K01127587

Page 1

International Journal of Engineering Inventions ISSN: 2278-7461, ISBN: 2319-6491 Volume 1, Issue 12 (December 2012) PP: 75-87

Periodic Solution for Nonlinear System of Differential Equations with Pulse Action of Parameters Dr. Raad. N. Butris Department of Mathematics Faculty of Science, University of Zakho

Abstract:- In this paper we study the existence of a periodic solution for nonlinear system of differential equations with pulse action of parameters. The numerical-analytic method has been used to study the periodic solutions of the nonlinear ordinary differential equations that were introduced by Somioleko And the result of this study which is the using the periodic solutions on a wide range in difference processes in industry and technology.

I.

INTRODUCTION

There are many subjects in physics and technology using mathematical methods that depends on the nonlinear differential equations, and it became clear that the existence of the periodic solutions and it's algorithm structure from more important problems in the present time. Where many of studies and researches dedicates for treatment the autonomous and non-autonomous periodic systems and specially with the integral equations and differential equations and the linear and nonlinear differential and which is dealing in general shape with the problems about periodic solutions theory and the modern methods in its quality treatment for the periodic differential equations. Somioleko [6] assumes the numerical analytic method to study the periodic solutions for the ordinary differential equations and its algorithm structure and this method include uniformly sequences of the periodic functions and the result of that study is the using of the periodic solutions on a wide range for example see [4, 5, 6]. Consider the following system of nonlinear differential equation, which has the form: ฤ‘?‘‘ฤ‘?‘ฤฝ = ฤ‘?œ†ฤ‘?‘ฤฝ + ฤ‘?‘“ ฤ‘?‘ฤ„, ฤ‘?‘ฤฝ, ฤ‘?‘ลš , ฤ‘?‘ฤ„ รข‰ ฤ‘?‘ฤ„ฤ‘?‘– , รขˆ†ฤ‘?‘ฤฝ = ฤ‘??ลบฤ‘?‘– ฤ‘?‘ฤฝ, ฤ‘?‘ลš ฤ‘?‘ฤ„ = ฤ‘?‘ฤ„ฤ‘?‘– ฤ‘?‘‘ฤ‘?‘ฤ„ . . . (1) ฤ‘?‘‘ฤ‘?‘ลš = ฤ‘?›หฤ‘?‘ฤฝ + ฤ‘?‘” ฤ‘?‘ฤ„, ฤ‘?‘ฤฝ, ฤ‘?‘ลš , ฤ‘?‘ฤ„ รข‰ ฤ‘?‘ฤ„ฤ‘?‘– , รขˆ†ฤ‘?‘ลš = ฤ‘??ลŸฤ‘?‘– ฤ‘?‘ฤฝ, ฤ‘?‘ลš ฤ‘?‘ฤ„ = ฤ‘?‘ฤ„ฤ‘?‘– ฤ‘?‘‘ฤ‘?‘ฤ„ Where ฤ‘?‘ฤฝ รขˆˆ ฤ‘??ห‡ฤ‘?œ† รขŠ† ฤ‘?‘…ฤ‘?‘› , ฤ‘?‘ลš รขˆˆ ฤ‘??ห‡ฤ‘?›ห รขŠ† ฤ‘?‘…ฤ‘?‘› , ฤ‘??ห‡ฤ‘?œ† is a closed and bounded domain. The vector functions ฤ‘?‘“ ฤ‘?‘ฤ„, ฤ‘?‘ฤฝ, ฤ‘?‘ลš ฤ‘?‘Žฤ‘?‘›ฤ‘?‘‘ ฤ‘?‘” ฤ‘?‘ฤ„, ฤ‘?‘ฤฝ, ฤ‘?‘ลš are defined on the domain: ฤ‘?‘ฤ„, ฤ‘?‘ฤฝ, ฤ‘?‘ลš รขˆˆ ฤ‘?‘…1 ฤ‚— ฤ‘??ห‡ฤ‘?œ† ฤ‚— ฤ‘??ห‡ฤ‘?›ห = รขˆ’รขˆž, รขˆž ฤ‚— ฤ‘??ห‡ฤ‘?œ† ฤ‚— ฤ‘??ห‡ฤ‘?›ห . . . (2) Which are continuous inฤ‘?‘ฤ„ , ฤ‘?‘ฤฝ , ฤ‘?‘ลšand periodic in t of period T, where ฤ‘??ห‡ฤ‘?›ห is bounded domains subset of Euclidean spaces ฤ‘?‘…ฤ‘?‘š , and the functions ฤ‘??ลบฤ‘?‘– ฤ‘?‘ฤฝ, ฤ‘?‘ลš , ฤ‘??ลŸฤ‘?‘– ฤ‘?‘ฤฝ, ฤ‘?‘ลš are continuous in the domain (2), where ฤ‘??ลบฤ‘?‘–+ฤ‘?‘? ฤ‘?‘ฤฝ, ฤ‘?‘ลš = ฤ‘??ลบฤ‘?‘– ฤ‘?‘ฤฝ, ฤ‘?‘ลš , ฤ‘??ลŸฤ‘?‘–+ฤ‘?‘? ฤ‘?‘ฤฝ, ฤ‘?‘ลš = ฤ‘??ลŸฤ‘?‘– ฤ‘?‘ฤฝ, ฤ‘?‘ลš ฤ‘?‘Žฤ‘?‘›ฤ‘?‘‘ ฤ‘?‘ฤ„ฤ‘?‘–+ฤ‘?‘? ฤ‘?‘ฤฝ, ฤ‘?‘ลš = ฤ‘?‘ฤ„ฤ‘?‘– + ฤ‘?‘‡ for p is a positive integer and ฤ‘?‘ฤ„ฤ‘?‘– is finite positive sequence of numbers. Suppose that the vector functions in (1)are satisfying the following inequalities: max ฤ‘?‘“(ฤ‘?‘ฤ„, ฤ‘?‘ฤฝ, ฤ‘?‘ลš) รข‰ยค ฤ‘?‘€1 , max ฤ‘?‘”(ฤ‘?‘ฤ„, ฤ‘?‘ฤฝ, ฤ‘?‘ลš) รข‰ยค ฤ‘?‘€2 . . . (3) ฤ‘?‘ฤฝ,ฤ‘?‘ลš รขˆˆฤ‘??ห‡ ฤ‘?œ† ฤ‚— ฤ‘??ห‡ ฤ‘?›ห ฤ‘?‘ฤ„รขˆˆ[0,ฤ‘?‘‡]

max

ฤ‘?‘ฤฝ,ฤ‘?‘ลš รขˆˆฤ‘??ห‡ ฤ‘?œ† ฤ‚— ฤ‘??ห‡ ฤ‘?›ห 1รข‰ยคฤ‘?‘–รข‰ยคฤ‘?‘?

ฤ‘?‘ฤฝ ,ฤ‘?‘ลš รขˆˆฤ‘??ห‡ ฤ‘?œ† ฤ‚— ฤ‘??ห‡ ฤ‘?›ห ฤ‘?‘ฤ„รขˆˆ[0,ฤ‘?‘‡]

ฤ‘??ลบฤ‘?‘– (ฤ‘?‘ฤฝ, ฤ‘?‘ลš) รข‰ยค ฤ‘?‘€3

,

max

ฤ‘?‘ฤฝ,ฤ‘?‘ลš รขˆˆฤ‘??ห‡ ฤ‘?œ† ฤ‚— ฤ‘??ห‡ ฤ‘?›ห 1รข‰ยคฤ‘?‘–รข‰ยคฤ‘?‘?

ฤ‘??ลŸฤ‘?‘– (ฤ‘?‘ฤฝ, ฤ‘?‘ลš) รข‰ยค ฤ‘?‘€4

. . . (4)

ฤ‘?‘“ ฤ‘?‘ฤ„, ฤ‘?‘ฤฝ1 , ฤ‘?‘ลš1 รขˆ’ ฤ‘?‘“ ฤ‘?‘ฤ„, ฤ‘?‘ฤฝ2 , ฤ‘?‘ลš2 รข‰ยค ฤ‘??ลพ1 ฤ‘?‘ฤฝ1 รขˆ’ ฤ‘?‘ฤฝ2 + ฤ‘??ลพ2 ฤ‘?‘ลš1 รขˆ’ ฤ‘?‘ลš2 . . . (5) ฤ‘?‘” ฤ‘?‘ฤ„, ฤ‘?‘ฤฝ1 , ฤ‘?‘ลš1 รขˆ’ ฤ‘?‘” ฤ‘?‘ฤ„, ฤ‘?‘ฤฝ2 , ฤ‘?‘ลš2 รข‰ยค ฤ‘??ลผ1 ฤ‘?‘ฤฝ1 รขˆ’ ฤ‘?‘ฤฝ2 + ฤ‘??ลผ2 ฤ‘?‘ลš1 รขˆ’ ฤ‘?‘ลš2 . . . (6) ฤ‘??ลบฤ‘?‘– ฤ‘?‘ฤฝ1 , ฤ‘?‘ลš1 รขˆ’ ฤ‘??ลบฤ‘?‘– ฤ‘?‘ฤฝ2 , ฤ‘?‘ลš2 รข‰ยค ฤ‘??ลพ3 ฤ‘?‘ฤฝ1 รขˆ’ ฤ‘?‘ฤฝ2 + ฤ‘??ลพ4 ฤ‘?‘ลš1 รขˆ’ ฤ‘?‘ลš2 . . . (7) ฤ‘??ลŸฤ‘?‘– ฤ‘?‘ฤฝ1 , ฤ‘?‘ลš1 รขˆ’ ฤ‘??ลŸฤ‘?‘– ฤ‘?‘ฤฝ2 , ฤ‘?‘ลš2 รข‰ยค ฤ‘??ลผ3 ฤ‘?‘ฤฝ1 รขˆ’ ฤ‘?‘ฤฝ2 + ฤ‘??ลผ4 ฤ‘?‘ลš1 รขˆ’ ฤ‘?‘ลš2 . . . (8) Where ฤ‘?‘ฤ„ รขˆˆ ฤ‘?‘…รข€ห› , x ,ฤ‘?‘ฤฝ1 ,ฤ‘?‘ฤฝ2 รขˆˆ ฤ‘??ห‡ฤ‘?œ† , y ,ฤ‘?‘ลš1 ,ฤ‘?‘ลš2 รขˆˆ ฤ‘??ห‡ฤ‘?›ห and ฤ‘?‘€1 , ฤ‘?‘€2 , ฤ‘?‘€3 , ฤ‘?‘€4 , ฤ‘??ลพ1 , ฤ‘??ลพ2 , ฤ‘??ลพ3 , ฤ‘??ลพ4 , ฤ‘??ลผ1 , ฤ‘??ลผ2 , ฤ‘??ลผ3 , ฤ‘??ลผ4 are a positive constant , . = max0รข‰ยคฤ‘?‘ฤ„รข‰ยคฤ‘?‘‡ . . Let, ฤ‘?›ห are a positive parameter defined in (2), continuous and periodic at ฤ‘?œ? , ฤ‘?‘ , ฤ‘?‘ฤ„ and satisfy both following inequalities:

www.ijeijournal.com

P a g e | 75

Turn static files into dynamic content formats.

Createย aย flipbook
Issuu converts static files into: digital portfolios, online yearbooks, online catalogs, digital photo albums and more. Sign up and create your flipbook.