Periodic Solutions for Nonlinear System of Differential Equations of Third Order with Boundary Conditions

Raad N Butris,Ghada Shuker Jameel

doi:10.33899/edusj.2013.89560

Abstract

In this study we investigate the existence and approximation of the periodic solutions for nonlinear system of differential equations of third order with boundary conditions. The numerical-analytic method has been used to study the periodic solutions of the ordinary differential equations that were introduced by Samoilenko. Also these investigation lead us to the improving and extending the above method and the results of Butris.

Highlights

There are many subject in physics and technology using mathematical methods that depends of nonlinear differential equations and boundary value problems and it became clear that the existence and uniqueness of periodic solutions and its algorithm structure from more important problems in the present time 1,4,6 .Periodic Solutions for Nonlinear System of Differential Equations of Third ...The periodic solutions for some nonlinear systems of differential equations and boundary value problems have been used to study many problems for example 2,3,5,7 .In this paper we use the above method for investigating the periodic solution for nonlinear system of differential equations of third order with boundary conditions.Our work is to extend the results of Butris 3
We prove that the sequence of functions (2.1) is uniformly convergent in (2.2)
Proof: We have to show to that x(t, x0, x 0, x 0 ) is a unique solution of problem (1.1), (1.2)

Summary

Introduction

There are many subject in physics and technology using mathematical methods that depends of nonlinear differential equations and boundary value problems and it became clear that the existence and uniqueness of periodic solutions and its algorithm structure from more important problems in the present time 1,4,6. Periodic Solutions for Nonlinear System of Differential Equations of Third. The periodic solutions for some nonlinear systems of differential equations and boundary value problems have been used to study many problems for example 2,3,5,7. In this paper we use the above method for investigating the periodic solution for nonlinear system of differential equations of third order with boundary conditions. I.e. Lemma 1: Let f (t, x, x , x ) be continuous vector function in the interval 0,T : t s, x, x , x

Approximate Solution

Uniqueness solution

Existence of Solution

K2U4 1 U2 K2U 4 K1