pages, 4427 KiB  
Article
Numerical Investigation of Cavitation Models Combined with RANS and PANS Turbulence Models for Cavitating Flow Around a Hemispherical Head-Form Body
by Hyeri Lee, Changhun Lee, Myoung-Soo Kim and Woochan Seok
J. Mar. Sci. Eng. 2025, 13(4), 821; https://doi.org/10.3390/jmse13040821 - 21 Apr 2025
Viewed by 699
Abstract
Accurate prediction of cavitating flows is essential for improving the performance and durability of marine and hydrodynamic systems. This study investigates the influence of different cavitation models—Kunz, Merkle, and Schnerr–Sauer—on the numerical prediction of cavitation around a hemispherical head-form body using computational fluid [...] Read more.
Accurate prediction of cavitating flows is essential for improving the performance and durability of marine and hydrodynamic systems. This study investigates the influence of different cavitation models—Kunz, Merkle, and Schnerr–Sauer—on the numerical prediction of cavitation around a hemispherical head-form body using computational fluid dynamics (CFD). Additionally, the effects of turbulence modeling approaches, including Reynolds-averaged Navier–Stokes (RANS) and partially averaged Navier–Stokes (PANS), are examined to assess their capability in capturing transient cavitation structures and turbulence interactions. The results indicate that the Schnerr–Sauer model, which incorporates bubble dynamics based on the Rayleigh–Plesset equation, provides the most accurate prediction of cavitation structures, closely aligning with experimental data. The Merkle model shows intermediate accuracy, while the Kunz model tends to overpredict cavity closure, limiting its ability to capture unsteady cavitation dynamics. Furthermore, the PANS turbulence model demonstrates superior performance over RANS by resolving more transient cavitation phenomena, such as cavity shedding and re-entrant jets, leading to improved accuracy in pressure distribution and vapor volume fraction predictions. The combination of the PANS turbulence model with the Schnerr–Sauer cavitation model yields the most consistent results with experimental observations, highlighting its effectiveness in modeling highly dynamic cavitating flows. Full article
(This article belongs to the Section Ocean Engineering)
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35 pages, 13648 KiB  
Article
Parameterizing the Tip Effects of Submerged Vegetation in a VARANS Solver
by Lai Jiang, Jisheng Zhang, Hao Chen, Chenglin Liu and Mingzong Zhang
J. Mar. Sci. Eng. 2025, 13(4), 785; https://doi.org/10.3390/jmse13040785 - 15 Apr 2025
Viewed by 378
Abstract
This paper presents an experimental and numerical investigation of submerged vegetation flow, with a particular focus on vegetation-related terms, especially in the vicinity of the free end. Experimental results indicate that substantial shear stress is observed near the top of vegetation, where the [...] Read more.
This paper presents an experimental and numerical investigation of submerged vegetation flow, with a particular focus on vegetation-related terms, especially in the vicinity of the free end. Experimental results indicate that substantial shear stress is observed near the top of vegetation, where the drag coefficient increases significantly due to the disturbance caused by the free end. Furthermore, wake generation is notably suppressed, particularly at heights where wake-generated turbulence dominates, leading to a reduction in turbulent kinetic energy (TKE). A numerical model based on the volume-averaged Reynolds-averaged Navier–Stokes (VARANS) equations was developed, incorporating a vertically varying drag coefficient. The two-scale kε turbulence model is further modified with the inclusion of a new damping function to capture the suppression of wake generation. The model accurately simulates both unidirectional and oscillatory flows, as well as the associated turbulence structures, with good agreement with experimental measurements. The influence of the tips on wave-induced currents, mass transport and TKE distribution is also investigated. It was found that the tip effects play a significant role in strengthening wave-induced currents at the top of loosely arranged, short, and sparse vegetation, with shear kinetic energy (SKE) serving as a critical component of TKE, contributing to the nonuniform distribution. Both Eulerian currents and Stokes drift contribute to streaming in the direction of wave propagation near the vegetation top, which intensifies with increasing solid volume fraction, while tip effects further enhance the onshore mass transport. Within the vegetation, mass transport is more sensitive to wave period and wave height, shifting from onshore to offshore as wavelength increases under constant water depth. Full article
(This article belongs to the Section Ocean Engineering)
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17 pages, 5387 KiB  
Article
Microscopic Effect of Mixed Wetting Capillary Characteristics on Spontaneous Imbibition Oil Recovery in Tight Reservoirs
by Yu Pu, Erlong Yang and Di Wang
Energies 2025, 18(2), 324; https://doi.org/10.3390/en18020324 - 13 Jan 2025
Viewed by 772
Abstract
The understanding of the mechanisms that govern water spontaneous imbibition in mixed wetting capillary channels plays a significant role in operating the oil extraction and energy replenishment for the tight oil reservoirs. In this work, the conservative form phase-field model together with the [...] Read more.
The understanding of the mechanisms that govern water spontaneous imbibition in mixed wetting capillary channels plays a significant role in operating the oil extraction and energy replenishment for the tight oil reservoirs. In this work, the conservative form phase-field model together with the Navier–Stokes equation is employed to investigate the influence of the mixed wetting distribution and the wetting degree on the imbibition oil recovery effects and microscopic flow characteristics. Results indicate that there exist different oil detachment modes of spontaneous imbibition, and these modes are determined by the coupled effect of mixed wetting fraction and contact angle size. For the mixed wetting capillary with strong oil wetting, when fw is low, spontaneous imbibition can only partially detach the oil. Low fw slows down the fluid flow velocity and leads to the small imbibition oil recovery rate. After that, the influence of the surface contact angle size of the mixed wetting capillary is discussed. For the complete detachment mode, the capillary tube presents a form of water phase saturated filling, achieving the optimal imbibition oil recovery effect. For the mixed wetting capillary tube with the combination of weak water wetting and strong oil wetting (i.e., θw = 75° and θo = 165°), local spontaneous imbibition turbulence can only detach very little oil at the inlet of the water wetting area, ultimately achieving a recovery efficiency of less than 10%. This work illuminates the spontaneous imbibition oil recovery mechanisms and flow potentiality for the different mixed wetting capillary channels. Full article
(This article belongs to the Special Issue Oil Recovery and Simulation in Reservoir Engineering)
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21 pages, 5722 KiB  
Article
Analytical Solutions of Time-Fractional Navier–Stokes Equations Employing Homotopy Perturbation–Laplace Transform Method
by Awatif Muflih Alqahtani, Hamza Mihoubi, Yacine Arioua and Brahim Bouderah
Fractal Fract. 2025, 9(1), 23; https://doi.org/10.3390/fractalfract9010023 - 31 Dec 2024
Cited by 3 | Viewed by 1126
Abstract
The aim of this article is to introduce analytical and approximate techniques to obtain the solution of time-fractional Navier–Stokes equations. This proposed technique consists is coupling the homotopy perturbation method (HPM) and Laplace transform (LT). The time-fractional derivative used is the Caputo–Hadamard fractional [...] Read more.
The aim of this article is to introduce analytical and approximate techniques to obtain the solution of time-fractional Navier–Stokes equations. This proposed technique consists is coupling the homotopy perturbation method (HPM) and Laplace transform (LT). The time-fractional derivative used is the Caputo–Hadamard fractional derivative (CHFD). The effectiveness of this method is demonstrated and validated through two test problems. The results show that the proposed method is robust, efficient, and easy to implement for both linear and nonlinear problems in science and engineering. Additionally, its computational efficiency requires less computation compared to other schemes. Full article
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23 pages, 613 KiB  
Article
Application of Triple- and Quadruple-Generalized Laplace Transform to (2+1)- and (3+1)-Dimensional Time-Fractional Navier–Stokes Equation
by Hassan Eltayeb Gadain and Said Mesloub
Axioms 2024, 13(11), 780; https://doi.org/10.3390/axioms13110780 - 12 Nov 2024
Viewed by 767
Abstract
In this study, the solution of the (2+1)- and (3+1)-dimensional system of the time-fractional Navier–Stokes equations is gained by utilizing the triple-generalized Laplace transform decomposition method (TGLTDM) and quadruple-generalized Laplace transform decomposition method (FGLTDM). In addition, the results of the offered methods match [...] Read more.
In this study, the solution of the (2+1)- and (3+1)-dimensional system of the time-fractional Navier–Stokes equations is gained by utilizing the triple-generalized Laplace transform decomposition method (TGLTDM) and quadruple-generalized Laplace transform decomposition method (FGLTDM). In addition, the results of the offered methods match with the exact solutions of the problems, which proves that, as the terms of the series increase, the approximate solutions are closer to the exact solutions of each problem. To verify the appropriateness of these methods, some examples are offered. The TGLTDM and FGLTDM results indicate that the suggested methods have higher evaluation convergence as compared to the ADM and HPM. Full article
(This article belongs to the Special Issue Advances in Differential Equations and Its Applications)
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15 pages, 3256 KiB  
Article
Mathematical Analysis of a Navier–Stokes Model with a Mittag–Leffler Kernel
by Victor Tebogo Monyayi, Emile Franc Doungmo Goufo and Ignace Tchangou Toudjeu
AppliedMath 2024, 4(4), 1230-1244; https://doi.org/10.3390/appliedmath4040066 - 8 Oct 2024
Cited by 2 | Viewed by 1589
Abstract
In this paper, we establish the existence and uniqueness results of the fractional Navier–Stokes (N-S) evolution equation using the Banach fixed-point theorem, where the fractional order β is in the form of the Atangana–Baleanu–Caputo fractional order. The iterative method combined with the Laplace [...] Read more.
In this paper, we establish the existence and uniqueness results of the fractional Navier–Stokes (N-S) evolution equation using the Banach fixed-point theorem, where the fractional order β is in the form of the Atangana–Baleanu–Caputo fractional order. The iterative method combined with the Laplace transform and Sumudu transform is employed to find the exact and approximate solutions of the fractional Navier–Stokes equation of a one-dimensional problem of unsteady flow of a viscous fluid in a tube. In the domains of science and engineering, these methods work well for solving a wide range of linear and nonlinear fractional partial differential equations and provide numerical solutions in terms of power series, with terms that are simple to compute and that quickly converge to the exact solution. After obtaining the solutions using these methods, we use Mathematica software Version 13.0.1.0 to present them graphically. We create two- and three-dimensional plots of the obtained solutions at various values of β and manipulate other variables to visualize and model relationships between the variables. We observe that as the fractional order β becomes closer to the integer order 1, the solutions approach the exact solution. Lastly, we plot a 2D graph of the first-, second-, third-, and fourth-term approximations of the series solution and observe from the graph that as the number of iterations increases, the approximate solutions become close to the series solution of the fourth-term approximation. Full article
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16 pages, 963 KiB  
Article
A Meshless Radial Point Interpolation Method for Solving Fractional Navier–Stokes Equations
by Arman Dabiri, Behrouz Parsa Moghaddam, Elham Taghizadeh and Alexandra Galhano
Axioms 2024, 13(10), 695; https://doi.org/10.3390/axioms13100695 - 7 Oct 2024
Cited by 2 | Viewed by 1253
Abstract
This paper aims to develop a meshless radial point interpolation (RPI) method for obtaining the numerical solution of fractional Navier–Stokes equations. The proposed RPI method discretizes differential equations into highly nonlinear algebraic equations, which are subsequently solved using a fixed-point method. Furthermore, a [...] Read more.
This paper aims to develop a meshless radial point interpolation (RPI) method for obtaining the numerical solution of fractional Navier–Stokes equations. The proposed RPI method discretizes differential equations into highly nonlinear algebraic equations, which are subsequently solved using a fixed-point method. Furthermore, a comprehensive analysis regarding the effects of spatial and temporal discretization, polynomial order, and fractional order is conducted. These factors’ impacts on the accuracy and efficiency of the solutions are discussed in detail. It can be shown that the meshless RPI method works quite well for solving some benchmark problems accurately. Full article
(This article belongs to the Special Issue Fractional Calculus and the Applied Analysis)
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12 pages, 258 KiB  
Article
Existence Result for a Class of Time-Fractional Nonstationary Incompressible Navier–Stokes–Voigt Equations
by Keji Xu and Biao Zeng
Axioms 2024, 13(8), 499; https://doi.org/10.3390/axioms13080499 - 25 Jul 2024
Viewed by 752
Abstract
We are devoted in this work to dealing with a class of time-fractional nonstationary incompressible Navier–Stokes–Voigt equation involving the Caputo fractional derivative. By exploiting the properties of the operators in the equation, we use the Rothe method to show the existence of weak [...] Read more.
We are devoted in this work to dealing with a class of time-fractional nonstationary incompressible Navier–Stokes–Voigt equation involving the Caputo fractional derivative. By exploiting the properties of the operators in the equation, we use the Rothe method to show the existence of weak solutions to the equation by verifying all the conditions of the surjectivity theorem for nonlinear weakly continuous operators. Full article
21 pages, 5776 KiB  
Article
Study on the Flow Velocity of Safe and Energy-Saving Transportation of Light-Particle Slurry
by Xiaochun Wang, Yue Wang, Dayun Hao, Haiqian Zhao and Zhipei Hu
Appl. Sci. 2024, 14(14), 6313; https://doi.org/10.3390/app14146313 - 19 Jul 2024
Cited by 1 | Viewed by 1075
Abstract
In order to determine the recommended flow velocity for the safe and energy-saving transport of ice-slurry-type light particle slurries, it is necessary to study the flow characteristics of light particle slurries, especially the critical flow velocity. Therefore, in this paper, a numerical simulation [...] Read more.
In order to determine the recommended flow velocity for the safe and energy-saving transport of ice-slurry-type light particle slurries, it is necessary to study the flow characteristics of light particle slurries, especially the critical flow velocity. Therefore, in this paper, a numerical simulation method based on the mixed turbulence model with the RANS (Reynolds averaged Navier Stokes) equation is used, and a new concentration distribution method is proposed for the first time to derive the critical flow velocity, as follows: the flow velocity of the light particle slurry when the ratio of the solid volume fraction vf at the position of 0.08D above the bottom of the pipeline to that at the center of the pipeline, vf/vf(y) = 0.75, is taken as the critical flow velocity. The flow changes in the slurry (polyethylene particles with a density of 922 kg/m3 and water) under 0.1–1.0 m/s (at intervals of 0.1 m/s) were investigated experimentally, and the pressure drop data obtained from the experiments were used to determine the recommended flow rate for safe and energy-saving transportation of the light particle slurry. The pipe diameter used for the experiments and simulations was 28 mm, and the solid-phase particle sizes were 0.3 mm, 0.4 mm, and 0.5 mm, with solid-phase contents of 5 vol%, 10 vol%, 15 vol%, and 20 vol%. In addition, the experimental and numerical simulation results show that an increase in solid-phase content and particle size leads to an increase in critical flow velocity. Full article
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18 pages, 3448 KiB  
Article
Employing the Laplace Residual Power Series Method to Solve (1+1)- and (2+1)-Dimensional Time-Fractional Nonlinear Differential Equations
by Adel R. Hadhoud, Abdulqawi A. M. Rageh and Taha Radwan
Fractal Fract. 2024, 8(7), 401; https://doi.org/10.3390/fractalfract8070401 - 4 Jul 2024
Cited by 1 | Viewed by 1073
Abstract
In this paper, we present a highly efficient analytical method that combines the Laplace transform and the residual power series approach to approximate solutions of nonlinear time-fractional partial differential equations (PDEs). First, we derive the analytical method for a general form of fractional [...] Read more.
In this paper, we present a highly efficient analytical method that combines the Laplace transform and the residual power series approach to approximate solutions of nonlinear time-fractional partial differential equations (PDEs). First, we derive the analytical method for a general form of fractional partial differential equations. Then, we apply the proposed method to find approximate solutions to the time-fractional coupled Berger equations, the time-fractional coupled Korteweg–de Vries equations and time-fractional Whitham–Broer–Kaup equations. Secondly, we extend the proposed method to solve the two-dimensional time-fractional coupled Navier–Stokes equations. The proposed method is validated through various test problems, measuring quality and efficiency using error norms E2 and E, and compared to existing methods. Full article
(This article belongs to the Section Numerical and Computational Methods)
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