Fractional Navier Stokes Equations Research Articles

Mathematical Analysis of a Navier–Stokes Model with a Mittag–Leffler Kernel

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.

A Meshless Radial Point Interpolation Method for Solving Fractional Navier–Stokes Equations

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.

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

The Navier–Stokes equations describe the behavior of viscous fluids and establish a fundamental connection between the application of external forces on fluid motion and the resulting pressure within the fluid. The objective of this study is to solve the two-dimensional time fractional Navier–Stokes equation through the utilization of the residual power series method together with the generalized Shehu transform. The method is called the generalized Shehu residual power series (GSHRPS) approach. The fractional derivative utilized in this research is the Katugampola derivative in the sense of Caputo. The effectiveness of this method is verified by demonstrating its convergence to the solution of the previously described problem. Furthermore, a practical example is presented to show the precision, accuracy, and efficiency of this approach in order to illustrate its effectiveness and benefits.

Stability Analysis According to the Regularity of External Forces of a Semi-Implicit Difference Scheme for Time Fractional Navier–Stokes Equations

Stability Analysis According to the Regularity of External Forces of a Semi-Implicit Difference Scheme for Time Fractional Navier–Stokes Equations

A new approach to solve time-fractional Navier–Stokes equation

This article presents a new technique for the solution of Time Fractional Navier–Stokes equation. The approach is a combination of Adomian decomposition method (ADM) with Kamal integral transform (KIT). The proposed method is implemented on time-fractional Navier–Stokes equation (TFNSEs) to yield the analytical solution. Illustrative examples in TFNSEs are discussed to validate the applicability of the technique. The graphical visualization of the solutions is also presented. Further, the obtained results are compared with the existing solution methods.

The Fractional Navier-Stokes Equations with delay conditions

Throught this paper, we study the cauchy problem for the conformable fractional Navier-Stokes Equations (FNSE) with finite delay external forces, contains some hereditary features, on a bounded domain. We prove that there exist a unique local mild solutions for the initial datum, by using semigroup theory, conformable fractional calculus and Banach contraction theorem.With more conditions on delay external forces we establish the globality and continuation of the mild solutions.

Analyticity for the fractional Navier-Stokes equations in critical Fourier-Besov-Morrey Spaces with variable exponents

In this paper, by using the Littlewood-Paley theory and the Fourier localization argument, we obtained the analyticity of the solution to the fractional Navier-Stokes equations in variable exponents Fourier-Besov-Morrey spaces when the initial data are small.

Phase transitions in the fractional three-dimensional Navier–Stokes equations

The fractional Navier–Stokes equations on a periodic domain [0,L]3 differ from their conventional counterpart by the replacement of the −νΔu Laplacian term by νsAsu , where A=−Δ is the Stokes operator and νs=νL2(s−1) is the viscosity parameter. Four critical values of the exponent s⩾0 have been identified where functional properties of solutions of the fractional Navier–Stokes equations change. These values are: s=13 ; s=34 ; s=56 and s=54 . In particular: (i) for s>13 we prove an analogue of one of the Prodi–Serrin regularity criteria; (ii) for s⩾34 we find an equation of local energy balance and; (iii) for s>56 we find an infinite hierarchy of weak solution time averages. The existence of our analogue of the Prodi–Serrin criterion for s>13 suggests the sharpness of the construction using convex integration of Hölder continuous solutions with epochs of regularity in the range 0<s<13 .

Density of weak solutions of the fractional Navier-Stokes equations in the smooth divergence-free vector fields

Density of weak solutions of the fractional Navier-Stokes equations in the smooth divergence-free vector fields

On a blow-up criterion for solution of 3D fractional Navier-Stokes-Coriolis equations in Lei-Lin-Gevrey spaces

Abstract In this article, we researched the existence of the solution to the fractional Navier-Stokes equations with the Coriolis force under initial data, which belong to the Lei-Lin-Gevrey spaces. Moreover, we showed a blow-up criterion, i.e., when the maximal time of existence T * {T}^{* } is finite, we proved that the norm of this same solution, in a specific Lei-Lin-Gevrey space, goes to infinity, as time tends to the maximal time of its existence.

A semi-analytical approach via yang transform on fractional-order navier-stokes equation

In this current, the applications of the Yang transformation technique are taken under consideration to deal with the non-linear fractional Navier–Stokes equation and fractional coupled Navier–Stokes equation. The suggested method produces approximate-analytical solutions in the form of a series that are correspondingly dependent on fractional-order derivative values and have modest, comprehensible mechanics and easy implementation. The Caputo fractional derivative is employed, and the numerical scheme’s stability and convergence are examined. Numerical examples demonstrate the analytical solution of the technique and it is examined that the proposed techniques are robust, efficient and reduce the number of numerical computations. The current technique’s results are compatible with the theoretical analysis, and the suggested technique can be extended to solve numerous higher-order non-linear dynamics.

Globally Well-Posedness Results of the Fractional Navier–Stokes Equations on the Heisenberg Group

Globally Well-Posedness Results of the Fractional Navier–Stokes Equations on the Heisenberg Group

An Approach for Approximating Analytical Solutions of the Navier-Stokes Time-Fractional Equation Using the Homotopy Perturbation Sumudu Transform’s Strategy

In this study, we utilize the properties of the Sumudu transform (SuT) and combine it with the homotopy perturbation method to address the time fractional Navier-Stokes equation. We introduce a new technique called the homotopy perturbation Sumudu transform Strategy (HPSuTS), which combines the SuT with the homotopy perturbation method using He’s polynomials. This approach proves to be powerful and practical for solving various linear and nonlinear fractional partial differential equations (FPDEs) in scientific and engineering fields. We demonstrate the efficiency and simplicity of this method through examples, showcasing its ability to approximate solutions for FPDEs. Additionally, we compare the numerical results obtained using this technique for different values of alpha, showing that as the value moves from a fractional order to an integer order, the solution becomes increasingly similar to the exact solution. Furthermore, we provide the tabular representations of the solution for each example.

Well-Posedness of Mild Solutions for the Fractional Navier–Stokes Equations in Besov Spaces

Well-Posedness of Mild Solutions for the Fractional Navier–Stokes Equations in Besov Spaces

Non‐uniqueness of Leray–Hopf solutions to the forced fractional Navier–Stokes equations in three dimensions, up to the J. L. Lions exponent

AbstractIn this paper, we show that for , there exists a force and two distinct Leray–Hopf flows solving the forced fractional Navier–Stokes equation starting from rest. This shows that the J. L. Lions exponent is sharp in the class of Leray–Hopf solutions for the forced fractional Navier–Stokes equation.

A note on energy equality for the fractional Navier-Stokes equations

This paper proves the energy equality for distributional solutions to fractional Navier-Stokes equations, which gives a new proof and covers the classical result of Galdi [Proc. Amer. Math. Soc. 147 (2019), 785–792].

Existence of Global and Local Mild Solution for the Fractional Navier–Stokes Equations

Navier–Stokes equations (NS-equations) are applied extensively for the study of various waves phenomena where the symmetries are involved. In this paper, we discuss the NS-equations with the time-fractional derivative of order β∈(0,1). In fractional media, these equations can be utilized to recreate anomalous diffusion equations which can be used to construct symmetries. We examine the initial value problem involving the symmetric Stokes operator and gravitational force utilizing the Caputo fractional derivative. Additionally, we demonstrate the global and local mild solutions in Hα,p. We also demonstrate the regularity of classical solutions in such circumstances. An example is presented to demonstrate the reliability of our findings.

Analysis of fractional Navier–Stokes equations

AbstractIn this study, we apply the fractional Laplace variational iteration method (FLVIM), a computer methodology for exploring fractional Navier–Stokes equation solutions. In light of the theory of fixed points and Banach spaces, this paper also explores the uniqueness and convergence of the solution of general fractional differential equations obtained by the suggested method. In addition, the fractional Laplace variational iteration method solution's error analysis is covered. The computational technique also clearly demonstrates the validity and dependability of the suggested method for solving fractional Navier‐Stokes equations. Furthermore, the obtained solutions are a perfect fit with previously established solutions.

A study of the time fractional Navier-Stokes equations for vertical flow

&lt;abstract&gt;&lt;p&gt;Navier-Stokes (NS) equations dealing with gravitational force with time-fractional derivatives are discussed in this paper. These equations can be used to predict fluid velocity and pressure for a given geometry. This paper investigates the local and global existence and uniqueness of mild solutions to NS equations for the time fractional differential operator. We also work on the regularity effects of such types of equations were caused by orthogonal flow.&lt;/p&gt;&lt;/abstract&gt;

Quantitative estimates for space-time analyticity of solutions to the fractional Navier-Stokes equations

Quantitative estimates for space-time analyticity of solutions to the fractional Navier-Stokes equations