CHEBYSHEV PSEUDOSPECTRAL METHOD FOR DUFFING NONLINEAR DIFFERENTIAL EQUATIONS

The system of Duffing nonlinear differential equations is often used in dynamics, which are known to describe many important oscillating phenomena in nonlinear engineering systems. This article presents the pseudospectral method to calculate numerical solutions for nonlinear Duffing differential equations on the interval [ – 1, 1]. This method is based on the differential matrix using the Chebyshev Gauss – Lobatto points. To find numerical solutions of the nonlinear Duffing differential equations, we have built an iterative procedure. The software is used in calculating in this study is Mathematica 10.4. The obtained results show that this method has high accuracy with very small errors.


Introduction
The Duffing equation was known in 1918 in the article with the title "Forced oscillations with variable natural frequency and their technical significance" of George Duffing [1]. Since the appearance of the paper, there are many authors have been studied, expanded and developed the Duffing equation [2][3][4][5][6]. Simultaneously, many numerical methods also were studied to solve that equation force [4][5][6][7].
Consider the most general forced form of the Duffing nonlinear differential equation (or Duffing oscillator) here, the numbers are given constants, in which: is controls the amount of damping;controls the amount of non-linearity in the restoring force;controls the linear stiffness; is the amplitude of the periodic driving force; is the angular frequency of the periodic driving force [4][5][6].
Depending on the choice of the and , we had some the following special cases: these special cases had been studied in the literature of Richard [5,Chapter 8].
Besides, depending on the parametric values and  , the problem (1) may be become other problems, for example: If and then (1) becomes the cubic free undamped Duffing oscillator [2][3] Several numerical solutions have been studied so far dealing with the Duffing differential equation such as the modified differential transform method to obtain the approximate solutions of the nonlinear Duffing oscillator [11]; the collocation method is based on the radial basis functions to approximate the solution of the nonlinear controlled Duffing oscillator [12]; M. A. Al-Jawary proposed the Daftardar-Jafari method to solve the Duffing equations and to find the exact solution and numerical solutions [13]; M. Gorji-Bandpy applied Modified Homotopy Perturbation Method and the Max-Min approach to study the generalized Duffing equation [14]; in [15], the authors employed the new perturbation technique to solve strongly nonlinear Duffing oscillators; in [16], the authors used the Taylor Expansion to find approximate solution of Nonlinear Duffing Oscillator; to find numerical solution of the Duffing oscillator, the authors in [17][18] used the Legendre pseudospectral method, the authors in [19] used the spectral method, the authors in [20] used the Taylor matrix method; in [21], the authors proposed the post-verification method for solving the forced Duffing oscillator problems without prescribed periods; the analytical approximation technique basing on the energy balance method was used to determine approximate solutions for highly nonlinear Duffing oscillator [22][23]; the block multistep method is integrated with a variable order step size algorithm to find numerical solutions of the nonlinear Duffing oscillator [24].
This article uses the pseudospectral method based on Chebyshev differential matrix [25] to determine approximate solutions with the boundary conditions on the interval   1,1  take the form ( 1) , (1) .

Chebyshev differentiation matrix for Chebyshev Gauss -Lobatto points
The following can be shown: As we know the values of ( ) p x at 1 N  points, we would like to find approximately the values of the derivative of ( ) p x at those points, Evidently, the derivative of ( ) We has the entries (1) , (1) , where k c is determined by the formula (8).
with the entries , or they are identified as follows [30][31]: The entries of the second-order differentiation matrix satisfy the identity (2)

Chebyshev pseudospectral method Suppose that
and the collocation points { } i t so that We know that Therefore, problem (13) becomes 2 , Alternately, we partition the matrix C D into matrices.
Consider the matrix D C , we cut off the first-row (1) 0,i d and last-row (1) , N i d with e i J  , then we partition that matrix into three matrices: (1) (1) 0 .
And we can rewrite in the matrix form Using the Chebyshev pseudospectral method to solve number problems (13), the simple second-order differential equation (13) can rewrite in the matrix form (17). Thence, we find numerical solution x . Section 4 and 5 will present its application for the Duffing nonlinear differential equations.

Applications
Consider the Duffing nonlinear differential equation with the boundary conditions We apply the Section 3 to the equation (18),

( ) d x t dt
can be written in the matrix form (1) ( where P denotes the diagonal matrice with elements   cos Similarly, the cubic free undamped Duffing oscillator (2) can rewrite in the matrix form as follows: where P 1 denotes the diagonal matrice with elements   2 ( ) The Ueda oscillator (3) in the matrix form: where P 2 denotes the diagonal matrice with elements   The matrix form of the Duffing -Holmes nonautonomous oscillator (4) as follows:  (20), (21) and (22) we may be able to approach it with an iterative procedure has the following:

Procedure FindSolution;
Begin set (old) : T u I  ; : 1   ; 8 10 er here I is the unit vector, er is the error that might change.

Numerical results
With the equation (2), we utilize the inhomogeneous boundary conditions, which mean In figures, dots illustrate numerical solutions of CPSM and solid lines illustrate numerical results of Mathematica's NDSolve. The Fig. 1 and 2 illustrate numerical solutions of equations (1) and (2) (3) and (4) Table 2.   Table 2.
The obtained results of the equations (1), (2), (3) and (4) shown in Table 1 and 2 show that this method has high accuracy with very small errors.

Conclusion
We present the pseudospectral method basing on the differentiation matrix using the Chebyshev Gauss -Lobatto points to calculate numerical solutions for nonlinear Duffing differential equations on the interval [-1, 1]. We use the iterative procedure to find numerical solutions of the Duffing nonlinear differential equations and consider four special cases of the Duffing differential equations system. The numerical results demonstrate the efficiency and of the reliable method for solving this problem.