Computational Methods for Solving Nonlinear Ordinary Differential Equations Arising in Engineering and Applied Sciences

In this paper, the computational method (CM) based on the standard polynomials has been implemented to solve some nonlinear differential equations arising in engineering and applied sciences. Moreover, novel computational methods have been developed in this study by orthogonal base functions, namely Hermite, Legendre


Introduction
In the classical theories of the various branches of science, differential equations are mainly linear. In modern science, when certain phenomena cannot be explained by linear differential equations, it is inevitable to resort to nonlinear differential equations to obtain the desired information [1]. Solution methods for these types of equations are of great importance and have appeared in the mathematical formulation of many phenomena, including engineering, fluid mechanics, flow models, and mathematical physics [2]. Therefore, the need for reliable and effective numerical or approximate methods to solve these types of equations has become a very important requirement [3].
In 1973, Corrington [19] showed that linear differential and integral equations can be converted into a system of algebraic linear equations with a least-squares approximation and repeated integrations of Walsh functions. On the other hand, the orthogonal polynomials are characterized, above all, by the fact that they effectively simplify the required solution by transforming the nonlinear differential equations into nonlinear algebraic systems of equations using the operational matrices technique, where they can be solved simply by using any computational program. In addition, the classical operational matrix method based on orthogonal polynomials such as Legendre polynomials [20], Bernstein polynomials [21], and Hermite polynomials [22] attracted great interest from the authors as they were very useful techniques for solving many different problems in approximation theory and numerical analysis [3].
In 2013, Turkyilmazoglu [23] proposed an analytic approximate method, namely the effective computational method, and used it to solve various types of problems, for more details, see [24][25][26][27]. Moreover, the approach depends upon standard base functions of the general type, such as the standard polynomials [1, , 2 , … ], and the exact solutions are given under certain conditions. In addition, the solution of the nonlinear equations is converted into a nonlinear algebraic system with unknown standard polynomial coefficients, which can be solved numerically or analytically using modern software.
The current aim of this paper is to implement CM based on the standard polynomials to solve three applications involving well-known nonlinear problems: the Darcy-Brinkman-Forchheimer equation, the Blasius equation, and the Falkner-Skan equation, which appeared in engineering and applied sciences. The main goals are to develop the CM by introducing various orthogonal polynomials, such as Hermite, Legendre, and Bernstein polynomials, and to form a novel D-CMs collection. The ultimate objective is to apply the D-CMs to solve these problems.
The outline of the paper is as follows: Section two describes the mathematical formulation of three nonlinear models. Section three presents the basic concepts of the proposed methods.
In section four, the convergence of the proposed methods will be given, and the problems will be solved using the proposed methods, with a discussion of the numerical results. Finally, in section five, the conclusions will be presented.

The mathematical formulation of nonlinear models 2.1 The Darcy-Brinkman-Forchheimer equation
Consider a steady-state, pressure-driven, fully developed parallel flow through a horizontal channel filled with a porous medium [28], as shown in Figure 1:  [29]. The bottom and the top plates are located at = ℎ and = − ℎ, respectively. The flow is in the direction of the -axis and the velocity is also of the form = ( ( ), 0,0). It is known that the flow in the channel is determined by the Darcy-Brinkman-Forchheimer equation, which is as follows [30]: with boundary conditions: (2) where represents the Forchheimer number, represents the porous medium shape parameter, and is the viscosity ratio. Several analytical and approximate methods have been presented for solving the Darcy-Brinkman-Forchheimer equation, for instance, the finite difference method [31], the Tau homotopy analysis method [28], the optimal Galerkin homotopy asymptotic method [30], and the homotopy analysis method [32]. In particular, Motsa et al. [29] implemented the spectral homotopy analysis approach to obtain an accurate result for the model. Adewumi et al. [33] applied the hybrid method in combination with the Chebyshev collocation method with Laplace and differential transform methods to obtain approximate solutions for the model. In addition, Abbasbandy et al. [34] obtained a closed-form solution of forced convection in a porous saturated channel.

The Blasius equation
The Blasius equation is a well-known third-order nonlinear ordinary differential equation that appeared in certain boundary layer problems of the two-dimensional laminar viscous flow of a fluid over a flat plate. It is the governing equation for fluid dynamics and is represented by the following equation [35]: with boundary conditions: (0) = ′ (0) = 0 , ′ (∞) = 1, (4) The second derivative of ( ) at zero is important in the Blasius equation to determine the shear stress on the plate. Many authors have tried to solve this equation and obtained different numbers for this value. More details can be found in [36][37][38].
The Blasius equation has been solved by various numerical and analytical methods like the Adomian decomposition method [39], the variational iteration method [40], the optimal homotopy asymptotic method [41], and the homotopy analysis method [42]. Moreover, Khataybeh et al. [36] employed the classical operational matrices of the Bernstein polynomials method to solve the Blasius equation. Parand and Taghavi [43] used a collocation method based on a rational scaled generalized Laguerre function to solve this equation.

The Falkner-Skan equation
The boundary layer equations are an important type of nonlinear ordinary differential equations with various applications in physics and fluid mechanics [44]. The stationary Falkner-Skan boundary layer equation is one type of these equations. Falkner and Skan [45] first proposed the Falkner-Skan equation in 1931. This equation has an important role in a variety of applications, such as fluid mechanics, aerospace, heat transfer, glass applications, and polymer studies [3].

The basic idea of the proposed methods
This section presents the basic concepts of the proposed techniques. Moreover, orthogonal polynomials and operational matrices will be discussed as tools for developing the CM technique to achieve approximate solutions to specific nonlinear models presented in section two.
where ( ) is a known function and , , , are constants.
Consider the Hilbert space = 2 [0,1], in which the inner product is defined as follows: Moreover, the set of functions = { 0 , 1 … , }, are linearly independent in , where = , 0 ≤ ≤ , is the base function of standard polynomials [23,24]. Thus, performing the inner product of the set of base functions with the left and right sides of the Eq. (14) in the manner of the Eq. (17), we obtain the following matrix equation [25]: Eventually, some entries in this matrix equation change when the initial or boundary conditions for the Eqs. (15) and (16) are substituted into the Eq. (18). As a result, a system of ( + 1) nonlinear algebraic equations with their unknown coefficients, , is created. These algebraic equations can be solved numerically with available programs to obtain the values of the coefficients . These values are then substituted into the Eq. (12) to produce the approximate solution of the Eq. (9).

The Legendre polynomials with their operational matrices
The Legendre polynomials, ( ), of ℎ -order on the interval [−1,1], are defined as [20,56]: Also, the analytical formula of the Legendre polynomials is obtained by the following: Furthermore, any function ( ) can be expressed by the ( + 1)-terms of the Legendre polynomials presented below: The derivatives of ( ) can be regarded as: is the operational matrix of the derivatives and is defined by Otherwise. Therefore, the derivatives of the function ( ) can be written as follows:

The Bernstein polynomials with their operational matrices
The Bernstein polynomials , ( ) of ℎ -degree on the interval [0,1] are defined by [57] as follows: In general, ( ) can be approximated by the linear combination of the Bernstein polynomials shown in the following formula: Moreover, ( ) can be defined as follows [57]: Thus, the derivatives of ( ) can be defined by: is the operational matrix of the derivatives and is defined by .
Therefore, the derivatives of the function ( ) can be expressed as below:

The convergence of the proposed methods and numerical results
This section presents the convergence for the proposed methods. In addition, the CM and D-CMs proposed methods will be applied to find the approximate solutions and discuss the numerical results for the problems.

Convergence Analysis of the proposed methods
This subsection will discuss the convergence analysis of the proposed methods and fundamental theorem.

Application of the CM and D-CMs and numerical results
In this subsection, the proposed methods of CM and D-CMs will be implemented to find the approximate solutions and present the numerical results for the three problems: the Darcy-Brinkman-Forchheimer equation, the Blasius equation, and the Falkner-Skan equation.
The D-CMs are based on the base functions of diverse polynomials such as Hermite, Legendre, and Bernstein polynomials, presented in the Eqs. (20), (25), and (28), respectively, with related operational matrices. These polynomials are executed in two steps of the proposed methods techniques to improve the accuracy of the CM. Firstly, to represent a function ( ) and its derivatives; and secondly, to compute the inner product to solve the left and right sides of the matrix equation shown in the Eq. (18). Furthermore, by substituting the initial or boundary conditions, as given in the Eqs. (15) and (16), some entries of the Eq. (18) are adjusted. Then we get ( + 1) nonlinear algebraic equations for the unknown . By solving this system numerically by Mathematica ® 12, we get unique values for the unknown elements 0 , 1 , 2 , … to achieve the best approximate solution to the problems.

Solving the Darcy-Brinkman-Forchheimer equation by the CM and D-CMs
The procedures of CM and D-CMs that are presented in section three are applied to solve the first problem with the boundary conditions shown in the Eqs. (1) and (2). To be more precise, we substitute the Eqs. (12) and (13) into the Eqs. (1) and (2) for the technique CM, converting the function ( ) and its derivatives as matrices. Thus, we obtain the following result:     Moreover, Table 1 shows the values of the for the approximate solution using the CM and D-CMs with = 10 and parameters = = 1, versus the value of , which shows the efficiency of these methods. In addition, it can be observed that the D-CMs based on the Hermite polynomials method provide slightly better accuracy with the lowest number of errors compared to other methods. , as shown in Table 2. The results of the values for all ≥ 2 and 0 ≤ ≤ 1, are less than one. Therefore, the approximate solutions obtained by the proposed methods CM and D-CMs converge.

Solving the Blasius equation by the CM and D-CMs
The procedures of CM and D-CMs that are presented in section three can be utilized to solve the second problem illustrated in the Eqs. (3) and (5). To do this, the Eqs. (12) and (13)  (43) Also, by implementing the procedures as given in the Eqs. (18) and (19), it follows that: The exact solution to this problem is not available. Therefore, the maximum error remainder ( ) has been computed to verify the accuracy and efficiency of the approximate solution obtained by the proposed methods. The is computed by [15]:  [37]. The accuracy of these methods can be shown by observing the error values for , as we observed that the error becomes smaller as the value of is increased.    Table 3 shows the values for the approximate solution using the CM and D-CMs with = 10, demonstrating the effectiveness of these approaches. In addition, it can be observed that the D-CMs based on the Hermite polynomials method provide better accuracy with less errors compared to the other methods. , as presented in Table 4. The results of the values for all ≥ 3 and 0 ≤ ≤ 1, are less than one. Therefore, the approximate solutions achieved by the proposed methods CM and D-CMs converge.

Solving the Falkner-Skan equation by the CM and D-CMs
The procedures of CM and D-CMs presented in section three can be implemented to solve the third problem explained in the Eqs. (6) and (8). To be more specific, for the CM approach, we transform the function ( ) and its derivatives into matrices by substituting the Eqs. (12) and (13) Also, by implementing the techniques that are given in Eqs. (18) and (19), the following is obtained: 〈 ( ), ( * ) 3 ( ) + ( ( ))( ( * ) 2 ( )) + [−( * ( )) 2 ] 〉 = 〈 ( ), − 2 〉, ∀ 0 ≤ ≤ . (52) Moreover, using the D-CMs based on the Bernstein polynomials by substituting the Eqs. (29) and (30) Then, the processes have been utilized as given in the Eqs. (18) and (19), which will be shown:  Figure 6 shows the logarithmic plots for the values obtained by the CM based on the standard polynomials and by the D-CMs based on the Hermite, Legendre, and Bernstein polynomials for the parameters = 0.5 and = 0.1, according to studies [47], which show the reliability and efficiency of these methods by observing the error values for = 2 to 8. We find that the error decreases with increasing values of . Also, it can be observed that the D-CMs based on the Hermite polynomials method provide better accuracy with less errors compared to the other methods. Moreover, Figure 7 demonstrates the comparison between the approximate solutions computed by the proposed methods for = 8, = 0.5, and = 0.1. As can be seen from the figure, good agreement was obtained for all the proposed methods. In addition, Figures 8 and 9 show the logarithmic plots of the for the approximate solution of the Falkner-Skan equation with = 2 to 8, using the CM and D-CMs when fixed the pressure gradient parameter = 0.5, and increasing the values of the velocity ratio parameter as = 0.1, 0.2, 0.3, and 0.4, as chosen in [47]. In Figures 8 and 9, the errors decrease when the value of is increased.   , as indicated in Table 5. The results of the values for all ≥ 2 and 0 ≤ ≤ 1, are less than one. Therefore, the approximate solutions obtained by the proposed methods CM and D-CMs converge.

Conclusions
In this paper, the computational method (CM) based on standard polynomials and the novel computational methods (D-CMs) based on different types of orthogonal polynomials, Hermite, Legendre, and Bernstein polynomials have been presented and implemented to solve three nonlinear problems, the Darcy-Brinkman-Forchheimer equation, the Blasius equation, and the Falkner-Skan equation. The nonlinear problems are reduced to a nonlinear algebraic system of equations, which is solved using Mathematica ® 12. The approximate solutions were obtained and appeared to be accurate and efficient even within polynomials of low orders. Moreover, the was computed for the proposed methods. The results show that the proposed methods have better accuracy with lower errors. In addition, it is observed that the results of the by the proposed methods D-CMs decreased significantly compared to the CM. Therefore, the suggested novel methods D-CMs have better accuracy than the CM. It can be concluded that the D-CMs based on the Hermite polynomials are better than the other methods for the three nonlinear problems.