Effective Computational Methods for Solving the Jeffery-Hamel Flow Problem

In this paper, the effective computational method (ECM) based on the standard monomial polynomial has been implemented to solve the nonlinear Jeffery-Hamel flow problem. Moreover, novel effective computational methods have been developed and suggested in this study by suitable base functions, namely Chebyshev, Bernstein, Legendre, and Hermite polynomials. The utilization of the base functions converts the nonlinear problem to a nonlinear algebraic system of equations, which is then resolved using the Mathematica ® 12 program. The development of effective computational methods (D-ECM) has been applied to solve the nonlinear Jeffery-Hamel flow problem, then a comparison between the methods has been shown. Furthermore, the maximum error remainder ( 𝑀𝐸𝑅 𝑛 ) has been calculated to exhibit the reliability of the suggested methods. The results persuasively prove that ECM and D-ECM are accurate, effective, and reliable in getting approximate solutions to the problem.


Introduction:
In several fields of engineering and applied sciences, nonlinear ordinary differential equations (NODE) play a significant role in simulating many real-life issues. Many phenomena, including engineering, fluid mechanics, physics, chemical matters, biology, and electrostatics, have been mathematically formulated using these types of equations. The exact solution for nonlinear problems is difficult or sometimes cannot be obtainable. Therefore authors want to develop efficient either numerical or approximate methods to solve these types of problems [1][2][3][4] .
Several analytical and approximate methods have been proposed by researchers to solve nonlinear differential equations, such as the Adomian decomposition method (ADM) and Direct Homotopy Analysis Method (DHAM) 5 , the Bernoulli collocation method 6 , the Hemite polynomial method 7 , the Taylor collocation method 8 , and the Gegenbauer wavelet method 9 . In particular, Singh 10 has used the Jacobi collocation method to solve the fractional advection-dispersion equation. Ganji et al. 11 have used the fifth-kind Chebyshev polynomials to solve differential equations with multiple variable orders and nonlocal and non-singular kernels. Also, Singha et al. 12 used Boubaker polynomials to solve a class of fractional optimal control problems. Yuttanan et al. 13 solved the non-linear distributed fractional differential equations using the Legendre wavelets method and some other approximation methods, see [14][15][16] .
One of the most important applications in fluid mechanics and biomechanical engineering is the flow between two nonparallel plates 17 . Jeffery 18 and Hamel 19 introduced incompressible viscous fluid movement in convergent and divergent channels, and this is known as Jeffery-Hamel flow.
More recently, the Turkyilmazoglu has proposed an analytic approximate method namely the effective computational method (ECM), and implemented it to solve various types of problems. For example, Lane-Emden-Fowler singular nonlinear equations 31 , high-order Fredholm integro-differential equations 32 , highorder Volterra-Fredholm-Hammerstein integrodifferential equations 33 , heat transfer of fin problems 34 , and initial and boundary value problems for linear differential equations of any order with difficult exact solutions 35 . The approach was based on well-chosen general-type basis functions, such as classical polynomials, and that exact solution is obtained under particular conditions. A nonlinear equation's solution is also converted into a nonlinear algebraic equations system that can be solved numerically.
Recently, orthogonal functions and polynomials have received a lot of attention from researchers since they are very useful tools and techniques in dealing with many different problems in approximation theory as well as numerical analysis 30 . On the other hand, these techniques are mainly characterized by simplifying the required solution effectively by transforming the problem into a system of algebraic equations, where it can be solved simply by using any computational program [36][37][38][39] . Accordingly, the problems are simplified substantially and the unknown function is approximated using a series of powers of polynomials. Thus, all integrals and differentials are eliminated by using the operational matrices procedure. Furthermore, the literature is full of the applications that have been discussed by OMM of orthogonal polynomials, for instance, see [40][41][42][43] .
The motivation for this research work is our great interest in finding the approximate solutions of the nonlinear ordinary differential equations, in particular the Jeffery-Hamel flow problem, which is one of the most important applications in fluid mechanics and biomechanics. Moreover, this study aims to implement the ECM based on the standard polynomial to solve the Jeffrey-Hamel problem, and another aim is to develop and suggest a novel ECM based on various orthogonal polynomials such as Chebyshev, Bernstein, Legendre, and Hermite polynomials, and then D-ECM has been applied to solve the Jeffery-Hamel flow problem. This paper is organized as follows: The mathematical description of the Jeffery-Hamel flow problem is presented in section two. Section three explains the basic concepts of the proposed methods. Solving the Jeffery-Hamel flow problem by the proposed methods will be given in section four. In section five, the numerical results will be displayed and explained. Finally, in section six, a conclusion will be presented.

The Mathematical Formulation of Jeffrey Hamel's Flow Problem
The Jeffrey-Hamel flow problem represented by the NODE is the steady flow of a viscous, conductive, incompressible fluid in two dimensions at the intersection of two plane rigid and non-parallel walls that get together at an angle 2α 21 . It is assumed that the flow is perfectly radial and symmetric. Therefore, the velocity field is only along the radial direction and depends on and , so it can be given by ( ( , ), 0), as illustrated in (Fig. 1) 30 . where ( , ) is the radial velocity, 0 is denoted by the electromagnetic induction and is a fluid's conductivity, is the pressure of the fluid, is the fluid density constant, and is the kinematic viscosity parameter. Eq.1 can be written as: ( ) = ( , ), 4 By using dimensionless parameters 29 , so where, = . 5 By eliminating term from Eq.2 and Eq.3, and using the formulas given in Eq.4 and Eq.5, a nonlinear third-order ODE is obtained: ′′′ ( ) + 2 ( ) ′ ( ) + (4− ) 2 ′ ( ) = 0, 6 with the boundary conditions as follows: (0) = 1, ′ (0) = 0, (1) = 0, 7 where, = , and 2 = 0 2 , are the Reynolds number and the Hartmann number's square, respectively.

The Basic Concepts of the Proposed Methods
A description of the suggested methods will be presented in this section. Also, orthogonal polynomials and the operational matrices will be offered, which are used in the development of the ECM algorithm to get the approximate solution to the problem.

The Basic Concepts of ECM
Consider ℎ -order non-linear ODE as follows 34 , ( , , ′ , ′′ , … , ( ) ) = ℎ( ), ≤ ≤ . 8 with either the I.C: ( ) ( ) = , 0 ≤ ≤ − 1, 9 or the following B.C: where ℎ( ) is a function that is known and , , , are constants. The essential assumption is that Eq.8 has a unique solution with the initial or boundary conditions given in Eq.9 or Eq.10. Moreover, a function ( ) ∈ 2 [0, 1] can be expressed by a linear combination of ℎ -order function series based on the classical standard monomial polynomials as: where , are the coefficients whose values will be found by giving the following definitions where represents the base functions from the classical polynomials 31 . By using the dot product, the ℎ order approximation of the series solution provided in Eq.11 is as follows: Assume that the derivative of vector will be defined as below where ( +1)×( +1) is the operational auxiliary matrix with the given entries in classical monomials: Also, the higher derivatives can be written as, [ ] = where = 1,2, … 13 Therefore, Eq.13 can be used to write the derivatives in the following format: ( ) ( ) = ≥ 1. 14 Now, substituting the Eqs.12, and 14 in Eqs.8-10, the matrix equation with the restrictions 31 , can be obtained: Consider the Hilbert space = 2 [0,1], which has the inner product as follows: Assume a set of functions that are linearly independent in = { 0 , 1 , … , }, 18 where be the base function of a standard monomial polynomials , ∀ = 0,1,2, … , or any other type of polynomial 31,32 . Then, by applying the inner product given in Eq.17 with the elements of defined in Eq.18, the following matrix equation 33 will be shown: The i th row of and E, respectively, is made up of: In addition, by applying the initial or boundary conditions in Eqs.15, and 16, some entries of Eq.19 are modified from the left-hand side and the corresponding right-hand side 35 . Thus, a system of ( + 1) nonlinear algebraic equations for unknown will be obtained. By solving the resulting system numerically or sometimes analytically, unique values can be obtained for unknown elements 0 , 1 , 2 , … , this will be substituted in Eq.12 to obtain an approximate solution to Eq.8.

First Kind Chebyshev Polynomials
The first kind of Chebyshev polynomials ( ) of degree i is defined by: The unknown function ( ) can be represented as: where, = 〈 , 〉 = (2 + 1) ∫ ( ) 1 0 ( ) ; ≥ 0. In general, only the first ( + 1) terms of the Chebyshev polynomials have been expressed 39 , so Moreover, the derivatives of ∅( ) can be considered as: , is the operational matrix of the provided derivative, which is defined as follows: if is odd, 0 = 2, and = 1 for all ≥ 1. For example, if is even then the is expressed as follows: In addition, if is odd then the matrix is defined as follows: Hence, the derivatives can be written by using Eq.23 in the following form:

Legendre Polynomials
The Legendre polynomials, ( ), on [−1,1] of ℎ -order are defined as 41,42 : Also, the Legendre polynomials ( ) can be obtained in the analytical formula by the following: 31 Furthermore, the( + 1) −terms of polynomials ( ) can be used to approximate the function ( ) as: 859 in two steps of the suggested approach procedure. Firstly, to describe the unknown function ( ) and its derivatives; secondly, to process of calculating the inner product to solve the left and right sides of the matrix equation, which are given in Eq.19. By substituting the initial or boundary conditions in Eqs.15, and 16, some entries of Eq.19 are modified. Thereafter, ( + 1) nonlinear algebraic equations for unknown C can be obtained by solving this system numerically by Mathematica ® 12, where unique values are given for unknown elements 0 , 1 , 2 , … , to achieve the approximate solution to the problem.
The ECM and D-ECM procedures can be used to solve Eq.6 with boundary conditions Eq.7, by using Eqs.12, 14, replacing unknown function ( ) with its derivatives as matrices, for ECM:

The Numerical Results and Discussion:
In this section, an example is presented when the value of = 3, = 5°, = 10, and = 0, to illustrate the approach of the proposed methods to solve the Jeffery-Hamel flow problem.
Furthermore, the maximal error remainder has been introduced in this section because there is no exact solution available to the problem, as well as to verify the accuracy and reliability of the approximate solution obtained by ECM and D-ECM. The is calculated by: = 0≤ ≤1 | ′′′ ( ) + 2 ( ) ′ ( ) + (4− ) 2 ′ ( )| Fig. 2 presents the logarithmic plots for the values, obtained by the ECM based on the standard monomial polynomial, as well as, by the D-ECM based on the Chebyshev, Bernstein, Legendre, and Hermite polynomials, for parameters = 10, = 0 and = 5 ∘ according to previous studies 30 , which showed the efficiency of these methods by observing the error values for = 4 to 12, the error was observed to be lower when the value of increased.

Figure 2. Logarithmic plots for by proposed methods.
A comparison of the approximate solutions obtained using the proposed techniques is also shown in Fig. 3 for = 12, = 10, = 0, and = 5 ∘ , as is evident from the figure, good agreements have been obtained for all proposed methods. Moreover, in Table 1 the values of for the approximate solution is given by using ECM and D-ECM with = 12 and parameters = 10, = 0 and versus the value of , which appears the efficiency of these methods. In addition, it can be noted that D-ECM based on the Hermite polynomials method produces better accuracy with the lowest errors compared to the other methods. Furthermore, in Table 2 the comparisons of   12 values are presented when = 10, = 0, = 5 ∘ , for the solutions by proposed methods and by the Chebyshev and the Bernstein operational matrices methods according to previous studies 30 . Better accuracy can be realized by using the suggested methods.

Conclusion:
The effective computational method and novel computational methods with suitable base functions, namely Chebyshev, Bernstein, Legendre, and Hermite polynomials, have been presented in this paper for solving the Jeffery-Hamel problem. The nonlinear problems are reduced to the solution of a nonlinear algebraic system of equations, which is processed using Mathematica ® 12. The approximate solution is accurate and efficient even within a few orders of polynomials. In addition, the has been calculated for the proposed methods and compared with the Chebyshev and the Bernstein operational matrices methods that are available in the literature, the results obtained showed that the proposed methods have produced better accuracy with less errors. Moreover, it can be concluded that the results of the by the proposed methods D-ECM decreased significantly compared to ECM, which gives higher accuracy and efficiency. Furthermore, it was found that the results of D-ECM based on the Hermite polynomials are better than the other methods.
The present methods can also be extended to partial differential equations and fractional differential equations, which certainly require extensive further analysis.