A unique approach for solving the fractional Navier–Stokes equation

Analysis of fractional Navier–Stokes equations

“…Thus, the approximate solution of (40) can be written $$$u(x,t)=1-{x}^{2}+(p-4)\left(1-\alpha +\frac{{t}^{\alpha }}{{\rm{\Gamma }}({\rm{\alpha }})}\right),$$$when choosing $$\alpha =1$$, it becomes $$$u(x,t)=1-{x}^{2}+(p-4)t,$$$which is the same solution given by RPSTM 38 and VIM 39 . In Figure 1, the approximate solutions and exact solutions are plotted for different values of $$\alpha $$ when $$=0.5$$.…”

“…which is the same solution given by RPSTM 38 and VIM 39 . In Figure 1, the approximate solutions and exact solutions are plotted for different values of α when = 0.5.…”

“…Thus, the approximate solution of (40) can be written $$$u(x,t)=x+\frac{1}{x}\left(1-\alpha +\frac{{t}^{\alpha }}{\Gamma (\alpha )}\right)+\frac{1}{{x}^{3}}\left((1-{\alpha )}^{2}+2(1-\alpha )\frac{{t}^{\alpha }}{\Gamma (\alpha )}+\frac{{\alpha }^{2}{t}^{2\alpha }}{\Gamma (2\alpha +1)}\right),$$$for $$\alpha =1$$, it becomes $$$u(x,t)=x+\sum _{n=1}^{{\rm{\infty }}}\frac{{{1}^{2}\times {2}^{2}\times {3}^{2}\times \text{\unicode{x02026}}\text{\unicode{x02026}}..(2n-3)}^{2}}{{x}^{2n-1}}.\frac{{t}^{n}}{n! },$$$which is the same solution given by RPSTM 38 and VIM 39 . In Figure 5, the approximate solutions and exact solutions are plotted for different values of $$\alpha $$ when $$t=0.5$$.…”

“…Thus, the approximate solution of (40) can be written which is the same solution given by RPSTM 38 and VIM. 39 In Figure 5, the approximate solutions and exact solutions are plotted for different values of α when t = 0.5. Also, In Figures 6-8, u x t ( , ) are plotted for various values of α.…”

“…Recently, several mathematicians have applied various strategies for the solutions of fractional differential equations, including the fractional variational iteration method, fractional differential transform method, fractional Sumudu variational iteration method, fractional homotopy perturbation method, fractional Sumudu decomposition method, fractional Laplace homotopy perturbation method, fractional Sumudu homotopy perturbation method, and fractional Sumudu homotopy analysis method 5–39 . In this paper, the Laplace variational iteration technique is implemented to analyze the solution of fractional‐order Navier–Stokes equations.…”