Two meshless methods for solving nonlinear ordinary differential equations in engineering and applied sciences

Abstract In this paper, two meshless methods have been introduced to solve some nonlinear problems arising in engineering and applied sciences. These two methods include the operational matrix Bernstein polynomials and the operational matrix with Chebyshev polynomials. They provide an approximate solution by converting the nonlinear differential equation into a system of nonlinear algebraic equations, which is solved by using Mathematica® 10. Four applications, which are the well-known nonlinear problems: the magnetohydrodynamic squeezing fluid, the Jeffery-Hamel flow, the straight fin problem and the Falkner-Skan equation are presented and solved using the proposed methods. To illustrate the accuracy and efficiency of the proposed methods, the maximum error remainder is calculated. The results shown that the proposed methods are accurate, reliable, time saving and effective. In addition, the approximate solutions are compared with the fourth order Runge-Kutta method (RK4) achieving good agreements.


Introduction
Nonlinear ordinary di erential equations (NODE) play a signi cant role in all branches of science and engineering. Many material events can be formulated by the di erential equation. These types of equations appear in a wide range of problems including, but not limited to, uid mechanics, chemical matters, electricity and electronics. Therefore, the need for reliable and e ective method to solve this kind of equation has become a very important requirement [1].
Orthogonal functions and polynomials are instruments extremely helpful in approximation theory and numerical analysis [2].The main feature of this technique is to simplify the solution by converting the equation into a system of algebraic equations. Accordingly, simplifying these problems substantially, and approximate the unknown function by using the polynomial series and then using the operational matrices to get rid of the integration and di erentiation.
In the last few years, there was much research done by using the Bernstein polynomials. Bernstein polynomials were introduced in [2], which are used to solve the differential equations. In Ref. [3] authors proposed the collocation method by Bernstein polynomials to solve the variational problems. Also, Alshbool et al. presented the approximate solution of singular nonlinear di erential equations by using a collocation method and Bernstein polynomials [4]. Moreover, the Bernstein polynomials have been successfully employed for a class of boundary value problems [5]. In addition, Khataybeh et al. havesolved directly third-order ODEs by using the Bernstein operational matrices method [6].
On the other hand, there are several works dealing with the use of the Chebyshev operational matrix method. Sharma et al. have implemented the Chebyshev operational method to solve Lane-Emden problem [7]. Also, Öztür was studied the solution of the system of di erential equations by using the Chebyshev operational matrix method. Rajeev and Raigar have used a numerical approach based on the Chebyshev wavelets operational matrix to solve Stefan problem [9]. Recently, Hashemizadeh, and Mahmoudi have used shifted Chebyshev operational matrix to solve the Physiology problems [10].
The main objective of this paper is to implement two meshless methods the operational matrices methods based on Bernstein polynomials and the Chebyshev polynomials to solve some NODE that appear in engineering and applied sciences.
This paper has been organized as follows: In Section 2 the Bernstein polynomials and their operational matrices are introduced. In Section 3 the rst kind shifted Chebyshev polynomials and their operational matrices are considered. In Section 4, some nonlinear problems will be introduced and solved by using the proposed methods. Finally, the conclusion will be given in Section 5.

Bernstein polynomials
The nth degree of Bernstein Polynomials on the interval [ , ] are de ned by [11]: where There are n + , nth degree Bernstein Polynomials. For mathematical convenience, the equation B i,n (x) = if i < or n < i is often used. These polynomials have many useful and important properties making it bene cial, some of these properties in the following list [11] (a) Positivity property: (c) Recursion's relation property: It is easily to approximate any polynomial of degree n to the form of linear combination as given below, where C T = [C , C , . . . , Cn] and Φ(x) = [B ,n , B ,n , . . . , Bn,n] T Also, we could make the disintegration of the vector Φ(x) as a multiplication of a square matrix of size (n + )×(n + ) and vector (n De ne vector A i+ The matrix A is an upper triangular matrix and |A| = n i= n i , so that A is an invertible matrix.

. Operational matrix of product for Bernstein polynomials
In Section 2, we obtain a clear formula for the Bernstein polynomials of n-th degree, now let us introduce the operational matrix of product. Let D is an (n + ) × (n + ) operational matrix of the derivative, then The matrix D is given by D = AσA − , where σ is (n + ) × (n) matrix [2] Equation (5) can be written for higher derivative as follows Now, we can use this operational matrix to solve the equation by approximate the unknown function y(x) as y (x) = C T Φ(x) and we have This approximation is applied to all conditions of the equation as well.
We select the Chebyshev roots as the collocation node, We will change the unknown functions and their derivatives using Eqs. (2) and (8), and then replace the nodes in its. The resulting were compensated in the NODE to obtain a system of algebraic equations that we so resolve by computer programs such as Mathematica or MATLAB. The convergent analysis of this method was illustrated by [2].

First kind Chebyshev polynomials and the operational matrices
The shifted Chebyshev polynomials (the rst kind) Pn(x) of degree n is de ned by [12] Pn The unknown function y(x) ∈ L [ , ] can be approximated as the form where . . In general, we consider only the rst (n+1)-terms of the shifted Chebyshev polynomials. Hence, we write where where D * is (n + )× (n + ) operational matrix of derivative given by where η = and η k = for k ≥ [7]. If n is even the matrix D * de ned as fallows Also, if n is odd then k=1,3,5,. . . , n and the matrix D * can be written as By using Eq. (13), we can write the n-th derivative as Thus, the operational matrix can be applied to solve the NODE by approximate the unknown function y (x) = C T Φ(x) and their derivatives by We will replace unknown functions and derivatives using Eqs. (12) and (17), and then they are compensated in NODE and their conditions, we then assign the contract in these equations to obtain a system of nonlinear equations (n + ) which can be solved using by the computer programs Mathematica or MATLAB to get the coe cients of vector C T .

Test problems
In this section, the Bernstein and Chebyshev operational matrices methods will be implemented to solve some problems of NODEs that appear in engineering and applied sciences.

. Magnetohydrodynamic (MHD) squeezing fluid flow
The squeezing uid ow in a porous medium with magnetic eld in uence shows in Figure 1 has the governing equations [13]: where W is the velocity vector, J the electric current density and B = B + b is the total magnetic eld and B , b are the imposed and induced magnetic elds, respectively. T = −pI+µA is the Cauchy stress tensor with A = ∇W+ (∇W) t and r is the Darcy's resistance.
The MHD force J × B can be written as follows: Suppose that the magnetic eld is enforced along zaxis and the plates are non conducting. For small velocity w, the gap distance L between the panels slowly changes over time t so that it can be used as a constant.
For axial symmetry, the components W for the present case are W = (wr , , wz). If the generalized pressure given by P = ρ w r + w z + p and the ow are steady then the Navier-Stokes Eq. (18) can be written as Using the transformation ψ (r , z ) = r f (z) [14], eliminating the pressure P from Eqs. (19) and (20), we get with boundary conditions Using the following dimensionless parameters and omitting the (*) then the Eqs. (21) and (22) become In our work, we take m = R = .

. . Solving the MHD squeezing fluid by Bernstein and Chebyshev operational matrices methods
The two meshless methods will be applied to solve the rst problem by using the Bernstein and rst kind shifted Chebyshev operational matrices. The rst step, let us write the function F(z) and its derivatives as matrices by using Eqs. (2) and (8), for the Bernstein polynomial, we get Also, by using the Eqs. (12) and (17) for the Chebyshev polynomial, we obtain To nd the values of the unknown coe cients C T = [c , c , . . ., cn] we get the algebraic system by substitute the collocation nodes in Eq. (9) into Eq. (26) or Eq. (27) and solve it.
The following approximate polynomial for this problem when n = , R = , m = , will be obtained: • By using Bernstein polynomial operational matrices In order to inspect the accuracy for the solution obtained by suggested methods for Eq. (24), we calculate the maximal error remainder MERn since the exact solution is unknown.
The error remainder function for MHD squeezing uid can be de ned as follows and the MERn is Figure 2 shows that the logarithmic plots for MERn of the approximate solution obtained by the methods of Bernstein and Chebyshev operational matrices which indicate the e ciency of these methods. In Figure 2 we taken n=6, 7,8,9,10,11, we can see the e ciency by increasing n, the errors will be decreasing, this can be described in both cases.
Moreover, Figure 2 presents the method of the operational matrix which depends on Bernstein polynomial is more e ective than the Chebyshev polynomial. Further numerical investigation can be made, a numerical comparison has been made between the solutions obtained the suggested methods and the Range-Kutta (RK4) method when n = . This comparison is illustrated in Figure 3, it can be seen a good agreements between the approximate solutions and RK4.

. Je ery-Hamel fluid flow
The ow between two nonparallel walls is one of the most important problems in uid mechanics because of the wide range of applications [15].
The governing equations for the Je ery-Hamel ow incompressible viscous uid that is present at the intersection of two rigid, nonparallel plane walls; angle between walls is 2α. Flow is supposed to be purely radial and symmetric. Thus the velocity eld can be denoted by V = [ur , , ] where ur = u(r, θ) is the velocity along radial direction, see Figure 4 [16].
The continuity and Navier-Stokes equations in the polar coordinates can be written as follows where B represents the electromagnetic induction and σ, p , ρ, v are the conductivity of the uid, pressure gradient, constant density of uid and kinematic viscosity respectively. We can written Eq.(30) as the form Dimensionless parameters are de ned by [15] By eliminating pressure terms from Eqs. (31) and (32) and using the formulations in Eqs. (33) and (34), we obtain a third-order di erential equation:  .

. Solving Je ery-Hamel fluid flow by Bernstein and Chebyshev operational matrix methods
The procedures for the methods of the operational matrices based on Bernstein and Chebyshev polynomials can be applied to solve the Eq. (35) with boundary conditions Eq. (36). Replace the F (η) and its derivatives as matrices by using Eqs. (2) and (8), for the Bernstein polynomial, we get If we use the Eqs. (12) and (17) for the Chebyshev polynomial, we obtain Calculate the values of C T = [c , c , . . ., cn] by solving the algebraic system obtained by substitute the collocation nodes on the Eqs. (37) and (38).
The following approximate polynomial for this problem when n = , α = • , Re = , Ha = will be obtained • By using Bernstein polynomial operational matrices • By using shifted Chebyshev polynomial operational matrices To examine the accuracy of the obtained approximate solution by using the Bernstein and Chebyshev polynomials with operational matrices to solve this problem, we dened the maximal error remainder MERn as the form and the MERn is Figure 5 illustrates that the logarithmic plots for MERn of the approximate solution gotten by solving this problem by using the operational matrix methods based on the Bernstein or Chebyshev polynomials which indicate the efciency of these methods. The values of n has been considered n=4 to 11, the e ciency has been seen by the errors will be decreasing when n increasing.
In additional, numerical comparison between the solutions calculated by the suggested methods and the RK4 method when n = . This comparison is presented in Figure 6, the good agreements between the approximate solutions and RK4 can be clearly seen.

. Straight n problem
Nonlinear n problems are important in engineering and applied sciences because of their applications and the wide uses in the di erent eld of science and technology [17].
The 1D straight n is presented in this form where k(T) is the temperature with heat transfer coecient or the thermal conductivity, T is the distribution of the heat on the n, S is some arbitrary constant area (crosssectional), L is the n length, p is the perimeter, h represents the heat transfer coe cient, Ta and T b are temperatures of base surface and surrounding uid, in a respective way, see Figure 7 [17]. The thermal conductivity of the n material is assumed to be a linear function of temperature as where ka is the thermal n conductivity at Ta and γ is a parameter without dimensions which represented the variance in the temperature conductivity [18].
The nonlinear equation of the straight energy described as a balance of ns extending to the surface under the in uence was obtained by temperature-dependent thermal conductivity.
with boundary conditions: where u is an unknown function which represents the temperature without dimensions, µ the parameter of the thermal n, β parameter of the thermal conductivity, a is a parameter can be evaluated, see [19]. .

. Solving straight n problem by Bernstein and Chebyshev operational matrices methods
The operational matrices based on Bernstein and Chebyshev polynomials can be applied to solve the Eq. (44) with boundary conditions Eq. (45). We will replace the unknown function u (x) and their di erentials as an operational matrices form, Eqs. (2) and (8) will be used for the Bernstein polynomial, we get Eqs. (12) and (17) for the Chebyshev polynomial can be used in Eqs. (44) and (45), we obtain (47) The algebraic system obtained by substitute the collocation nodes on the Eqs. (46) and (47).
, the value of a is taking from [19]. • By using Bernstein polynomial operational matrices • By using shifted Chebyshev polynomial operational matrices To examine the accuracy of the obtained approximate solution by using the Bernstein and Chebyshev polynomials with operational matrices to solve this problem, we dened the maximal error remainder MERn as the form and the MERn is The logarithmic plots for MERn of the approximate solution has been shown in Figure 8 by solved this problem using the operational matrix methods for both Bernstein and Chebyshev polynomials which indicate the e ciency of these methods. The values of n has been presented in Figure 8, we taken n=4 till 10, the e ciency has been seen by the errors will be decreasing when n increasing. Also, it can be seen clearly the error of using the Bernstein polynomial is less than using the Chebyshev polynomial this indicates the operational matrix method with Bernstein polynomial provides better accuracy. Also, the numerical comparison between the solutions calculated by the suggested methods and the RK4 method when n = is given in Figure 9. The good agreements between the approximate solutions and RK4 can be noticed. .

. The Falkner-Skan equation
The Falkner-Skan equation is classi ed as one of the nonlinear ordinary di erential equations of the third order which has a large number of applications, such as insulation materials, glass applications and polymer studies [20]. The Falkner-Skan equation represents in the study of laminar boundary layers displaying similarity. The problem is given by with boundary conditions as the following: where β is the pressure gradient parameter and ϵ is velocity ratio parameter. When β = 0, Eq (50) is called the Blasius, when β = , Eq. ( ) is called the Hiemenz ow problem and when β = , Eq. (50) represent to the Homann ow problem.
To solve this problem we can nd the numerical value of the missing boundary conditions by using the Padé approximate method. For more details see [21].
Thus, the boundary conditions become We will use the value of a = − . as given in [21].

. . Solving Falkner-Skan equation by Bernstein and Chebyshev operational matrices methods
The operational matrices based on Bernstein and Chebyshev polynomials will be used to solve the Eq. (50) with boundary conditions represented in Eq. (52). The unknown function y (x) and their di erentials have been exchanged as an operational matrices form, Eqs. (2) and (8) for the Bernstein polynomial has been used to get equations as the form follows Eqs. (12) and (17) for the Chebyshev polynomial can be used in Eqs.(50) and (52), we obtain (54) The algebraic system has been achieved by substituting the collocation nodes on the Eqs. (53) and (54).
We consider n= 4 to 12 in this problem. If n = , β = . , ∈ = . , a = − . , (for calculate the value of a, see [21]) the following approximate polynomial has been gotten • By using Bernstein polynomial operational matrices • By using shifted Chebyshev polynomial operational matrices To check the accuracy of the gotten approximate solution by using the Bernstein or Chebyshev polynomials with operational matrices to solve this problem, the maximal error remainder MERn has been de ned as the form and the MERn is The logarithmic plots for MERn of the approximate solution by solved this problem by using the operational matrix methods based on the Bernstein or Chebyshev polynomials have been given in Figure 10 which indicates the e ciency of these methods. We can see the e ciency for these methods when n is increasing, the errors reminders are decreasing. Moreover, the operational matrix method based on the Bernstein polynomial provides better accuracy.
In addition, the comparison numerically between the solutions calculated by the suggested methods and the RK4 method when n=12, has been presented in Figure 11. Furthermore, the good agreements between the approximate solutions and RK4 can be seen clearly.
Also, we have calculated the run time of the proposed methods and RK4. It can be seen from Table 1 that running time for the proposed methods are faster compared to the RK4 method in seconds.

Conclusion
In this paper, the methods of operational matrices based on the two di erent types of Bernstein and Chebyshev polynomials to solve several problems containing NODEs have been used. Mentioned problems arise in engineering and applied sciences. Each problem has been solved and an approximate solution has been obtained by approximate the unknown function by the polynomial series and uses the operational matrices to eliminate of the di erentiation from the equation. Moreover, the problems have been solved numerically by using the RK4 method to compare the numerical result with the approximate solutions and the compatibility was good among them. Furthermore, both methods have provided a good accuracy, however, the operational matrix method based on the Bernstein polynomial provides better accuracy. Mathematica ® 10 software has been used for calculations on this study.