Fractional Boundary Layer Flow: Lie Symmetry Analysis and Numerical Solution

Abstract

In this paper, we present a fractional version of the Sakiadis flow described by a nonlinear two-point fractional boundary value problem on a semi-infinite interval, in terms of the Caputo derivative. We derive the fractional Sakiadis model by substituting, in the classical Prandtl boundary layer equations, the second derivative with a fractional-order derivative by the Caputo operator. By using the Lie symmetry analysis, we reduce the fractional partial differential equations to a fractional ordinary differential equation, and, then, a finite difference method on quasi-uniform grids, with a suitable variation of the classical L1 approximation formula for the Caputo fractional derivative, is proposed. Finally, highly accurate numerical solutions are reported.

1. Introduction

Fractional differential equations (FDEs), with extensive applications across a wide class of problems that appear in many fields of the applied sciences, have recently been studied in various research areas. In the literature, it is possible to find some extensions of differential equations to fractional differential equations, for example, the fractional Biswas–Milovic model, the fractional Riccati differential equation, the time-fractional $K(m,n)$ equation, the time–fractional $B(m,n)$ equation, etc. [1,2,3,4]. A widely used class of FDEs is represented by linear and nonlinear boundary value problems of fractional order (FBVPs), and a lot of these models are defined on unlimited domains.

Recently, it has been demonstrated that mathematical problems describing the boundary layer flow of viscous fluid are well-modeled by fractional derivatives. The time–space-dependent fractional boundary layer flow of Maxwell fluid on an unsteady stretching surface is studied by Chen et al. [5].

Recently, the Lie symmetry analysis has been applied to the nonlinear space-fractional boundary layer equation by Pan et al. [6]. Mohammadein et al. [7] found a similarity solution for a viscous fluid flow on an infinite vertical plate with a fractional laminar boundary layer by means of fractional power series technique, and, recently, in [8], artificial boundary conditions on an unlimited domain were considered.

In fluid mechanics, the classical Sakiadis problem [9,10] is considered a variant of the well-known Blasius model [11], with the boundary layer flow in a quiescent fluid. The main aspect of this problem is to provide the fluid motion via a moving flat plate. In this context, a large number of research studies deal with this problem in the classical formulation [12,13,14], but no study has been performed for its fractional formulation. In this paper, starting from the classical Prandtl boundary layer equations for a Newtonian fluid and following the recently proposed theories, we propose a fractional version of the boundary layer Sakiadis model in terms of the fractional Caputo derivative. The fractional derivative allows for the modeling of some anomalous diffusion processes, and it is very efficient at describing certain real-world phenomena, especially when the dynamic is affected by the structure inherent to the system. Furthermore, the Caputo definition of the fractional derivative enables the definition of an initial value problem whose initial conditions are given in terms of the field variable and its integer-order derivatives. This is in agreement with the clear physical meaning of most of the processes that arise in the real world.

The governing nonlinear fractional partial differential equations are reduced to a fractional ordinary differential equation by means of the extended Lie symmetry analysis for FDEs with Caputo or Riemann–Liouville derivatives, developed by [15,16,17] and applied in [18,19]. The classical numerical approach to solving the fractional ordinary differential equation is not applicable, and, therefore, some new techniques are needed. Thus, the resulting fractional ordinary differential equation has been solved numerically by means of a finite difference method implemented on quasi-uniform meshes for FBVPs. The Caputo derivative is approximated by a suitable variant of the well-known L1 formula, and then a finite difference operator is found. Next, we consider a smooth, strictly monotonic function to build a quasi-uniform map, and then we implement, on the original semi-infinite domain, an implicit finite difference method defined on a quasi-uniform grid that permits requiring the given asymptotic boundary conditions exactly. The obtained numerical results show the reliability of the proposed fractional formulation of the Sakiadis model, and the numerical advantage of the used approach is to solve the FBVP defined on a semi-infinite interval by using a stencil built such that the boundary conditions are exactly assigned at infinity. The paper is structured as follows. In Section 2, we present the derivation of the fractional Sakiadis model, the Lie Group analysis, the Lie transformation of the variables and the reduced fractional ordinary differential model. In Section 3, we show the numerical method and results, and, in Section 4, we offer our concluding remarks.

2. Derivation of the Fractional Sakiadis Model

In order to derive the fractional formulation of the mathematical model, the object of this study, we follow the theory recently proposed in [20]. We derive the fractional Sakiadis model from the classical Prandtl boundary layer equations, which are of particular interest since they are considered a simplification of the original Navier–Stokes equations and describe various physical problems of fluid mechanics. In particular, in the equation of the moment, we replace the second derivative of integer order with a fractional-order Caputo derivative and proceed using the Lie symmetry analysis that allows us to reduce the fractional partial differential equations to a fractional ordinary differential one.

The system of boundary layer equations for a planar steady flow of liquid with constant coefficients of the kinematic viscosity $\nu$ and density $\rho$ has the form

$$\begin{array}{ccc}\hfill & & u_x+v_y=0,\hfill \\ & & u{u}_x+v{u}_y=\nu u_{yy}-\frac{1}{\rho }{p}_x,\hfill \\ & & p_y=0\hfill \end{array}$$

(1)

and is known as the Prandtl model [21,22]. The flow is two dimensional where the x-axis is along the plane of the moving plate and the y-axis is normal to it. $u(x,y)$ and $v(x,y)$ are velocity components, and $p_x$ and $p_y$ are the components of the pressure gradient in the x- and y-directions.

In this paper, we consider the problem with the surface moving continuously at a constant velocity U. We neglect the pressure gradient as it is sufficiently small that it can be assumed to be equal to zero. We propose a model obtained by replacing, in the equation of the moment of the classical model, the second derivative of integer order $u_{yy}$ with a fractional $(\alpha +1)$-order Caputo derivative [23,24,25] $D_y^{\alpha +1}u$ defined as follows

$$D_y^{\alpha +1}u=\frac{1}{\Gamma (1-\alpha )}{\int }_0^y\frac{u_{ss}(x,s)}{{(y-s)}^{\alpha }}ds,0<\alpha <1.$$

Thus, we obtain the following fractional formulation of the model for the laminar boundary layer flow on a semi-infinite flat plate

$$\begin{array}{cc}\hfill & u_x+v_y=0,\\ & u{u}_x+v{u}_y=\nu D_y^{\alpha +1}u\end{array}$$

(2)

subject to the boundary conditions

$$\begin{array}{ccc}\hfill & & u(x,0)=Uv(x,0)=0x>0,\hfill \\ & & u(x,\infty )=0x>0\hfill \end{array}$$

(3)

with $U>0$ as the constant surface velocity.

By defining the stream function $\psi =\psi (x,y)$ such that

$$u={\psi }_y,v=-{\psi }_x$$

we obtain the $(\alpha +2)$-order nonlinear fractional partial differential equation (FPDE)

$$\begin{array}{c}\hfill {\psi }_y{\psi }_{xy}-{\psi }_x{\psi }_{yy}=\nu D_y^{\alpha +2}\psi \end{array}$$

(4)

where

$$\begin{array}{c}\hfill D_y^{\alpha +2}\psi =\frac{1}{\Gamma (1-\alpha )}{\int }_0^y\frac{{\psi }_{sss}(x,s)}{{(y-s)}^{\alpha }}ds.\end{array}$$

The boundary conditions become

$$\begin{array}{ccc}\hfill & & \psi (x,0)=0{\psi }_y(x,0)=Ux>0,\hfill \\ & & {\psi }_y(x,\infty )=0x>0.\hfill \end{array}$$

(5)

In the next section, we perform the Lie Group analysis for Equation (4) and for boundary conditions (5), which will allow us to obtain a transformation of the variables that reduces the above fractional partial differential equation into a fractional ordinary differential model.

Lie Symmetry Method

In this section, we briefly recall the Lie Symmetry theory for FDEs involving the Caputo derivative, developed in [16,17]. We consider an FPDE of $(\alpha +k)$-order ($k\in N$),

$$\Delta =f(x,y,u,u_x,u_y,u_{xx},u_{xy},u_{yy},\dots ,D^{\alpha +k}u)=0,$$

(6)

where $D^{\alpha +k}$ is the Caputo derivative with respect to x or y. According to the theory, the invertible transformations of the variables $x,y$ and u, represented as

$$X=X(x,y,u,a),Y=Y(x,y,u,a),U=U(x,y,u,a),$$

(7)

which depend on a continuous parameter a, are one parameter Lie point symmetry transformations of Equation (6) if Equation (6) preserves its form in the new variables $X,Y$ and U. The set G of all such transformations forms a continuous group, defined as the group admitted by Equation (6). By expanding (7) in a Taylor series around $a=0$, we obtain the infinitesimal transformations

$$\begin{array}{c}\hfill X=x+a{\xi }^x(x,y,u)+o\left(a\right),Y=y+a{\xi }^y(x,y,u)+o\left(a\right),U=u+a\zeta (x,y,u)+o\left(a\right)\end{array}$$

where ${\xi }^x$, ${\xi }^y$ and $\zeta$, the infinitesimals, are given by

$$\begin{array}{ccc}& & {\xi }^x(x,y,u)={\left.\frac{\partial X}{\partial a}\right|}_{a=0},{\xi }^y(x,y,u)={\left.\frac{\partial Y}{\partial a}\right|}_{a=0},\zeta (x,y,u)={\left.\frac{\partial U}{\partial a}\right|}_{a=0}.\hfill \end{array}$$

and the corresponding infinitesimal operator of the group G is

$$\Xi ={\xi }^x(x,y,u){\partial }_x+{\xi }^y(x,y,u){\partial }_y+\zeta (x,y,u){\partial }_u.$$

(8)

The infinitesimals ${\xi }^x$, ${\xi }^y$ and $\zeta$ are obtained by solving the invariance condition:

$${\Xi }^{\alpha +k}{\Delta =0|}_{\Delta =0},$$

(9)

where the ${\Xi }^{\alpha +k}$ is the $(\alpha +k)$-order prolongation of operator (8) acting on (6), given by

$$\begin{array}{ccc}& & {\Xi }^{\alpha +k}={\xi }^x{\partial }_x+{\xi }^y{\partial }_y+\zeta {\partial }_u+{\zeta }^x{\partial }_{u_x}+{\zeta }^y{\partial }_{u_y}\hfill \\ \hfill & & + {\zeta }^{xx}{\partial }_{u_{xx}}+{\zeta }^{xy}{\partial }_{u_{xy}}+{\zeta }^{yy}{\partial }_{u_{yy}}+\cdots +{\zeta }^{\alpha +k}{\partial }_{D^{\alpha +k}u}.\hfill \end{array}$$

(10)

where ${\zeta }^x$,${\zeta }^y$, ${\zeta }^{xx}$, ${\zeta }^{xy}$ and ${\zeta }^{yy}$ are the extended infinitesimals given by well-known recursion relations [26], whereas ${\zeta }^{\alpha +k}$ is a new extended infinitesimal with, as in the classical theory, a recursion form [15,16,17].

Invariance condition (9) is a linear FDE for the unknown infinitesimals ${\xi }^x(x,y,u)$, ${\xi }^y(x,y,u)$ and $\zeta (x,y,u)$. Splitting the coefficients of all derivatives of u, including the fractional ones $D^{\alpha +k}u$, we obtain an overdetermined set of linear differential equations (determining equations) for the infinitesimals, which (by integration) leads us to find the generators of the Lie point symmetries admitted by Equation (6). When boundary value conditions are assigned to model (6), in accordance with the invariance principle [26], the invariance with respect to the operator $\Xi$ of the assigned boundary conditions must be satisfied.

Now, we determine the Lie symmetries of Equation (4). Invariance condition (9) applied to Equation (4) reads as

$$\begin{array}{c}\hfill k{\zeta }^{\alpha +2}+{\psi }_x{\zeta }_{{\psi }_{yy}}-{\psi }_y{\zeta }_{{\psi }_{xy}}+{\psi }_{yy}{\zeta }_{{\psi }_x}-{\psi }_{xy}{\zeta }_{{\psi }_y}{=0|}_{\Delta =0}.\end{array}$$

and the extended fractional infinitesimal reads as [15,16,17,27]

$$\begin{array}{c}\hfill \begin{array}{cc}& {\zeta }^{\alpha +2}=D_y^{\alpha +2}(\zeta -{\xi }^x{\psi }_x-{\xi }^y{\psi }_y)+{\xi }^x{D}_y^{\alpha +2}{\psi }_x+{\xi }^y{D}_y^{\alpha +2}{\psi }_y.\hfill \end{array}\end{array}$$

As usual, we introduce the generalized Leibnitz rule to simplify the extended infinitesimal ${\zeta }^{\alpha +2}$, assuming $\zeta$ to be linear in u. After combining sums and factorizing, and neglecting terms of higher powers of the group parameter a, with the support of the computer algebra software Wolfram Mathematica v.12 (Wolfram Research, Inc., Champaign, IL, USA), we obtain

$${\xi }^x=x(c_2+\alpha c_3)+c_4,{\xi }^y=c_{3}y,\zeta =c_2\psi .$$

(11)

The obtained infinitesimals satisfy, in accordance with the extension of the Lie symmetry method to FPDEs [16,17], the following invariance condition to conserve the structure of the fractional derivative: ${\xi }^y{(x,y,\psi )|}_{y=0}=0.$

The invariance of the boundary value conditions on $y=0$ of stream function (5) reads as

$${\Xi }^{\alpha +2}\left(\psi \right){{|}_{\psi =0}=0,{\Xi }^{\alpha +2}({\psi }_y-U)|}_{{\psi }_y=U}=(c_2-c_3)U$$

(12)

which leads to obtaining

$$c_2=c_3$$

and, to simplify the calculations, we neglect the x-translation, setting $c_4=0$, which may be included again in the solution by replaying $x\to x+c_4/(1+\alpha )$. The transformation is

$$\psi =x^{\frac{1}{1+\alpha }}f\left(\eta \right),\eta =y{x}^{-\frac{1}{1+\alpha }}.$$

(13)

In terms of the new variables, $f\left(\eta \right)$ and $\eta$, the Caputo derivative $D_y^{\alpha +2}\psi$ reads as

$$\begin{array}{ccc}\hfill D_y^{\alpha +2}\psi & =& \frac{1}{\Gamma (1-\alpha )}{\int }_0^y{x}^{\frac{-2}{1+\alpha }}{(y-s)}^{-\alpha }{f}^{\prime \prime \prime }\left(\tau \right)ds\hfill \\ & =& \frac{1}{\Gamma (1-\alpha )}{\int }_0^{\eta }{x}^{\frac{-2-\alpha +1}{1+\alpha }}{(\eta -\tau )}^{-\alpha }{f}^{\prime \prime \prime }\left(\tau \right)d\tau \hfill \\ & =& x^{-1}\frac{1}{\Gamma (1-\alpha )}{\int }_0^{\eta }{(\eta -\tau )}^{-\alpha }{f}^{\prime \prime \prime }\left(\tau \right)d\tau \hfill \\ & =& x^{-1}{D}_{\eta }^{\alpha +2}f\left(\eta \right).\hfill \end{array}$$

Finally, by using the above relation and transformation (13), Equation (4) is reduced to the following fractional ordinary differential equation

$$D_{\eta }^{\alpha +2}f\left(\eta \right)+\frac{1}{1+\alpha }f\left(\eta \right)f^{\prime \prime }\left(\eta \right)=0$$

(14)

with $\nu =1$ and the boundary conditions given by

$$\begin{array}{ccc}\hfill & & f\left(0\right)=0,f^{\prime }\left(0\right)=1\underset{\eta \to \infty }{lim}{f}^{\prime }\left(\eta \right)=0,\hfill \end{array}$$

(15)

where we set $U=1$. Equation (14) with boundary conditions (15) is the fractional formulation of classical Sakiadis equation for $\alpha =1$ for a continuous flat surface. However, Equation (14) is the same as the well-known Blasius one, but the boundary conditions are different. As a consequence, the solution to Equation (14) with (15) will be different from the Blasius solution for the flat plate of a finite length.

In the next section, we find the numerical solution to fractional boundary value problems (14) and (15) by means of a finite difference method.

3. Numerical Method and Results

In this section, in order to solve fractional model (14), subject to boundary conditions (15), we implement the recently proposed finite difference method that is able to impose the given asymptotic boundary conditions exactly. An analysis of the stability, consistency and convergence properties of the proposed method is conducted in the paper [20]. In order to develop the numerical method, we first discretize the infinite domain by a quasi-uniform grid, and then we define suitable approximation formulas for the field variable, its first derivative and its fractional derivative.

In order to discretize the infinite domain, we propose the following map, $\eta =\eta \left(\xi \right)$, with $\xi \in \left[0,1\right]$,

$$\eta =-c·ln(1-\xi ),$$

(16)

with $c>0$ as the control parameter that, by a uniform distribution of the grid points ${\xi }_n=n/N$ in $\left[0,1\right]$, defines a quasi-uniform distribution of the grid points ${\eta }_n\in [0,\infty )$ for $n=0,\cdots ,N$. In this way, we use a map in order to discretize the infinite domain by a finite number of intervals, where the last interval is of infinity length since the last node is placed at infinity: ${\eta }_N=\eta \left({\xi }_N\right)=\infty$. Moreover, we define the mesh points, ${\eta }_{n+j}$, internal to the single interval, as follows

$${\eta }_{n+j}=\eta \left(\frac{n+j}{N}\right).$$

(17)

The main features of the proposed mesh discretization are:

(1)

The last interval is an infinite one;

(2)

The last point ${\eta }_N=\infty$;

(3)

The mid-point ${\eta }_{N-1/2}$ of the last infinite interval is finite.

Now, we need to define the approximations of a scalar function $u\left(\eta \right)$, its first derivative $u^{\prime }\left(\eta \right)$ and the Caputo derivative $D_{\eta }^{\alpha }u\left(\eta \right)$ at mid-points of the grid ${\eta }_{n+1/2}$. To approximate the function and its first derivative with respect to $\eta$, we use the following finite difference discretization formulas [28]

$$\begin{array}{ccc}\hfill u({\eta }_{n+\frac{1}{2}})& \approx & \frac{{\eta }_{n+\frac{3}{4}}-{\eta }_{n+\frac{1}{2}}}{{\eta }_{n+\frac{3}{4}}-{\eta }_{n+\frac{1}{4}}}u\left({\eta }_n\right)+\frac{{\eta }_{n+\frac{1}{2}}-{\eta }_{n+\frac{1}{4}}}{{\eta }_{n+\frac{3}{4}}-{\eta }_{n+\frac{1}{4}}}u\left({\eta }_{n+1}\right)\hfill \\ & \approx \hfill & h_n{u}_n+h_{n_1}{u}_{n_1}\hfill \\ \hfill \end{array}$$

(18)

$$\begin{array}{ccc}\hfill u^{\prime }({\eta }_{n+\frac{1}{2}})& \approx & \frac{u\left({\eta }_{n+1}\right)-u\left({\eta }_n\right)}{2({\eta }_{n+\frac{3}{4}}-{\eta }_{n+\frac{1}{4}})}\approx \frac{u_{n+1}-u_n}{h_0}.\hfill \end{array}$$

(19)

where $u_n\approx u\left({\eta }_n\right)$ and the grid points ${\eta }_{n+p}$, with $p=\frac{1}{4},\frac{1}{2}$ and $\frac{3}{4}$, are defined by (17).

In order to integrate the Caputo derivative $D_{\eta }^{\alpha }u$, we propose a suitable modification of the following classical well-known L1 formula, [29], evaluated at the mesh point ${\eta }_{n+1}$,

$$\begin{array}{ccc}\hfill D_{\eta }^{\alpha }u\left({\eta }_{n+1}\right)& =& \frac{1}{\Gamma (1-\alpha )}{\int }_0^{{\eta }_{n+1}}{u}^{\prime }\left(s\right){({\eta }_{n+1}-s)}^{-\alpha }ds\hfill \\ & \approx & \frac{1}{\Gamma (1-\alpha )}{\int }_0^{{\eta }_{n+1}}{u}^{\prime }\left({\eta }_{n+1}\right){({\eta }_{n+1}-s)}^{-\alpha }ds\hfill \\ & \approx & \frac{1}{\Gamma (1-\alpha )}\sum _{k=0}^n\frac{u_{k+1}-u_k}{({\eta }_{k+1}-{\eta }_k)}{\int }_{{\eta }_k}^{{\eta }_{k+1}}{({\eta }_{n+1}-s)}^{-\alpha }ds,\hfill \end{array}$$

by evaluating it at the mesh mid-point ${\eta }_{n+\frac{1}{2}}$ and not at the mesh point ${\eta }_{n+1}$, as follows

$$\begin{array}{ccc}\hfill D_{\eta }^{\alpha }u({\eta }_{n+\frac{1}{2}})& =& \frac{1}{\Gamma (1-\alpha )}{\int }_0^{{\eta }_{n+\frac{1}{2}}}{u}^{\prime }(s){({\eta }_{n+\frac{1}{2}}-s)}^{-\alpha }ds\hfill \\ & \approx & \frac{1}{\Gamma (1-\alpha )}{\int }_0^{{\eta }_{n+\frac{1}{2}}}{u}^{\prime }({\eta }_{n+\frac{1}{2}}){({\eta }_{n+\frac{1}{2}}-s)}^{-\alpha }ds\hfill \\ & \approx & \frac{1}{\Gamma (1-\alpha )}\sum _{k=0}^n\frac{u_{k+1}-u_k}{2({\eta }_{k+\frac{3}{4}}-{\eta }_{k+\frac{1}{4}})}{\int }_{{\eta }_{k-\frac{1}{2}}}^{{\eta }_{k+\frac{1}{2}}}{({\eta }_{n+\frac{1}{2}}-s)}^{-\alpha }ds\hfill \end{array}$$

where the first derivative $u^{\prime }({\eta }_{n+\frac{1}{2}})$ is approximated by (19). By using $u_n$ and $u_{n+j}$, with $0\le j\le 1$, as the numerical approximations of the function $u\left(\eta \right)$ at the mesh points ${\eta }_n$ and ${\eta }_{n+j}$, $u\left({\eta }_n\right)$ and $u\left({\eta }_{n+j}\right)$, respectively, we obtain

$$\begin{array}{c}\hfill D_{\eta }^{\alpha }u({\eta }_{n+\frac{1}{2}})\approx \frac{1}{\Gamma (2-\alpha )}\sum _{k=0}^n{T}_{n+\frac{1}{2},k}(u_{k+1}-u_k)\end{array}$$

(20)

where

$$\begin{array}{c}\hfill T_{n+\frac{1}{2},k}=\frac{{({\eta }_{n+\frac{1}{2}}-{\eta }_{k-\frac{1}{2}})}^{1-\alpha }-{({\eta }_{n+\frac{1}{2}}-{\eta }_{k+\frac{1}{2}})}^{1-\alpha }}{2({\eta }_{k+\frac{3}{4}}-{\eta }_{k+\frac{1}{4}})}\end{array}$$

being

$$\begin{array}{c}\hfill {\int }_{{\eta }_{k-\frac{1}{2}}}^{{\eta }_{k+\frac{1}{2}}}{({\eta }_{n+\frac{1}{2}}-s)}^{-\alpha }ds=\frac{1}{1-\alpha }\left({({\eta }_{n+\frac{1}{2}}-{\eta }_{k-\frac{1}{2}})}^{1-\alpha }-{({\eta }_{n+\frac{1}{2}}-{\eta }_{k+\frac{1}{2}})}^{1-\alpha }\right).\end{array}$$

As usual, we write FBVPs (14) and (15) as a system of fractional nonlinear ordinary differential equations, and, by setting ${}^{1}u=f\left(\eta \right)$, ${}^{2}u=f^{\prime }\left(\eta \right)$ and ${}^{3}u=f^{\prime \prime }\left(\eta \right)$ and using formulas (18)–(20) for ${}^{\mathsf{\ell }}u$, for $\mathsf{\ell }=1,2,3$, we have

$$\begin{array}{ccc}\hfill & & {}^1{u}_{n+1}={}^1{u}_n+h_0{}^2{u}_{n+\frac{1}{2}}\hfill \\ & & {}^2{u}_{n+1}={}^2{u}_n+h_0{}^3{u}_{n+\frac{1}{2}}\hfill \\ & & {}^3{u}_{n+1}={}^3{u}_n-\frac{h_0}{h_1}\left(\sum _{k=0}^{n-1}{T}_{n+\frac{1}{2},k}({}^3{u}_{k+1}-{}^3{u}_k)+\frac{\Gamma (2-\alpha )}{1+\alpha }{}^1{u}_{n+\frac{1}{2}}{}^3{u}_{n+\frac{1}{2}}\right)\hfill \end{array}$$

(21)

for $n=0,\cdots ,N-1$, where

$$\begin{array}{c}\hfill {}^l{u}_{n+\frac{1}{2}}=h_n{}^l{u}_n+h_{n_1}{}^l{u}_{n+1},l=1,2,3,\end{array}$$

(22)

and where the boundary conditions read as

$$\begin{array}{ccc}\hfill & & {}^1{u}_0=0,{}^2{u}_0=1{}^2{u}_N=0.\hfill \end{array}$$

(23)

Moreover,

$$h_n=\frac{{\eta }_{n+\frac{3}{4}}-{\eta }_{n+\frac{1}{2}}}{{\eta }_{n+\frac{3}{4}}-{\eta }_{n+\frac{1}{4}}},h_{n_1}=\frac{{\eta }_{n+\frac{1}{2}}-{\eta }_{n+\frac{1}{4}}}{{\eta }_{n+\frac{3}{4}}-{\eta }_{n+\frac{1}{4}}}.$$

Additionally, $h_0=2({\eta }_{n+\frac{3}{4}}-{\eta }_{n+\frac{1}{4}})$, $h_1={({\eta }_{n+\frac{1}{2}}-{\eta }_{n-\frac{1}{2}})}^{1-\alpha }$ and

$$T_{n+\frac{1}{2},n}=\frac{{({\eta }_{n+\frac{1}{2}}-{\eta }_{n-\frac{1}{2}})}^{1-\alpha }}{2({\eta }_{n+\frac{3}{4}}-{\eta }_{n+\frac{1}{4}})}.$$

In the proposed finite difference formulas, (18)–(20), used to construct the finite difference method in (21), the value of the solution at infinity, denoted as $u\left({\eta }_N\right)=u(\infty )$, appears instead of the infinity grid point ${\eta }_N=\infty$, which, in general, cannot be used from the numerical point of view. In this way, the boundary conditions at infinity are assigned in a natural way.

As usual, the FBVP is reformulated as an equivalent fractional initial value problem (FIVP). By setting the missing initial condition, ${}^3{u}_0=\beta$, we define an equation that is solved by means of the shooting implemented with the secant method. Setting two suitable values, ${\beta }_0=-0.4$ and ${\beta }_1=-0.5$, the FIVP is solved numerically with a fixed tolerance $\mathrm{TOL}=1\times {10}^{-12}$. Figure 1 shows the numerical solution obtained with $N=100$ grid points.

In

Table 1. Numerical values of ${}^{3}u$ at point $\eta =0$ for increasing values of $\alpha$ and $N=100$.

$\mathit{\alpha }$	${}^3{\mathit{u}}_0={\mathit{f}}^{\prime \prime }\left(0\right)$
0.1	−0.283185584345104
0.2	−0.294051822111457
0.3	−0.305387847908001
0.4	−0.318365424665809
0.5	−0.333032626256035
0.6	−0.349515965244018
0.7	−0.368055913603354
0.8	−0.389099343907209
0.9	−0.413554792825066
1	−0.443786571474161

, we report the initial missing value for ${}^{3}u$ at the point $\eta =0$ for different values of $\alpha$. We find that ${}^3{u}_0=-0.443786571474161$ for the missing initial conditions ${}^{3}u$ at the point $\eta =0$ for $\alpha =1$. The missing value, calculated by Sakiadis [10], is ${}^3{u}_0=-0.44375$.

In Figure 2, we show the numerical solution obtained for $\alpha =0.999$ and the numerical solution for $\alpha =1$ obtained by using the Matlab solver ode23s, implemented via the shooting approach on a truncated boundary of the length $[0,13.8]$. It is important to note that, in the literature, the benchmark values of the missing initial condition ${}^3{u}_0=f^{\prime \prime }\left(0\right)$ exist only for $\alpha =1$. Then, to validate the fractional formulation of the proposed model and perform a comparison, we can only show how the numerical solution of the fractional model for $\alpha \to 1$ tends to the numerical solution of the classical model. Therefore, we can assert that the proposed model represents a reliable fractional formulation of the classical Sakiadis flow.

We can conclude that the proposed method, implemented on the logarithmic map, proves to be a reliable and efficient tool for solving the Sakiadis problem. The quasi-uniform distribution of grid points allows the solution to be approximated with a higher accuracy. The grid is denser at the beginning of the process where the solution shows a particular behavior.

4. Concluding Remarks

The objective of this paper is to derive the fractional formulation, in terms of the Caputo derivative, of the nonlinear boundary value Sakiadis problem defined on a semi-infinite interval starting from the classical Prandtl equations. To this end, the system of fractional partial differential equations is mapped into a nonlinear ordinary differential equation by means of the Lie symmetry analysis. Then, the recently proposed finite difference method on a quasi-uniform grid is used for the numerical integration of the reduced model. The highly accurate numerical results are reported.