GP-Stability of Linear Multistep Methods for Delay Retarded Differential Equations with Several Delays

where     , , , 2 1  are positive constants,  and f denote given vector-valued functions. The stability behaviors of linear multistep method for RDEs is studied and it is proved that the linear multistep method is GP-stable if it is A-stable for Ordinary Differential Equations (ODEs).


-Introduction
In this paper, we are concerned with stability behaviors in numerical methods for the solution of initial-value problems for systems of Delay Differential Equations (DDEs) with several delays: are positive constants, f and  denote given vector-valued functions   with   ,  ) , , , is unknown for 0  t In particular, stability properties of linear multistep methods (LMMs) will be investigated based on the following test system: . The solution of (2) is called asymptotically stable if For any matrix , X denote its determinant by   X det , its spectrum by   X  and its spectral radius by   X 

-The Stability of The Tested Systems:
Define a function of two complex variables z , and for a system of several delays, the function becomes: , then all exact solutions to the system: are asymptotically stable if: (C1) Every eigenvalue of the matrix A has negative real part For any pure imaginary  , the spectral radius of is the inner product. (2) Under condition (C1), the identity w F1(z, w): is a function of two complex variables z, w. [6] Lemma 2.2: [6] Let 1  C , then all exact solution of equation (5) is asymptotically stable if (C1), ) ( 2 C , (C3) are satisfied. Lemmas, 2.1 and 2.2, deals with Neutral Differential Equations (NDEs) type systems. Putting 0 = C transforms the attention from NDEs systems to RDEs systems, the treatment will then be generalized to RDEs with one delay by making it deals with several delays. By these assumptions and the first two conditions only, it will be shown that the linear multistep method for the system (2) is GP-stable.

New Investigations:
in Lemmas 2.1 and 2.2, it will be shown that all exact solutions to (2) preserve the asymptotical stability if: is the inner product

Proof:
To prove that where (S1) is satisfied, then ).
By the definition of spectral radius [3]: w By Assumption 2.3 above and under the condition (S1) we have: , then all exact solution of equation (5) with 0 = C and several delays (that is equation (2)) will preserve the asymptotical stability if (S1) & ) ( 2 S are satisfied.

Proof:
According to Lemma 2.2, we have , so equation (5) with and for several delays we get test system (2).
Thus, according to Lemma 2.2, this system preserves asymptotical stability if it is proven that ).
and for several delays system we have Also for one delay system we get: And for equation (5) we have: which is in contradiction to the condition ) ( 2 C . # Theorem 2.6: then all exact solutions of equation (5) with 0 = C and several delays (that is equation (2)) will preserve the asymptotical stability if: 1 -(S1) and (S2) are satisfied. or 2 -(S1) and ) ( * 2 S are satisfied.

Proof:
According to Lemma 2.5, one needs only to prove that: which is in contradiction to the condition ) ( * 2 S , this completes the proof of theorem 2.6. #

Illustrative Example 2.7:
To illustrate the main theorem stated in this paper, we give an example of RDE. Recall equation: This example, see [3], is the case for: It easy to calculate the spectrum of A, that is 2 , 1 6603 .
Henceforth, conditions (S1) and ) ( * 2 S are satisfied, and the system (2) is analytically asymptotically stable. Now, Consider the ordinary differential equations: f(t, x(t)) are vector-valued functions. If 0  h denotes a given stepsize, the gridpoint tn is given by tn=nh, and xn denotes an approximation to x(tn). A linear multistep method can be written as

Remark 2.8:
Then we easily obtain the following result.

The linear multistep method is A-stable if and only if
In section 3, we will present adaptation of the linear multistep methods for the numerical solution of (1). In section 4, we investigate the stability of LMM and we will show that the LMM is GP-stable if it is A-stable for ODEs. Where GP stands for the general P-stability of DDEs.

-Linear Multistep Method for DDEs:
In order to make the linear multistep method (8) adapt to(1), we introduce unknowns ) ( , ), ( ), Then (1) can be converted into the following form: Then the characteristic equation of (14) can be written as: In view of the assumption discussed in section 2 and definition 3 of reference [5], we introduce the set:

Definition 4.1:
Let ) , , , The stability region S of the process (14) is defined by:

 
is A-stable for ODEs, then the process (14) is GP-stable.

Proof:
In order to proof that the process (14) is GP-stable we have to show that: