The Numerical Solution of a Class of Delay Differential Equations Boundary

In this study, we propose a new method for the Cauchy solution of the Common Delay Differential Equations ?̇?x (t) = ƒ(t, x(t), x(t τ1 (t)),..., x(t τm (t))). Moreover, they exist only and approximation of equations boundary were especially proposed. From our analyze, we could get the results that Cauchy solution of the Common Delay Differential Equations play an important role in the network information processing.


INTRODUCTION
It is well known that a dynamical system is usually expressed as the form of a calculus equation: the delay of dynamical systems is inevitable, even through information systems based on the speed of light is no exception.In this sense, the above equation is an approximate description of dynamical systems, which delay factor, is omitted.But he necessary precision will not be reached even leads to the wrong system or do not establish the mathematical model system, so as to use the theories and methods of delay Differential Equations to solve a variety of forms.Henry and Penney (2004) study the differential equations and boundary value problems computing and modeling.(Guo, 2001) make a detailed research of the nonlinear functional analysis.Shi et al. (2005) analyze the differential equation theory and its application.Kuang (1999) analyze the differential equation theory and its application.Sun (1997) have a research of the measurement of the motor winding temperature rise.Dong (2011) analyzes the measurement and calculation of the motor winding temperature rise.
In this study, we have a research of the numerical solution of a class of delay differential equations boundary.We propose a new method for the Cauchy solution of the Common Delay Differential Equations.Moreover, they exist only and approximation of equations boundary were especially proposed.From our analyze, we could get the results that Cauchy solution of the Common Delay Differential Equations play an important role in the network information processing.

PROBLEM FORMULATION
Common Delay Differential Equations can be summarized in two forms; the 1 st form is as follows: ) in this equation, τ i (t) 0, i = 1, 2,…, m which reflects the lagged effect of the limited time at the state of the current system.Make τ(t) = max τ (t), when τ (t) r ∞, Eq. (2) known as functional differential equations with finite delay.Otherwise, known as functional differential equations with infinite delay.The 2 nd form is as follows: We call Eq. (3) functional differential equations with distributed delay.When |σ (t)| r ∞, functional differential Eq. (3) known as functional differential equations with finite delay.Otherwise, known as functional differential equations with infinite delay.
Equation (3) reflects the state of a system affect the current state of the system.The above two methods can be integrated to a very general form within the framework.
Definition 1: High-level overview of a variety of Retarded Functional Differential Equations, just select the appropriate operator ƒ(t, φ), Eq. (1.3) can represent a variety of Retarded Functional Differential Equations.
For example, if we can take ƒ(t, φ) = ƒ(t, φ (0), φ(τ 1 (t)), … φ(-τ m (t))).Then(4)become( 2).If we want to determine the solution of Delay Differential Equations, only give the state of the system at a given moment is not enough, we need to give the state of the system at some point before this moment for some time that is, given an initial function.So, Delayed Functional Differential Equations Cauchy problem in the form as follows (Fig. 1): ( ) ( , ) where, σ R is the initial moment and φ C is the initial function.
Here is the definition of Delayed Functional Differential Equation: • If x(t) satisfies the Eq. ( 3), then X(t) is called a solution when the Eq. ( 4) in the interval [σ -r, σ + A) • If X (t) is called a solution when the Eq. ( 4) in the interval [σ -r, σ + A) and x σ = φ, then x (t) is called a solution through (σ, φ), as know as the solution of Delayed Functional Differential Equations Cauchy problem ( 5), (2) form of Cauchy problem numerical solution method for solving, especially working on the equation has an important role in the network information processing:

THE SOLUTION EXISTENCE AND UNIQUENESS THEOREM
Assume that R, C, Ω R×C is an open set, operator ƒ C(Ω, R n ), hen following conclusions are about Cauchy problems: The Cauchy Problem(5) for any ( , ) Ω, the solution exists.Lemma 2: Assume that Ω R×C is an open set, W Ω is a compact set, ƒ 0 C (Ω, R n ) given.Then there exists a neighborhood of W V Ω, Make ƒ 0 C 0 (V, R n ).Move forward a single step, there exists a neighborhood of ƒ 0 U C 0 (V, R n ), for ƒ U, ( , ) Ω, problem (5) existence of the solution x( , , ƒ) (t).Lemma 3: Assume that Ω R×C is an open set, ƒ C (Ω, R n ) and ƒ(t, ) each compact subset in Ω about Satisfies the Lipschitz condition: Here, L R + is Lipschitz constant.Then problem (5) exists a unique solution.
The following discussion based on ( 7) is satisfied under the premise that the existence and uniqueness is guaranteed.Numerical method our idea is into long differential equations appropriate qualitative considerations.So that you can facilitate the use of classical numerical methods have been established to solve problems, for example, Runge-Kutta method.The following 2 cases to consider Introduction 1: The 1 st situation: Here x(t) = (x 1 (t), x 2 (t),…, x n (t)) T is the ndimensional vector function: (t) = ( 1 (t), 2 (t),…, n (t)) T is the Known n-dimensional vector function: monotonically decreasing and: Let S(t) = t -(t), and (t) monotonically decreasing makes S(t) monotonically increasing, therefore, there is a continuous inverse function t = t(S).Substituted into the (8) then we got: where, (S) = (t)., ƒ (S, x(S)) = ƒ (t(S)). .Because r r 0 = max t 0 τ(t), so A , as a matter of fact, S 0 = S(0) = -(0) [-r, 0], the, (8) conversion for the following appropriate qualitative issues: at this point, we can choose the following Runge-Kutta method of Simpson formula to solve the problem.