Bernstein Collocation Method for Solving the First Order Nonlinear Differential Equations with the Mixed Non-linear Conditions

In this study, we present the Bernstein matrix method to solve the first order nonlinear ordinary differential equations with the mixed non-linear conditions. By using this method, we obtain the approximate solutions in form of the Bernstein polynomials [1,2,16,17]. The method reduces the problem to a system of the nonlinear algebraic equations by means of the required matrix relations of the solutions form. By solving this system, the approximate solution is obtained. Finally, the method will be illustrated on the examples.

The equation defined by (1) is a class of the first order nonlinear differential equation.This is an important branch of modern mathematics and arises frequently in many applied areas which include engineering, ecology, economics, biology and astrophysics.That is why these methods are important to Engineers and scientists.Our purpose in this study is to develop a new matrix method, obtain the approximate solution of the problem (1)-( 2) in the Bernstein polynomial form, 0 , ( ) ( ) are the coefficients to be determined and

 
, nN Bx is the Bernstein polynomial of degree N .

FUNDAMENTAL MATRIX RELATIONS
Let us consider the nonlinear differential equation ( 1) and find the matrix forms of each term in these equations.Firstly, we consider the solution   yx defined by a truncated series (3) and then we can convert it to the matrix form If we differentiate equation ( 4) with respect to x, we obtain On the other hand, the matrix form of the equation   By using the equations (4), ( 5) and ( 6) we obtain Following a similar way to (6), we have where where

Let us use the collocation points defined by
By using the collocation points (10) into Eq.( 1), we obtain the system By using the relations (4), ( 5), ( 6), ( 7) and (8); the system (11) can be written in the matrix form Consequently, the fundamental matrix equations of ( 12) can be written in the following compact form where Or it can be written shortly as

METHOD OF SOLUTION
The fundamental matrix equation ( 13) corresponding to Eq. ( 1) can be written as We can find the corresponding matrix equation for the condition (2), using the relation ( 4) and ( 6), as follows: so that  To obtain the approximate solution of Eq. ( 1) with the mixed condition (2) in the terms of Bernstein polynomials, by replacing the row matrix ( 16) by the last row of the matrix (13), we obtain the required augmented matrix: or the corresponding matrix equation y can be determined from the nonlinear system (17).As a result, we can obtain approximate solution in the truncated series form (3).

ACCURACY OF SOLUTION
We can check the accuracy of the method.The truncated Benstein series in (3) have to be approximately satisfying Eq. ( 1).For each k is any positive integer) is prescribed, then the truncation limit N is increased until the difference   i Ex at each of the points i x becomes smaller than the prescribed 10 k  [3-17].

NUMERICAL EXAMPLES
In this section, three numerical examples are given to illustrate the accuracy and efficiency of the presented method.

CONCLUSION
In this paper, we have presented a suggested method to solve second order nonlinear ordinary differential equations with mixed non-linear conditions using the matrix method based on collocation points on any interval  0, R .The matrix method avoids the difficulties and massive computational work by determining the analytic solution.
On the other hand, the numerical results show that the accuracy improves when N is increased.N increases indicate that as N increases, the errors decrease more rapidly; hence for better results, using large number N is recommended.A considerable advantage of the method is that Bernstein coefficients of the solution are found very easily by using the computer programs.Besides, our Nth order approximation gives the exact solution when the solution is polynomial of degree equal to or less than N.If the solution is not polynomial, Bernstein series approximation converges to the exact solution as N increases.

2 2 3 P
x y x Q x y x R x y x S x y x y x T x y x U x y x g x given together with the mixed non-linear conditions defined as follows the corresponding matrix form(15) for the mixed condition (2) in the augmented matrix form as

Example 6 . 1 .
Let us first consider the nonlinear differential equation forms of the conditions are

Example 6 . 3 .
Substituting the coefficients into equation (3), we obtain the solution   y x x  which is the exact solution.Our last example is the nonlinear differential equation exact solution () =   .The solutions obtained for  = 2, 5, 7 are compared with the exact solution is x

Table 1 .
Comparison of the numerical errors of Example 6.3 Table and Figure indicate that as