Taylor Collocation Method for Solving a Class of the First Order Nonlinear Differential Equations

In this study, we present a reliable numerical approximation of the some first order nonlinear ordinary differential equations with the mixed condition by the using a new Taylor collocation method. The solution is obtained in the form of a truncated Taylor series with easily determined components. Also, the method can be used to solve Riccati equation. The numerical results show the effectuality of the method for this type of equations. Comparing the methodology with some known techniques shows that the existing approximation is relatively easy and highly accurate.


INTRODUCTION
Nonlinear ordinary differential equations are frequently used to model a wide class of problems in many areas of scientific fields; chemical reactions, spring-mass systems bending of beams, resistor-capacitor-inductance circuits, pendulums, the motion of a rotating mass around another body and so forth   1, 2 .These equations here also demonstrated their usefulness in ecology, economics, biology, astrophysics and engineering.Thus, methods of solution for these equations are of great importance to engineers and scientists   3, 4 .In this paper, for our aim we consider the first order nonlinear ordinary differential equation of the form

FUNDAMENTAL MATRIX RELATIONS
Our aim is to find the matrix form of each term in the nonlinear equation given by Eq. ( 1).Firstly, we consider the solution () yx defined by a truncated series (3) and then we can convert to the matrix form If we differentiate expression (4) with respect to x , we obtain ( ) ( ) On the other hand, the matrix form of expression 2 () yx is obtained as By using the expression (4), ( 5) and ( 6) we obtain Following a similar way to (6), we have

MATRIX RELATIONS BASED ON COLLOCATION POINTS
Let us use the collocation points defined by , 0,1,..., in order to 01 .

METHOD OF SOLUTION
The fundamental matrix equation ( 11) corresponding to Eq. ( 1), can be written as where RXX SXX B TXBX B .We can find the corresponding matrix equation for the condition (2), using the relation (4), as follows: We can write the corresponding matrix form (13) for the mixed condition (2) in the augmented matrix form as

 
; : where To obtain the approximate solution of Eq. ( 1) with the mixed condition (2) ; : g(x ) ;: ;: The unknown coefficients set   0 1 N y , y , , y  can be determined from the nonlinear system (15).As a result, we can obtain approximate solution in the truncated series form (3).

ACCURACY OF SOLUTION
We can check the accuracy of the solution by following procedure ] 12 9 [  : The truncated Taylor series in (3) have to be approximately satisfying Eq. ( 1); that is, for each k is any positive integer) is prescribed, then the truncation limit N is increased until the difference () Ex at each of the points i x becomes smaller than the prescribed 10 k  .

NUMERICAL EXAMPLES
In this section, two numerical examples are given to illustrate the accuracy and efficiency of the presented method.
and the approximate solution () yx by the truncated Taylor polynomial . From the fundamental matrix equations for the given equation and condition respectively are obtained as * T y y y y y y y y y y y y y y y y y y  Y .
The solutions obtained for

CONCLUSION
In this study, a new Taylor approximation method for the solution of a class of first order nonlinear differential equations has been presented.The principal advantage of this method, at the around xc  , is the capability to succeed in the solution up to all term of Taylor expansion.It is seen from Example 6.2 that Taylor collocation method gives well results for the different values N .Also it is important to note that Taylor coefficients of the solution are found very simply by using the computer programs.
x Q x y x R x y x S x y x y x T x y Taylor polynomial of degree N at xc  , where coefficients ,  and  are appropriate constants.Note that, if( )

Example 6 . 1 .
Let us first consider the first-order nonlinear differential equation

2 yExample 6 . 2 .
Consider the following nonlinear differential equation (Riccati Equation ) given by 22 x xy xy y e    

Table 1 .
Comparison of the absolute errors of Example 6.2