Novel Approximate Solutions for Nonlinear Initial and Boundary Value Problems

This paper investigates an Effective Computational Method (ECM) based on the standard polynomials used to solve some nonlinear initial and boundary value problems in engineering and applied sciences. Moreover, the effective computational methods in this paper were improved by suitable orthogonal base functions, especially the Chebyshev, Bernoulli, and Laguerre polynomials, to obtain novel approximate solutions for some nonlinear problems. These base functions enable the nonlinear problem to be effectively converted into a nonlinear algebraic system of equations, which are then solved using Mathematica ® 12. The Improved Effective Computational Methods (I-ECMs) have been implemented to solve three applications involving nonlinear initial and boundary value problems: the Darcy-Brinkman-Forchheimer equation, the Blasius equation, and the Falkner-Skan equation, and a comparison between the proposed methods has been presented. Furthermore, the Maximum Error Remainder ( 𝑀𝐸𝑅 𝑛 ) has been computed to prove the proposed methods' accuracy. The results convincingly prove that ECM and I-ECMs are effective and accurate in obtaining novel approximate solutions to the problems.


Introduction
There are many problems in engineering and applied sciences, such as fluid flow models, mechanical engineering, and mathematical physics, which can be described by nonlinear ordinary differential equations [1].This leads to significant computational difficulties for nonlinear boundary conditions, particularly for nonlinear initial value problems [2].Since the exact solutions to these problems are often complicated or occasionally may not be available.Therefore, there is a great need to develop efficient, novel approximate, and numerical methods to solve these problems [3 and 4].
Numerous analytical and approximate methods for solving nonlinear differential equations have been introduced and developed by authors around the world, such as the advanced Adomian decomposition method [5], the Variational Iteration Method (VIM), the Differential Transformation Method (DTM) [6], the finite difference methods [7], the optimal quartic Bspline collocation method [8], the homotopy analysis method with Padé approximations [9], the Chebyshev operational matrix method [10], the Bernoulli matrix method [11], the Laguerre collocation method [12].In particular, AL-Jawary et al. [13] have applied the Daftardar-Jafari Method (DJM), the Temimi-Ansari Method (TAM), and the Banach Contraction Method (BCM) to obtain the solution for the Jeffery-Hamel flow problem.Agom et al. [14] have implemented the Homotopy Perturbation Method (HPM) and the Adomian Decomposition Method (ADM) for solving the 12 ℎ -order boundary value problems in finite domains.Also, Singh [15] used the modified homotopy perturbation approach to solve a set of nonlinear Lane-Emden equations.
Ibraheem et al. [16] have recently implemented the operational matrix of Legendre polynomials to solve nonlinear thin-film flow problems.Gürbüz et al. [17] used the matrix relations between the Laguerre polynomials and their derivatives to study second-order nonlinear ordinary differential equations with quadratic and cubic terms and several other approximation methods for instance, see [18][19][20][21][22][23].
In recent years, approximation methods for analyzing linear systems of ordinary differential equations using orthogonal series have been widely developed.These are known as spectral methods, assuming that a truncated orthogonal series expansion can reasonably approximate the solution.Depending on the nature of the problem, a variety of orthogonal series have been used, such as the Walsh series, block-pulse, Laguerre, Chebyshev, Fourier series, and others [24].
Furthermore, orthogonal functions and polynomial series have attracted significant attention because they have been instrumental in treating various dynamical system problems.The main feature of this technique is that it reduces these problems to the solution of a system of algebraic equations by using the method of operational matrices based on orthogonal polynomials [25], such as Chebyshev polynomials [26], Bernoulli polynomials [27], and Laguerre polynomials [28], which significantly simplifies the problems and allows them to be solved by any computational program.
More recently, Turkyilmazoglu [29] has proposed and used an analytical approximation method, namely the ECM, to solve various types of problems, such as nonlinear Lane-Emden-Fowler equations [29], Fredholm integro-differential equations [30], Volterra-Fredholm-Hammerstein integro-differential equations [31], heat transfer of fin problems [32], and initial and boundary value problems with difficult exact solutions [33].Moreover, the approach depends on appropriate base functions, such as the standard polynomials.In addition, the solution of the nonlinear equations is transformed into a nonlinear algebraic system with unknown standard polynomial coefficients, which can be solved numerically or analytically with modern software.
The current paper aims to use the ECM based on the standard polynomials to solve three applications involving nonlinear initial and boundary value problems: the Darcy-Brinkman-Forchheimer equation, the Blasius equation, and the Falkner-Skan equation, which are found in engineering and applied sciences.The main goals are to develop the ECM by introducing various orthogonal polynomials, such as Chebyshev, Bernoulli, and Laguerre polynomials, and to form a novel collection of the I-ECMs.The final goal is to implement the I-ECMs to solve these problems.
The paper is organized as follows: Section two presents the mathematical formulations of three nonlinear models.Section three introduces the basic concepts of the proposed methods.Section four displays the implementation of the ECM and I-ECMs to solve three nonlinear problems and discusses the results.Finally, section five presents the conclusions.

The Mathematical Formulations of Nonlinear Models 2.1 The Darcy-Brinkman-Forchheimer Equation
Consider the following steady-state, pressure-driven, fully developed parallel flow over a horizontal channel filled with a porous medium [34], as demonstrated in Figure 1: The positions of the bottom and top plates are  = ℎ and  = − ℎ, respectively.The velocity takes the form  = ((), 0, 0) and the flow is in the -axis direction.The Darcy-Brinkman-Forchheimer equation, which has the following form [36], is known to determine the flow in the channel.
subjected to the boundary conditions: = 0.
(2) where  stands for the Forchheimer number,  for the shape parameter of the porous medium, and  for the viscosity ratio.
The Darcy-Brinkman-Forchheimer equation has been solved analytically and approximately using a variety of methods, such as the homotopy analysis method [37], the finite difference method [38], the optimal asymptotic Galerkin homotopy method [36], and the Tau homotopy analysis method [34].In particular, Adewumi et al. [39] obtained the approximate solutions for the model by using the hybrid method in combination with the Chebyshev collocation method with Laplace and differential transform methods.Motsa et al. [35] implemented the spectral homotopy analysis approach to obtain an accurate result for the model.In addition, Abbasbandy et al. [40] obtained a closed-form solution of forced convection in a porous saturated channel.

The Blasius Equation
The Blasius equation is the well-known third-order nonlinear ordinary differential equation that appeared in several boundary layer problems involving a fluid's two-dimensional laminar viscous flow through a flat plate.The following equation presents it as a governing equation for fluid dynamics [41]: subjected to the boundary conditions: The second derivative of () at zero is important in the Blasius equation to evaluate the shear stress on the plate.Numerous authors have tried to solve this problem and obtained various numbers for this value.For more details, see [42][43][44].Therefore, the boundary conditions of the Blasius equation become: (0) = 0,  ′ (0) = 0,  ′′ (0) = .
(5) The value of  = 0.3320573 will be utilized in the present work, as stated in [43].Several numerical and analytical techniques have been used to solve the Blasius equation, such as the homotopy analysis method [45], the optimal homotopy asymptotic method [46], the variational iteration method [47], and the Adomian decomposition method [48].In addition, Khataybeh et al. [42] applied the classical operational matrices of the Bernstein polynomial to solve the equation.Also, Parand and Taghavi [49] implemented a collocation method based on a rationally scaled generalized Laguerre function to solve the Blasius equation.

The Falkner-Skan Equation
The boundary layer equations are a significant class of nonlinear ordinary differential equations with several uses in fluid dynamics and physics [50].One of these equations is the stationary Falkner-Skan boundary layer equation.The Falkner-Skan equation was initially put out by Falkner and Skan in 1931 [51].This equation is essential for numerous applications, including fluid mechanics, aerospace, heat transfer, glass applications, and polymer investigations [20].
The velocity ratio parameter is denoted by , and the pressure gradient parameter by .The Equation ( 6) is known as the Blasius equation when  = 0 and  = and  = 1, and the Hiemenz flow problem when  = 1 and  = 1, see [20].
The initial condition  ′′ (0) = −0.832666has been derived by the authors from the boundary condition ′(∞) = , using the Padé approximation method [53], and this value will be utilized in the current paper.As a result, the following are the initial conditions for the Falkner-Skan equation: The Falkner-Skan equation has been solved by using a variety of techniques, like the homotopy perturbation method [54], the homotopy analysis method [55], the Adomian decomposition method [56], the differential transformation method [57], the iterative transformation method [58], the Legendre rational polynomials method [59], the shifted Chebyshev collocation method [60], and the modified rational Bernoulli functions [61].

The Basic Concepts of the Proposed Methods
The fundamental ideas of the suggested methods are presented in this section.In addition, the orthogonal polynomials and operational matrices will be introduced as instruments for improving the ECM approach to obtain novel approximate solutions to specific nonlinear initial and boundary value problems described in section two.

The Basic Concepts of the ECM and Their Operational Matrices
Consider the following  ℎ -order ordinary differential equation [29]: subjected to the initial condition: or with the boundary conditions: Where   ,   , and   are constants and () is a known function.
The essential assumption is that the Equation ( 9) has a unique solution when the initial or boundary conditions are determined in the Equations ( 10) or (11).Moreover, a linear combination of  ℎ -order functional series based on standard polynomials may be used to represent the unknown function () as follows: where () = [1   2  3 …   ] and  = [ 0  1  2 …   ]  , such that   ,  = 0, … , , are the coefficients, whose values will be specified later.
Assume () has the following derivatives:  , where  * (+1)×(+1), is the operational matrix and its entry values are from the following in the standard polynomials: Thus, the forms presented below can be used to define the derivatives of the function ():  () () = () ( * )  , where,  ≥ 1.
(13) Then, the Equations ( 12) and ( 13) are substituted into the Equations ( 9), (10), and (11), to provide the following result: (, () , ()  * , () ( * ) 2 , … , () ( * )  ) = (), ( 14) and, ( Moreover, the inner product in the Hilbert space  =  2 [0,1], is defined as follows: Also, the set of functions  = { 0 ,  1 … ,   }, are linearly independent in , where   =   , 0 ≤  ≤ , is the base function of the standard polynomials [29]. Therefore, applying the inner product of the set of base functions  with the left and right sides of the Equation ( 14), as given in the Equation ( 17), yields the matrix equation shown below [31]: where the  ℎ row of  and  in the matrix equation given in the Equation ( 18) includes the following: Finally, some of the entries in the matrix equation (Equation( 18)) will be modified when the initial or boundary conditions from the Equations ( 15) and ( 16) are substituted.As a result, a system of ( + 1) non-linear algebraic equations are produced, with unknown coefficients .Then, solve these algebraic equations numerically with applicable programs or sometimes analytically.Unique values for the unknown coefficients  = [ 0  1  2 …   ] can be acquired, which are substituted into the Equation ( 12) to obtain the approximate solution of the Equation (9).
For instance, if  is even, the   * is written as follows: .
The matrix   * is also defined as follows if  is odd: Consequently, the derivatives of the function () have the following form:

The Implementation of the ECM and I-ECMs and Numerical Results
The proposed methods of the ECM and the I-ECMs will be applied in this section to find novel approximate solutions, and the numerical results will be presented for three nonlinear problems: the Darcy-Brinkman-Forchheimer equation, the Blasius equation, and the Falkner-Skan equation.
The I-ECMs are based on the base functions of diverse polynomials such as Chebyshev, Bernoulli, and Laguerre polynomials, introduced in Equations ( 20), (23), and (26), respectively, with relevant operational matrices.These polynomials are performed in two steps of the proposed method's procedures to improve the ECM's accuracy and reliability.First, describe the unknown function () and its derivatives; and second, calculate the inner product to solve the left and right sides of the matrix equation explained in Equation (18).Furthermore, the initial or boundary conditions are substituted, as specified in Equations ( 15) and ( 16), and some entries of Equation ( 18) are modified.Therefore, we obtain ( + 1) nonlinear algebraic equations for the unknown coefficients.By solving this system numerically using Mathematica ® 12, we get the values for the unknown coefficients  0 ,  1 ,  2 , … ,   to obtain a novel approximate solution to the nonlinear initial and boundary value problems.

Solving the Darcy-Brinkman-Forchheimer Equation by the ECM and I-ECMs
The ECM and the I-ECMs techniques are used to solve the first problem presented in the Equation ( 1) with boundary conditions in Equation (2).More precisely, for the ECM technique, we transform the function () and its derivatives into matrices by substituting Equations ( 12) and ( 13) into Equations ( 1) and ( 2).Thus, we get the following result: 0)  *  = 0, (1)  = 0. (29) Then, the procedures have been applied, as shown in Equations ( 18) and ( 19), leading to: Substituting Equations ( 21) and ( 22) into Equations ( 1) and ( 2) for the I-ECMs based on the first kind of the Chebyshev polynomials, the following result is obtained: * (0) = 0,   (1) = 0. (31) Additionally, the results of applying Equations ( 18) and ( 19) are as follows:  18) and ( 19), the following equation will be given: Moreover, applying the I-ECMs based on the Laguerre polynomials by substituting the Equations ( 27) and ( 28) into the Equations ( 1) and ( 2), we obtain: ) Subsequently, the procedures as specified in Equations ( 18) and ( 19) have been utilized, as will be illustrated: Additionally, the inner product for the left and right sides of Equations ( 30), ( 32), (34), and (36), respectively, is used to get the values of  = [ 0  1  2 …   ]  by solving the algebraic system of equations.Once the boundary conditions have been applied to Equations ( 29), ( 31), (33), and (35), respectively, the desired novel approximate solutions are obtained.
In addition, by utilizing the I-ECMs based on the Laguerre polynomials, we get: () ≈ 0.323852 − 0.285634  2 + 7.16738 × 10 −7  3 − 0.0392277  4  + 0.0000418568  5 + 0.000435078  6 + 0.0002453  7 + 0.000137559  8 + 0.000180872  9 − 0.0000324758  10 .Furthermore, since the exact solution to the Darcy-Brinkman-Forchheimer equation is unknown, the   has been calculated to determine the accuracy and reliability of the novel approximate solution produced by the proposed approaches.The   is calculated by:  Also, Figure 3 presents the comparison between the novel approximate solutions calculated by the proposed techniques for  = 10,  = 1,  = 1, and  = 1.It is evident that impressive agreements have been achieved for all the suggested methods.Moreover, the values of the   for the novel approximate solutions utilizing ECM and I-ECMs are also shown in Table 1 with  = 10 and parameters  =  = 1, versus the value of , which offers the accuracy of these techniques.In addition, it can be observed that the I-ECMs based on the Chebyshev polynomial method provide slightly better accuracy with the lowest number of errors compared to other techniques.
Table 1.The comparison between the  10 when  =  = 1, and versus the value of  for the Darcy-Brinkman-Forchheimer equation.

Solving the Blasius Equation by the ECM and I-ECMs
The ECM and the I-ECMs techniques are utilized to solve the second problem shown in the Equations ( 3) and (5).More precisely, we substitute Equations ( 12) and (13) into Equations ( 3) and ( 5) for the technique ECM, converting the function () and its derivatives as matrices.Thus, we obtain the following result: = 0, (0)  *  = 0, (0) ( * ) 2  = .(37) Then, the processes have been applied, as shown in the Equations ( 18) and ( 19), so: Substituting Equations ( 21) and (22) into Equations ( 3) and ( 5) for the I-ECMs based on the first kind of Chebyshev polynomials, it follows: (0) = 0,     * (0) = 0,   (  * ) 2 (0) = .(39) And, the results of implementing Equations ( 18) and ( 19) are as follows: In this problem, we consider the value of  = 0.3320573, as in [43] with  = 10.The novel approximate polynomials for the Blasius equation are: By using the ECM based on the standard polynomials: () ≈ 0.166029  2 + 3.40035 × 10 Figure 4 shows the logarithmic plots for the   values obtained by the ECM based on the standard polynomials and by the I-ECMs based on the Chebyshev, Bernoulli, and Laguerre polynomials, for  = 3 to 10, with a value of  = 0.3320573, according to previous studies [43].The accuracy and efficiency of these methods can be demonstrated by observing the error values for , as we observed that the error decreases as the value of  increases.
(45) Then, the processes have been used as presented in the Equations ( 18) and (19), so: 〈  , () ( * ) 3  + (() )(() ( * ) 2 ) +  [−(()  * ) 2 ] 〉 = 〈  , −   2 〉, ∀ 0 ≤  ≤ .(46) Substituting Equations ( 21) and ( 22) into Equations ( 6) and ( 8) for the I-ECMs based on the first kind of the Chebyshev polynomials yields the following:   (  The novel approximate polynomials for the Falkner-Skan equation when the parameter values are as follows:  = 0.1, β = 0.5, as in [53], with =8, will be: By implementing the ECM based on the standard polynomials: () ≈ 0.9  − 0.416333  2 + 0.0666511  3 + 0.0000592155  4 − 0.00313186  5 + 0.000639976  6 + 0.0000210854  7 − 0.0000188788  8 .Also, by applying the I-ECMs based on the first kind of the Chebyshev polynomials, we obtain: () ≈ 0.9  − 0.416333  2 + 0.0666646  3 + 0.0000169313  4 − 0.00305864  5  + 0.00056836  6 + 0.0000581686  7 − 0.0000267944  8 .In addition, by utilizing the I-ECMs based on the Bernoulli polynomials, we get: () ≈ 0.9  − 0.416333  2 + 0.0666648  3 + 0.0000164756  4 − 0.00305823  5 + 0.000568589  6 + 0.000057627  7 − 0.0000265718  8 .Moreover, by using the I-ECMs based on the Laguerre polynomials, we achieve: () ≈ −8.72066 × 10 −14 + 0.9  − 0.416333  2 + 0.0666602  3 + 0.0000532904  4  − 0.00316698  5 + 0.00071942  6 − 0.0000423225  7 − 9.73011 × 10 −7  8 .Since there is no exact solution to the Falkner-Skan equation, the   is computed in order to verify the efficiency and accuracy of the novel approximate solutions found by the ECM and the I-ECMs.The   is calculated by: Figure 6 exhibits the logarithmic plots for the   values obtained for the parameters  = 0.1, and  = 0.5, according to studies [53], by the ECM based on the standard polynomials and by the I-ECMs based on the Chebyshev, Bernoulli, and Laguerre polynomials, which demonstrate the accuracy and efficiency of these techniques by observing the error values for  = 2 to 8. We observe that when  is increased, the error decreased.Also, Table 3 shows the   values for the novel approximate solutions achieved with the ECM and the I-ECMs with  = 8, explaining the accuracy of these methods.Moreover, the I-ECMs based on the Bernoulli polynomials method, offer slightly better accuracy and fewer errors than the other methods.Moreover, Figure 7 shows the comparison between the novel approximate solutions calculated by the proposed techniques for  = 8,  = 0.1, and  = 0.5.The figure shows that all of the suggested approaches exhibited good agreement.In addition, Figures (8 and 9) explain the logarithmic plots of the   for the novel approximate solutions of the Falkner-Skan equation with  = 2 to 8, using the ECM and the I-ECMs 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 [53].In Figures (8 and 9), the errors decrease when the value of  is increased.Furthermore, Figures (10 and 11) illustrate the logarithmic plots of the   for the novel approximate solutions of the Falkner-Skan equation with  = 2 to 8, by using the ECM and the I-ECMs for different values of  when fixed the parameter  = 0.1.In Figures (10 and 11), it is evident that the errors increase as the values of  increase.

Conclusions
In this paper, the effective computational method based on standard polynomials and the novel effective computational methods based on the three different types of Chebyshev, Bernoulli, and Laguerre polynomials have been implemented to solve three nonlinear models involving initial and boundary value problems.Three models, which are well-known nonlinear problems: the Darcy-Brinkman-Forchheimer model, the Blasius model, and the Falkner-Skan model, have been presented and solved by using our suggested methods.The nonlinear problems are reduced to a nonlinear algebraic system of equations solved with Mathematica ® 12.The novel approximate solutions were obtained and proved accurate and reliable, even within a few polynomial orders.Moreover, the   for the proposed methods were calculated.The results show that the proposed approaches have higher accuracy and less error.It is also observed that the   results of the proposed methods I-ECMs decrease vastly compared to the ECM.Therefore, the proposed novel methods I-ECMs have better accuracy than the ECM.The main conclusion from the results is that the Chebyshev polynomials-based I-ECMs have slightly better accuracy than the other methods for solving the Darcy-Brinkman-Forchheimer equation.Moreover, the I-ECMs based on the Laguerre polynomials are more accurate than the other methods in solving the Blasius equation.In addition, the I-ECMs based on the Bernoulli polynomials are slightly more accurate than the other methods in solving the Falkner-Skan equation.

Figure 2
Figure 2 presents the logarithmic plots for the   values obtained by the ECM based on the standard polynomials and by the I-ECMs based on the Chebyshev, Bernoulli, and Laguerre polynomials, which prove the efficiency and accuracy of these techniques by observation of the error values for  = 2 to 10, as we found that the error decreases with increasing the values of .

Figure 3 .
Figure 3.The comparison of the solutions to the Darcy-Brinkman-Forchheimer equation by proposed methods.

Figure 4 .
Figure 4. Logarithmic plots of   for the Blasius equation.

Figure 5
Figure 5 also illustrates the comparison between the novel approximate solutions calculated by the proposed techniques for  = 10 and  = 0.3320573.The figure shows that all of the suggested methods have obtained good agreement.

Figure 5 .
Figure 5.The comparison of the solutions for the Blasius equation.

Figure 6 .
Figure 6.Logarithmic plots of   for the Falkner-Skan equation by proposed methods.

Figure 7 .
Figure 7.The comparison of the solutions to the Falkner-Skan equation by proposed methods.

Figure 8 .
Figure 8. Logarithmic plots of   for the Falkner-Skan equation by (a) ECM based on the standard polynomials and (b) I-ECMs based on the Chebyshev polynomials.

Figure 9 .
Figure 9. Logarithmic plots of   for the Falkner-Skan equation by (a) I-ECMs based on the Bernoulli polynomials and (b) I-ECMs based on the Laguerre polynomials.

Figure 10 .Figure 11 .
Figure 10.Logarithmic plots of   for the Falkner-Skan equation by (a) ECM based on the standard polynomials and (b) I-ECMs based on the Chebyshev polynomials.
The exact solution to the Blasius equation is not available.Hence, the   has been calculated to demonstrate the accuracy of the novel approximate solutions obtained by the proposed techniques.The   is calculated by:

Table 2 .
The comparison between the  10 for the Blasius equation by proposed methods.

Table 3 .
The comparison between the  8 for the Falkner-Skan equation by proposed methods.