Approximate Solution of Higher Order Linear Differential Equations by Means of a New Rational Chebyshev Collocation Method

In this paper, a new approximate method for solving higher-order linear ordinary differential equations with variable coefficients under the mixed conditions is presented. The method is based on the rational Chebyshev (RC) Tau, Chebyshev and Taylor collocation methods. The solution is obtained in terms of rational Chebyshev (RC) functions. Also, illustrative examples are given to demonstrate the validity and applicability of the method.


INTRODUCTION
Many problems arising in science and engineering are formulated in bounded and unbounded domains.Recently a number of different methods associated with orthogonal systems for solving higher-order differential equations, which are the Hermite spectral method [1,2], the Laguerre method [3,4], the Jacoby polynomials method [5], the methods based on rational Chebhshev (RC) functions [6,7], the Laguerre tau method [8] and the rational Chebyshev tau method [9], have been studied.
In this paper, the Chebyshev tau [9], the Taylor collocation [14,15] methods are developed and applied to the th m -order linear nonhomogenous differential equation with the mixed conditions , ) ( and the solution is expressed in terms of the rational Chebyshev functions [9] as follows:

PROPERTIES OF THE RATIONAL CHEBYSHEV (RC) FUNCTIONS [9]
In cases when errors near the ends of an interval .In other words a great variety of other types of least-square polynomial approximation can be formulated in terms of other weighting functions.In particular, for the weighting function The RC functions are defined by These functions are orthogonal with respect to the weight function

FUNDAMENTAL MATRIX RELATIONS
Let us first assume that the solution of Eq.( 1) can be expressed in the form (3), which is a truncated Chebyshev series in terms of RC functions.Then ) (x y and its derivative ) ( ) ( x y k can be put in the matrix forms are the RC functions defined in Eq.( 4); N a a a ..., , , 1 0 are coefficients defined in Eq.( 3).

If we use the expression
in the RC function (4), then the matrix so that In this case, we are going to use the last row for odd values of N , and othervise previous one as the last row of matrix.(5), from Eq (6), can be obtained as

For example, in the cases
= and thereby, from the expression (5) [ ] where [ ] , and

MATRIX RELATION BASED ON COLLOCATION POINTS
Now, let us define the collocation points as Then we substitute the collocation points (8) into Eq.( 1) to obtain the system The system (9) can be written in the matrix form where By putting the collocation points N r x r ..., , 2 , 1 , 0 , = in the relation (7) we have the matrix system [ ] Consequently, from the matrix forms (10) and (11), we obtain the fundamental matrix equation for Eq.( 1) as Next, we can obtain the corresponding matrix forms for the conditions (2) as follows: Using the relation (7) for

METHOD OF SOLUTION
The fundamental matrix equation (12) for Eq.( 1) corresponds to a system of ) We can obtain the matrix form for the mixed conditions (2), by means of Eq.( 13), briefly, as To obtain the solution of Eq.( 1) under the conditions (2), by replacing the rows of matrices (15) by the last m rows of the matrix ( 14), we have the required augmented matrix Thus the coefficients n a ; N n ..., , 1 , 0 = are uniquely determined by Eq.( 16).
Also we can easily check the accuracy of the obtained solutions as follows [13,15]: Since the obtained rational Chebyshev function expansion is an approximate solution of Eq.( 1), the resulting equation must be satisfied approximately; that is, for .
( k is any positive integer) is prescribed, then the truncation limit N is increased until the difference ) ( i x E at each of the points i x becomes smaller than the prescribed k − 10 .

ILLUSTRATIVE EXAMPLE
In this section, several numerical examples are given to illustrate the accuracy and effectiveness of properties of the method.All of them were performed on the computer using a program written in MATHEMATICA 5.
The augmented matrix forms of the conditions for 4 Then, we obtain the augmented matrix (16) as .
We obtain the solution Therefore, we find the solution which is exact solution of two-point boundary value problem [9].
We applied the RC collocation method and solved this problem.In Table    Example 4. Our simple example is the linear initial value problem as follows This expansion is approximate solution , that is, the first five terms of the Taylor series expansions of the Chebyshev solution given by Fox and Parker [17,p.137].In Figure 2, the results obtained by our method are compared with the results of Fox and Parker [17,p.137].The present method is also very effective and convenient.The errors in numerical solution of Example 4 are seen in Figure 3.

CONCLUSION
The rational Chebyshev collocation method based on the rational Chebyshev Tau and Chebyshev-Taylor collocation methods are used to solve the higher-order ordinary differential equations numerically.A considerable advantage of the method is that the rational Chebyshev coefficients of the solution are found very easily by using computer programs.For this reason, this process is much faster than the other methods.In addition, an interesting feature of this method is to find the analytical solutions if the equation has an exact solution that is a rational functions.Illustrative examples with the satisfactory results are used to demonstrate the application of this method.
The method can also be extended to the system of linear differential equations with variable coefficients, but some modifications are required.
determined with the aid of the recurrence formulae we have the fundemental matrix equation corresponding to the mixed conditions(2 the fundamental matrix equation of problem is

Example 2 .
([9], Example 2) Consider the differential equation 0 The present method is also very effective and convenient.The errors in numerical solution of Example 2 are seen in Figure1.The error decreases when the integer N is increased.
is the exact solution of Example 3.

) Using ( 17 )
to determine the individual terms of the RC collocation method,

Figure 2 .Figure 3 .
Figure 2. Numerical and Fox-Parker solution of the Example 4

Table 1 .
Approximates and exact values for Example 2 Figure 1.Exact and other method solutions of the Example 2