Abstract
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.
Highlights
In this study, modifying and developing methods in [3-17] and using matrix relations between Bernstein polynomials and their derivatives, we present an approach to numerical solutions of the first order nonlinear ordinary differential equation in the form
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
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, N
Summary
In this study, modifying and developing methods in [3-17] and using matrix relations between Bernstein polynomials and their derivatives, we present an approach to numerical solutions of the first order nonlinear ordinary differential equation in the form. 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, N y(x) ynBn,N (x) , n 0. 0 x R where yn , n 0,1,..., N are the coefficients to be determined and Bn,N x is the Bernstein polynomial of degree N
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