Numerical solution of Painleve ' equation by Chebyshev polynomials

In this paper, the Chebyshev polynomials and the collocation method are used for solving Painleve' equation. The accuracy of this method is shown by solving a sample example and the proposed scheme is compared with other methods to illustrate the efficiency of the method.


Introduction
The Painleve' equation and their solutions arise in parts of pure and applied mathematics and theoretical physics such as the second Painleve' equation as model for the electric field in a semiconductor [1].It is known that any rational differential equation of second order with the Painleve' property is reduced to one of the six Painleve' equations unless it can be integrated algebraically, or transformed into a simpler equation such as the linear differential equations or the differential equations of the elliptic functions.Generic solutions of the Painleve' equations are known to be very transcendental.For the general background of Painleve' equations [2][3][4][5][6][7][8][9].A lot of works have been done in order to find the numerical solution of this equation.For example, reduction KdV and cylindrical KdV equations to Painleve' equation [10], numerical studies of the fourth Painleve' equation [11], variational iteration method and homotopy perturbation method [12], the numerical solution of the second Painleve' equation [13], on comparison for series and numerical solutions for second Painleve' equation [14].In this paper, Chebyshev polynomials and the collocation methods are developed to solve the Painleve' equation as follows: with the initial conditions given by: (0) = 0,  ′ (0) = 1.To obtain the approximate solution of Eq. (1.1), by integrating two times from Eq. (1.1) with respect to  and using the initial conditions we obtain, () = () + 6 ∫ ∫ (()), The double integrals in (1.2) can be written as [15]:  0 Now, we decompose the unknown function () by following decomposition series: In (1.3), we assume () is bounded for all  in  = [0, ] and The paper is organized as follows: in section 2, we introduce Chebyshev polynomials.In section 3, Eq.(1.1) is solved by the Chebyshev polynomials and the collocation methods.In section 4, the proposed method is tested with an example and the results are compared with the results of other methods.Finally, the concluding remarks are determined in section 5.
Hence, the zeros of   (), are

Description of the method
In this section we are going to solve the Eq.(1.1) by using the Chebyshev polynomials method.To obtain the approximation solution of Eq.( 1.1), according to the Chebyshev polynomials method of the first kind   () we can write: () = () ∑      =0 () =   ().

Numerical example
In this section, a Painleve' equation is solved by applying the presented method.We compare our results with the results of other methods.
Example 4.1.Consider the Painleve' equation.The numerical results for solving this equations by Chebyshev polynomials and collocation method is presented in Table 1, also we presented some other method results for solving this equation.

Conclusion
In this study, the Chebyshev polynomials and collocation method have been presented to solve Painleve' equation.The method have been successfully employed to obtain the approximate solution of the Painleve' equation.Illustrative example was given to demonstrate the validity and applicability of proposed method by comparing with other methods.

Table 1 :
Numerical results for Example 4.1.