Numerical solution to the Falkner-Skan equation: a novel numerical approach through the new rational a-polynomials

The new rational a-polynomials are used to solve the Falkner-Skan equation. These polynomials are equipped with an auxiliary parameter. The approximated solution to the Falkner-Skan equation is obtained by the new rational a-polynomials with unknown coefficients. To find the unknown coefficients and the auxiliary parameter contained in the polynomials, the collocation method with Chebyshev-Gauss points is used. The numerical examples show the efficiency of this method.


Introduction
Nonlinear ordinary differential equations play an essential role in scientific, technical, economic, and mathematical modeling. Applications of nonlinear ordinary differential equations include the calculation of trajectories of space gliders and airplanes, forecast calculations, spread of AIDS, and automated parking maneuvering based on mechanical systems [1] . Boundary layer equations are an important class of nonlinear ordinary differential equations that have many applications in physics and fluid mechanics [2][3] . Applications of boundary layer equations include metal plate cooling in a cooling bath, aerodynamic extrusion of plastic sheets, plastic film design, metal spinning, metallic plates, insulation materials, and glass and polymer studies [4] . One class of these boundary layer equations is the stationary Falkner-Skan boundary layer equation. The problem is very old and has already been addressed with many different numerical methods. Furthermore, computational fluid dynamics have already evolved in more advanced topics. Efficient numerical methods for boundary layer equations were developed 50 years ago, and finding a new computational method will be useful for solving new problems. In this paper, the authors aim to present a new method for the Falkner-Skan equation, which may be applicable to other problems.
The Falkner-Skan equation plays an important role in fluid mechanics, aerospace, and heat transfer. In fluid dynamics, the Falkner-Skan boundary layer equation describes the twodimensional laminar boundary layer formed on a wedge, where the plate is not parallel to the flow. The Falkner-Skan equation is the generalized state of the Blasius boundary layer equations.
In this paper, new a-polynomials equipped with parameter a are used to solve this equation. These polynomials were used in Ref. [24] to solve the Benjamin-Bona-Mahony-Burgers (BBMB) equations and in Ref. [25] to solve an inverse problem.

Problem formulation
In this section, we explain how to obtain the Falkner-Skan ordinary differential equation. Assume the following boundary layer equations for steady incompressible flow with constant viscosity and density [26] : Here, the coordinate system is chosen with x pointing parallel to the plate in the direction of the flow and y-coordinate pointing towards the free stream. u and v are the xand yvelocity components, respectively. U is the inherent characteristic velocity. σ is the electrical conductivity. ρ is the fluid density. ν is the kinematic viscosity. We can generalize the Blasius boundary layer by considering a wedge at an angle of attack πβ/2 from some uniform velocity field U 0 . We then estimate the outer flow to be of the form where l is a characteristic length, and m is a dimensionless constant. In the Blasius solution, m = 0, corresponding to an angle of attack of zero radians. Thus, we can write β = 2m 2m + 1 .
In the Blasius solution, we use a similarity variable η to solve the boundary layer equations. We will obtain [27] η = y U 0 (m + 1) 2νl x l Therefore, Eqs. (1) and (2) can be transformed into the following third-order nonlinear Falkner-Skan differential equation: with the boundary conditions in which M 2 = σB 2 0 /(ρa). No exact analytical solution is known for Eq. (3).

Rational a-polynomials
In this section, the concept of a new class of rational polynomials is introduced. Also, by changing the coordinate, the unbounded domain is considered.
Definition [28] Assume that a is a constant parameter and A 0 (z) = 1. The a-polynomials as a combination of the Chebyshev polynomials of the second kind, U n (z), are defined by The following equations are also established: See Refs. [24] and [28] for more properties. By changing the variable z = η−L η+L for η ∈ [0, +∞), in which L is an arbitrary large positive number and will be chosen later, the following new rational a-polynomials are obtained from Eq. (4): with AL 0 (η) = 1. The constant L is called the mapping parameter. By using according to Eq. (6), we have

Proposition 1
The following recursive relation holds for the rational a-polynomials:

Proposition 2 Considering the inner product
where , |n − m| = 2, nm = 0, where δ n,m is the delta Kronecker number. Proposition 3 The system {AL 4n } ∞ n=0 is an orthogonal system with respect to the inner product defined in Proposition 2.

Proposition 4 U n is the eigenfunction of the singular Sturm-Liouville problem
and R n is the eigenfunction of the singular Sturm-Liouville problem Remark 1

The convergence theorem
Assume Λ = [0, +∞), ω is the weight function in Proposition 2, and L 2 ω (Λ) is the function Hilbert space with the inner product of Eq. (10). Let N be the positive integer, and we will consider which is the subspace of L 2 ω (Λ). We define the L 2 ω (Λ)-orthogonal projection as follows: To estimate P N u − u ω , we have the following space interpolation: H r ω,Q (Λ) = {u|u is measurable and u r,ω,Q < ∞}, where any real r > 0, Q is the Sturm-Liouville operator of Chebyshev rational polynomials, and u r,ω,Q is in the following norm: where q k is a rational function bounded uniformly on the whole interval Λ. The desired result is obtained from Eqs. (18) and (19).
Proof First, we assume that r = 2m. Due to Eqs. (11), (13), and (17) and the integration by parts, we obtain Now, according to Eqs. (19) and (21) and the definition of H r ω,Q (Λ), we have Next, we put r = 2m + 1. With Eq. (13) and the integration by parts, we have Now, with Eqs. (14) and (19), we obtain The general result follows from the previous results and space interpolation.

Method implementation
The solution to the Falkner-Skan equation is obtained by using the present method. To find the solution, the approximate solution to the equation is estimated as due to Remark 1 and the boundary condition f (∞) = 1. Considering the Chebyshev-Gauss points, the collocation points of the present method are defined as z j = cos(jπ) N for j = 0, 1, · · · , N. Now, we consider the collocation method for the residual of Eq. (3) with the collocation points η j (j = 0, 1, · · · , N), where z j = ηj −L ηj +L . Therefore, the unknown coefficients {c j } N j=0 and the parameters a and L are obtained by solving the following nonlinear system of equations: where In Table 1, the results of the method are compared with those of the spline function method [20] , the Chebyshev polynomial approximation [30] , the Chebyshev finite difference method [14] , and the piecewise linear functions method [12] . The results show good approximation for the solution to the equation. Table 1 Comparison of f (0) for different methods at f (0) = 0 and M = 0 Ref. [20] Ref. [30] Ref. [14] Ref. [ (27), instead of the initial condition f (0) = 0, put a more general condition f (0) = α. Table 2 provides the results of the method for the initial condition f (0) = α compared with the homotopy analysis method (HAM) [16] . The results are in good approximation for the solution to the equation .  Tables 3 and 4 show the results of the present method for the numerical solution of the MHD Falkner-Skan flow in β = 4 3 (m = 2) and β = −3 (m = − 3 5 ) cases, respectively. Table  3 compares the results of the present method for β = 4 3 (m = 2) by using the HAM [31] , the Sinc-collocation method [17] , and the Gegenbaer neural network method [22] . According to Table  3, we can conclude that the results are acceptable compared with other methods. In Table 4, the results of the present method are compared for β = −3 (m = − 3 5 ) by using the HAM [31] , the Sinc-collocation method [17] , and the Gegenbaer neural network method [22] .   To verify the accuracy and convergence of the method, R 2 is used. According to Table 5, the results of the present method are more accurate than those of the other methods. Moreover, the increase in the collocation points increases the accuracy of the solution and reduces R 2 . The results of Table 5 confirm the convergence Theorem 2.   In Table 6, the results of f (0) for the a-polynomial method at f (0) = α and M = 0 are displayed. In this table, the results of the residual error of the equation are compared in rational a-polynomials and a-polynomial methods, and it can be seen that the superiority of the rational a-polynomial method is evident. Besides, CPU time of the mentioned methods is acceptable.  (2) In Fig. 1, the f (η) curves are plotted for the values β = 0, 1, 2 under the initial condition f (η) = 0 and M = 0. It shows that the results are easily observable and comparable.

Conclusions
A novel method is presented for solving the classical Falkner-Skan equation. It uses two parameters to optimize and improve the approximation accuracy of the solution. The results are acceptable in comparison with other methods. In fact, in many cases, the results obtained by the proposed method are better than the findings of other papers. The present method includes all types of the Falkner-Skan equation. According to the theorems proved, the convergence of the method is guaranteed.