Hindawi Publishing Corporation
Abstract and Applied Analysis
Volume 2013, Article ID 679075, 13 pages
http://dx.doi.org/10.1155/2013/679075
Research Article
Nonlinear Stability and Convergence of Two-Step Runge-Kutta
Methods for Volterra Delay Integro-Differential Equations
Haiyan Yuan1 and Cheng Song2
1
2
Department of Mathematics, Heilongjiang Institute of Technology, Harbin 150050, China
School of Management, Harbin University of Commerce, Harbin 150028, China
Correspondence should be addressed to Haiyan Yuan; yhy82 47@163.com
Received 29 January 2013; Revised 11 March 2013; Accepted 15 March 2013
Academic Editor: Changsen Yang
Copyright © 2013 H. Yuan and C. Song. This is an open access article distributed under the Creative Commons Attribution License,
which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
This paper introduces the stability and convergence of two-step Runge-Kutta methods with compound quadrature formula for
solving nonlinear Volterra delay integro-differential equations. First, the definitions of (𝑘, 𝑙)-algebraically stable and asymptotically
stable are introduced; then the asymptotical stability of a (𝑘, 𝑙)-algebraically stable two-step Runge-Kutta method with 0 < 𝑘 < 1 is
proved. For the convergence, the concepts of D-convergence, diagonally stable, and generalized stage order are firstly introduced;
then it is proved by some theorems that if a two-step Runge-Kutta method is algebraically stable and diagonally stable and its
generalized stage order is p, then the method with compound quadrature formula is D-convergent of order at least min{𝑝, ]},
where ] depends on the compound quadrature formula.
1. Introduction
Volterra delay integro-differential equations (VDIDEs) arise
widely in the mathematical modeling of physical and biological phenomena. Significant advances in the theoretical
analysis and in the numerical analysis for these problems
have been made in the last few decades (see, e.g., [1, 2]).
For the case of linear stability and convergence for these
equations, numerical time-integration techniques of one-step
collocation and Runge-Kutta type were investigated in [3–8].
Linear multistep-based methods were studied in [9–12]. De
la Sen and Luo studied the uniform exponential stability of
a wide class of linear time-delay systems in [13]; De la Sen
considered the stability of impulsive time-varying systems in
[14].
For the case of nonlinear stability and convergence,
stability results were obtained in [15, 16], where the authors
investigated the nonlinear stability of continuous RungeKutta methods, discrete Runge-Kutta methods, and backward differentiation (BDF) methods, respectively. However,
most of these important results are based on the classical Lipschitz conditions, while the classical Lipschitz conditions are
so strong that there are few equations satisfying them. Most
of the Volterra delay integro-differential equations satisfy the
one-sided Lipschitz condition, while the studies focusing on
the stability and convergence of the numerical method for
nonlinear VDIDEs based on a one-sided Lipschitz condition
have not yet been seen in the literature. By means of a onesided Lipschitz condition, we will discuss the stability and
convergence of two-step Runge-Kutta (TSRK) methods for
nonlinear VDIDEs in the present paper.
The paper is organized as follows. In Section 2, a fairly
general class of VDIDEs is defined. We present a stability
criterion for such problems, which generalizes the criteria
in the above references. A class of two-step Runge-Kutta
methods is also derived for solving VDIDEs. They are
obtained by compound quadrature rules. In Sections 3 and
4, nonlinear stability and convergence of TSRK method for
NDDEs are derived and proved. In Section 5 we present
some numerical examples in order to illustrate the nonlinear
stability and convergence of a two-step Runge-Kutta method.
These numerical results show that the new methods are quite
effective.
2
Abstract and Applied Analysis
2. A Class of VDIDES and the Two-Step
Runge-Kutta Methods
In this paper, we are concerned with two-step RungeKutta (TSRK) methods of the form
𝑠
It is the purpose of this paper to investigate the nonlinear
stability and convergence properties of the following initialvalue problem VDIDEs:
󸀠
𝑦 (𝑡) = 𝑓 (𝑡, 𝑦 (𝑡) , 𝑦 (𝑡 − 𝜏) , ∫
𝑡
𝑡−𝜏
𝑗=1
𝑖 = 1, 2, . . . , 𝑠, (7a)
𝑠
𝑌𝑖(𝑛−1) = 𝑦𝑛−1 + ℎ ∑ 𝑎𝑖𝑗 𝑓 (𝑡𝑛−1 + 𝑐𝑗 ℎ, 𝑌𝑗(𝑛−1) ) ,
𝑗=1
𝑔 (𝑡, V, 𝑦 (V)) 𝑑V) ,
𝑡 ∈ [0, Τ] ,
𝑦 (𝑡) = 𝜑 (𝑡) ,
𝑌𝑖(𝑛) = 𝑦𝑛 + ℎ ∑ 𝑎𝑖𝑗 𝑓 (𝑡𝑛 + 𝑐𝑗 ℎ, 𝑌𝑗(𝑛) ) ,
(7b)
𝑖 = 1, 2, . . . , 𝑠,
(1)
𝑠
𝑦𝑛+1 = (1 − 𝜃) 𝑦𝑛 + 𝜃𝑦𝑛−1 + ℎ ∑ 𝑏𝑖 𝑓 (𝑡𝑛 + 𝑐𝑖 ℎ, 𝑌𝑖(𝑛) )
𝑖=1
𝑡 ∈ [−𝜏, 0] ,
(7c)
𝑠
+ ℎ ∑ ̃𝑏𝑖 𝑓 (𝑡𝑛−1 + 𝑐𝑖 ℎ, 𝑌𝑖(𝑛−1) ) ,
where 𝑓, 𝑔, and 𝜑 are smooth enough such that (1) has a
unique solution 𝑦(𝑡) and 𝜏 is a positive delay term. We assume
that there exist some inner product ⟨⋅, ⋅⟩ and the induced
norm || ⋅ || in 𝐶𝑁, such that 𝜑 : [−𝜏, 0] → 𝐶𝑁 and 𝑓 :
[0, Τ]×𝐶𝑁 ×𝐶𝑁 ×𝐶𝑁 → 𝐶𝑁 satisfy the following conditions:
𝑖=1
where
+ ∑𝑠𝑖=1 ̃𝑏𝑖 = 1 + 𝜃, 𝑐𝑖 = ∑𝑠𝑗=1 𝑎𝑖𝑗 , ℎ = 𝜏/𝑚 is a
stepsize, 𝑚 is an arbitrarily given positive integer, and 0 ≤ 𝜃 ≤
1. The above methods are studied in [17]. Now we consider the
adaptation of the two-step Runge-Kutta method to (1):
∑𝑠𝑖=1 𝑏𝑖
𝑠
(𝑛)
(𝑛)
̃ ),
𝑌𝑖(𝑛) = 𝑦𝑛 + ℎ ∑ 𝑎𝑖𝑗 𝑓 (𝑡𝑛 + 𝑐𝑗 ℎ, 𝑌𝑗(𝑛) , 𝑌𝑗 , 𝑌
𝑗
󵄩2
󵄩
Re ⟨𝑓 (𝑡, 𝑢1 , V, 𝑤) − 𝑓 (𝑡, 𝑢2 , V, 𝑤) , 𝑢1 − 𝑢2 ⟩ ≤ 𝛼󵄩󵄩󵄩𝑢1 − 𝑢2 󵄩󵄩󵄩 ,
(2)
𝑗=1
𝑖 = 1, 2, . . . , 𝑠,
󵄩
󵄩󵄩
󵄩
󵄩
󵄩󵄩𝑓 (𝑡, 𝑢, V1 , 𝑤1 ) − 𝑓 (𝑡, 𝑢, V2 , 𝑤2 )󵄩󵄩󵄩 ≤ 𝛽 󵄩󵄩󵄩V1 − V2 󵄩󵄩󵄩
(3)
󵄩
󵄩
+ 𝜎 󵄩󵄩󵄩𝑤1 − 𝑤2 󵄩󵄩󵄩 ,
󵄩
󵄩󵄩
󵄩
󵄩
󵄩󵄩𝑔 (𝑡, V, 𝑢1 ) − 𝑔 (𝑡, V, 𝑢2 )󵄩󵄩󵄩 ≤ 𝛾 󵄩󵄩󵄩𝑢1 − 𝑢2 󵄩󵄩󵄩 , (𝑡, V) ∈ 𝐷, (4)
(8a)
𝑌𝑖(𝑛−1)
= 𝑦𝑛−1
𝑠
(𝑛−1)
+ ℎ ∑ 𝑎𝑖𝑗 𝑓 (𝑡𝑛−1 + 𝑐𝑗 ℎ, 𝑌𝑗(𝑛−1) , 𝑌𝑗
𝑗=1
for 𝑡 ∈ [0, Τ], 𝐷 = {(𝑡, V) : 𝑡 ∈ [0, Τ), V ∈ [𝑡 − 𝜏, 𝑡]}, and for all
𝑢1 , 𝑢2 , V1 , V2 , 𝑤1 , 𝑤2 ∈ 𝐶𝑁(−𝛼), 𝛽, 𝜎, and 𝛾 are nonnegative
constants.
Throughout this paper, we assume that (1) has unique
solution 𝑦(𝑡) and denote the problem class 𝑅(𝛼, 𝛽, 𝜎, 𝛾) that
consist problem of type (1) with (2)–(4). In order to make the
error analysis feasible, we always assume that (1) has a unique
solution 𝑦(𝑡) which is sufficiently differentiable and satisfies
󵄩
󵄩󵄩 𝑖
󵄩󵄩 𝑑 𝑦 (𝑡) 󵄩󵄩󵄩
󵄩
󵄩󵄩
󵄩󵄩 𝑑𝑡𝑖 󵄩󵄩󵄩 ≤ 𝑀𝑖 ,
󵄩
󵄩
),
𝑖 = 1, 2, . . . , 𝑠,
(8b)
𝑦𝑛+1 = (1 − 𝜃) 𝑦𝑛
𝑠
(𝑛)
(𝑛)
̃ )
+ 𝜃𝑦𝑛−1 + ℎ ∑ 𝑏𝑖 𝑓 (𝑡𝑛 + 𝑐𝑖 ℎ, 𝑌𝑖(𝑛) , 𝑌𝑖 , 𝑌
𝑖
𝑖=1
(8c)
𝑠
(𝑛−1)
̃ (𝑛−1) ) .
+ ℎ ∑ ̃𝑏𝑖 𝑓 (𝑡𝑛−1 + 𝑐𝑖 ℎ, 𝑌𝑖(𝑛−1) , 𝑌𝑖 , 𝑌
𝑖
𝑖=1
𝑖 = 1, 2, . . . , 𝑡 ∈ [−𝜏, 𝑇] .
(5)
In particular, 𝑦0 = 𝜑(0), 𝑦𝑛 is the numerical approximation
(𝑛)
at 𝑡𝑛 = 𝑛ℎ to the analytic solution 𝑦(𝑡𝑛 ), the argument 𝑌𝑗 =
For function 𝑔(𝑡, 𝜃, 𝑦(𝜃)), we make the following assumptions, unless otherwise stated; all its partial derivatives used
later exist and satisfy the following:
󵄩󵄩 𝜕𝑖 𝑔 (𝑡, 𝜃, 𝑦 (𝜃)) 󵄩󵄩
󵄩󵄩
󵄩󵄩
󵄩󵄩 ≤ 𝑁𝑖 ,
󵄩󵄩
𝑖
󵄩󵄩
󵄩󵄩
𝜕𝜃
󵄩
󵄩
(𝑛−1)
̃
,𝑌
𝑗
𝑌𝑗(𝑛−𝑚) denotes an approximation to 𝑦(𝑡𝑛 + 𝑐𝑗 ℎ − 𝜏), and the
̃ (𝑛) denotes an approximation to ∫𝑡𝑛 +𝑐𝑗 ℎ 𝑔(𝑡𝑛 +
argument 𝑌
𝑗
𝑡 +𝑐 ℎ−𝜏
𝑛
𝑗
𝑐𝑗 ℎ, 𝜉, 𝑦(𝜉))𝑑𝜉 which are obtained by a convergent compound
quadrature (CQ) formula:
𝑚
𝑡 ∈ [0, 𝑇] , 𝜃 ∈ [−𝜏, 𝑇] .
(6)
̃ (𝑛) = ℎ ∑ 𝑝
̃ 𝑞 𝑔 (𝑡𝑛 + 𝑐𝑗 ℎ, 𝑡𝑛−𝑞 + 𝑐𝑗 ℎ, 𝑌𝑗(𝑛−𝑞) ) ,
𝑌
𝑗
𝑞=0
(9)
𝑗 = 1, 2, . . . , 𝑠,
Many numerical methods have been proposed for the numerical solution of (1).
(𝑛−𝑞)
using values 𝑌𝑗
= 𝜑(𝑡𝑛−𝑞 + 𝑐𝑗 ℎ) with 𝑛 ≤ 𝑞, 𝑡𝑛−𝑞 + 𝑐𝑗 ℎ ≤ 0.
Abstract and Applied Analysis
3
The class of Runge-Kutta methods with CQ formula has
been applied to delay-integro-differential equations by many
authors (c.f. [18, 19]). For the CQ formula (9), we usually
adopt the repeated trapezoidal rule, the repeated Simpson’s
rule, or the repeated Newton-cotes rule, and so forth, denote
̃1, . . . , 𝑝
̃ 𝑚 }. It should be pointed out that
̃0, 𝑝
𝜂 = max{𝑝
the adopted quadrature formula (9) is only a class of quad̃ (𝑛) , there also exist some other types of
rature formula for 𝑌
𝑖
quadrature formula, such as Pouzet quadrature (PQ) formula
and the quadrature formula based on Laguerre-Radau interpolations [20, 21]. It is the aim of our future research to
investigate the adaptation of PQ formula and the quadrature
formula based on Laguerre-Radau interpolations to VDIDEs.
3. The Nonlinear Stability Analysis
𝑡
𝑡−𝜏
𝑛 +𝑐𝑗 ℎ−𝜏
𝑔(𝑡𝑛 + 𝑐𝑗 ℎ, 𝜉, 𝑧(𝜉))𝑑𝜉 which are obtained by a
convergent compound quadrature (CQ) formula:
𝑚
̃(𝑛) = ℎ ∑ 𝑝
̃ 𝑞 𝑔 (𝑡𝑛 + 𝑐𝑖 ℎ, 𝑡𝑛−𝑞 + 𝑐𝑖 ℎ, 𝑍𝑖(𝑛−𝑞) ) ,
𝑍
𝑖
𝑞=0
(12)
𝑖 = 1, 2, . . . , 𝑠,
(𝑛)
and 𝑍𝑗 = 𝑍𝑗(𝑛−𝑚) denotes an approximation to 𝑧(𝑡𝑛 +𝑐𝑗 ℎ−𝜏),
and 𝑧𝑛 = 𝜓(𝑡𝑛 ) for 𝑛 = 0,
𝑍𝑖
= 𝜓 (𝑡𝑛−𝑞 + 𝑐𝑖 ℎ) ,
for 𝑡𝑛−𝑞 + 𝑐𝑖 ℎ ≤ 0.
̃ is a positive constant.
where 𝑝
Let
𝑡 ∈ [−𝜏, 0] ,
= 𝑧𝑛 + ℎ ∑ 𝑎𝑖𝑗 𝑓 (𝑡𝑛 +
𝑗=1
𝑤𝑛 = 𝑦𝑛 − 𝑧𝑛 ,
𝑊𝑗(𝑛) = 𝑌𝑗(𝑛) − 𝑍𝑗(𝑛) ,
̃(𝑛) ,
̃ (𝑛) = 𝑌
̃ (𝑛) − 𝑍
𝑊
𝑗
𝑗
𝑗
(𝑛)
(𝑛)
̃(𝑛) ) ,
𝑐𝑗 ℎ, 𝑍𝑗(𝑛) , 𝑍𝑗 , 𝑍
𝑗
(11a)
𝑍𝑖(𝑛−1) = 𝑧𝑛−1
(𝑛)
𝑠
+ ℎ ∑ 𝑎𝑖𝑗 𝑓 (𝑡𝑛−1 +
𝑗=1
(𝑛−1)
̃(𝑛−1) ) ,
𝑐𝑗 ℎ, 𝑍𝑗(𝑛−1) , 𝑍𝑗 , 𝑍
𝑗
𝑊𝑖(𝑛) = 𝑤𝑛 + ∑ 𝑎𝑖𝑗 𝑄𝑗(𝑛) ,
𝑖 = 1, 2, . . . , 𝑠,
(16a)
𝑖 = 1, 2, . . . , 𝑠,
(16b)
𝑗=1
𝑊𝑖(𝑛−1) = 𝑤𝑛−1 + ∑ 𝑎𝑖𝑗 𝑄𝑗(𝑛−1) ,
𝑗=1
𝑠
𝑧𝑛+1 = (1 − 𝜃) 𝑧𝑛 + 𝜃𝑧𝑛−1
𝑖=1
(𝑛)
It follows from (8a)–(8c) and (11a)–(11c) that
𝑠
𝑖 = 1, 2, . . . , 𝑠,
(11b)
(15)
𝑗 = 1, 2, . . . , 𝑠.
𝑠
+ ℎ ∑ 𝑏𝑖 𝑓 (𝑡𝑛 +
(𝑛)
̃ )] ,
− 𝑓 (𝑡𝑛 + 𝑐𝑗 ℎ, 𝑍𝑗(𝑛) , 𝑍𝑗 , 𝑍
𝑗
𝑖 = 1, 2, . . . , 𝑠,
𝑠
(14)
𝑞=0
̃ )
𝑄𝑗(𝑛) = ℎ [𝑓 (𝑡𝑛 + 𝑐𝑗 ℎ, 𝑌𝑗(𝑛) , 𝑌𝑗 , 𝑌
𝑗
𝑠
(13)
3.1. Some Concepts. For the stability analysis, we need the
compound quadrature formula (9) that satisfies the following
condition:
(10)
where 𝜓 : [−𝜏, 0] → 𝐶𝑁 is a given continuous function. The
unique exact solution of the problem (10) is denoted as 𝑧(𝑡).
Applying the two-step Runge-Kutta method (7a)–(7c) to
(10) leads to
𝑍𝑖(𝑛)
tion to ∫𝑡
𝑗
𝑚
󵄨 󵄨󵄨2
̃ 𝑞 󵄨󵄨 < 𝑝
̃2,
ℎ2 (𝑚 + 1)2 ∑ 󵄨󵄨󵄨󵄨𝑝
󵄨
𝑔 (𝑡, 𝜉, 𝑧 (𝜉)) 𝑑𝜉) ,
𝑡 ∈ [0, Τ] ,
𝑧 (𝑡) = 𝜓 (𝑡) ,
𝑡𝑛 +𝑐𝑗 ℎ
(𝑛−𝑞)
In this section, we will investigate the stability of the twostep Runge-Kutta methods for VDIDEs. In order to consider
the stability property, we also need to consider the perturbed
problem of (1):
𝑧󸀠 (𝑡) = 𝑓 (𝑡, 𝑧 (𝑡) , 𝑧 (𝑡 − 𝜏) , ∫
where 𝑧𝑛 and 𝑍𝑖(𝑛) denote approximations to 𝑧(𝑡𝑛 ) and 𝑧(𝑡𝑛 +
̃(𝑛) denotes an approxima𝑐𝑖 ℎ), respectively; the argument 𝑍
𝑤𝑛+1 = (1 − 𝜃) 𝑤𝑛 + 𝜃𝑤𝑛−1 + ∑ 𝑏𝑖 𝑄𝑖(𝑛)
𝑖=1
(𝑛)
̃(𝑛) )
𝑐𝑖 ℎ, 𝑍𝑖(𝑛) , 𝑍𝑖 , 𝑍
𝑖
𝑠
(16c)
+ ∑ ̃𝑏𝑖 𝑄𝑖(𝑛−1) .
𝑖=1
𝑠
(𝑛−1)
̃(𝑛−1) ) ,
+ ℎ ∑ ̃𝑏𝑖 𝑓 (𝑡𝑛−1 + 𝑐𝑖 ℎ, 𝑍𝑖(𝑛−1) , 𝑍𝑖 , 𝑍
𝑖
𝑖=1
(11c)
Now we will write the 𝑠-stage TSRK methods (7a)–(7c) as a
general linear method.
4
Abstract and Applied Analysis
Τ
Τ
Let 𝑉𝑖(𝑛) = (𝑊𝑖(𝑛) , 𝑊𝑖(𝑛−1) )Τ be the internal stages and
Τ
, 𝑤𝑛Τ )Τ the external vectors and 𝑃𝑖(𝑛) =
𝜇𝑛 = (𝑤𝑛+1
Τ
Τ
(𝑄𝑖(𝑛) ,
𝑄𝑖(𝑛−1) )Τ . Then we have a 2(𝑠 + 1)-stage partitioned general
linear method:
𝑠
𝑠
𝑗=1
𝑗=1
𝑠
𝑠
𝑉𝑖(𝑛) = ∑ 𝐶𝑖𝑗11 𝑃𝑖(𝑛) + ∑ 𝐶𝑖𝑗12 𝜇𝑛 ,
𝜇𝑛+1 = ∑ 𝐶𝑖𝑗21 𝑃𝑖(𝑛) + ∑ 𝐶𝑖𝑗22 𝜇𝑛 ,
𝑗=1
𝑖 = 1, 2, . . . , 𝑠,
(17a)
𝑖 = 1, 2, . . . , 𝑠,
(17b)
𝑗=1
where 𝐶𝑖𝑗11 = 𝑎𝑖𝑗 , 𝑖, 𝑗 = 1, 2, ⋅ ⋅ ⋅ , 𝑠, 𝐶𝑖𝑗11 = 0, 𝑖 = 1, 2, ⋅ ⋅ ⋅ , 𝑠, 𝑗 =
𝑠+1, 𝑠+2, ⋅ ⋅ ⋅ , 2𝑠, 𝐶𝑖𝑗11 = 0, 𝑖 = 𝑠+1, 𝑠+2, ⋅ ⋅ ⋅ , 2𝑠, 𝑗 = 1, 2, ⋅ ⋅ ⋅ , 𝑠,
𝐶𝑖𝑗11 = 𝑎𝑖𝑗 , 𝑖, 𝑗 = 𝑠 + 1, 𝑠 + 2, ⋅ ⋅ ⋅ , 2𝑠, 𝐶𝑖𝑗12 = 1, 𝑖, 𝑗 = 1, 2, ⋅ ⋅ ⋅ , 𝑠,
𝐶𝑖𝑗12 = 0, 𝑖 = 1, 2, ⋅ ⋅ ⋅ , 𝑠, 𝑗 = 𝑠 + 1, 𝑠 + 2, ⋅ ⋅ ⋅ , 2𝑠, 𝐶𝑖𝑗12 = 0, 𝑖 = 𝑠 +
1, 𝑠+2, ⋅ ⋅ ⋅ , 2𝑠, 𝑗 = 1, 2, ⋅ ⋅ ⋅ , 𝑠, 𝐶𝑖𝑗12 = 1, 𝑖, 𝑗 = 𝑠+1, 𝑠+2, ⋅ ⋅ ⋅ , 2𝑠,
𝐶21 = 𝑏 , 𝑖, 𝑗 = 1, 2, ⋅ ⋅ ⋅ , 𝑠, 𝐶21 = ̃𝑏 , 𝑖 = 1, 2, ⋅ ⋅ ⋅ , 𝑠, 𝑗 = 𝑠 +
𝑖𝑗
𝑗
𝑗
𝑖𝑗
1, 𝑠 + 2, ⋅ ⋅ ⋅ , 2𝑠, 𝐶𝑖𝑗21 = 0, 𝑖 = 𝑠 + 1, 𝑠 + 2, ⋅ ⋅ ⋅ , 2𝑠, 𝑗 = 1, 2, ⋅ ⋅ ⋅ , 𝑠,
𝑀 (𝑘, 𝑙) = (
Τ
Τ
𝐺𝐶22 − 2𝑙𝐶12
𝐷𝐶12
𝑘𝐺 − 𝐶22
𝐶𝑖𝑗21 = 0, 𝑖, 𝑗 = 𝑠 + 1, 𝑠 + 2, ⋅ ⋅ ⋅ , 2𝑠, 𝐶𝑖𝑗22 = 1 − 𝜃, 𝑖, 𝑗 = 1, 2, ⋅ ⋅ ⋅ , 𝑠,
𝐶𝑖𝑗22 = 𝜃, 𝑖 = 1, 2, ⋅ ⋅ ⋅ , 𝑠, 𝑗 = 𝑠 + 1, 𝑠 + 2, ⋅ ⋅ ⋅ , 2𝑠, 𝐶𝑖𝑗22 = 1, 𝑖 =
𝑠 + 1, 𝑠 + 2, ⋅ ⋅ ⋅ , 2𝑠, 𝑗 = 1, 2, ⋅ ⋅ ⋅ , 𝑠, 𝐶𝑖𝑗22 = 0, 𝑖, 𝑗 = 𝑠 + 1, 𝑠 +
2, ⋅ ⋅ ⋅ , 2𝑠, and 𝐴 = (𝑎𝑖𝑗 ), 𝑒 = (1, 1, . . . 1), 𝑏 = (𝑏1 , 𝑏2 , . . . , 𝑏𝑠 ),
̃𝑏 = (̃𝑏 , ̃𝑏 , . . . , ̃𝑏 ).
1 2
𝑠
We can identify the coefficient matrices
𝐴 0
𝐶11 = (𝐶𝑖𝑗11 ) = (
),
0 𝐴
𝑏 ̃𝑏
𝐶21 = (𝐶𝑖𝑗21 ) = (
),
0 0
𝑒 0
𝐶12 = (𝐶𝑖𝑗12 ) = (
),
0 𝑒
1−𝜃 𝜃
𝐶22 = (𝐶𝑖𝑗22 ) = (
).
1 0
(18)
Definition 1 (see [22]). Let 𝑘, 𝑙 be real constants, a TSRK
method is said to be (𝑘, 𝑙) algebraically stable if there exists
a diagonal matrix 𝐷 = diag(𝑑1 , 𝑑2 , . . . , 𝑑2𝑠 ) and a diagonal
nonnegative matrix 𝐺 such that 𝑀 = (𝑚𝑖𝑗 ) is nonnegative,
where
Τ
Τ
Τ
𝐶12
𝐷 − 𝐶22
𝐺𝐶21 − 2𝑙𝐶12
𝐷𝐶11
Τ
Τ
Τ
Τ
Τ
𝐷𝐶12 − 𝐶21
𝐺𝐶22 − 2𝑙𝐶11
𝐷𝐶12 𝐶11
𝐷 + 𝐷𝐶11 − 𝐶21
𝐺𝐶21 − 2𝑙𝐶11
𝐷𝐶11
In particular, a (1, 0)-algebraically stable method is called
algebraically stable.
Theorem 3. Assume that the TSRK method (7a)–(7c) is (𝑘, 𝑙)algebraically stable with 0 < 𝑘 ≤ 1, suppose that the quadrature
formula (9) satisfies the condition (14) and the conditions (2)–
(4) hold, then, method (17a) and (17b) satisfies the following:
󵄩
󵄩
󵄩
󵄩󵄩
󵄩󵄩𝑦𝑛 − 𝑧𝑛 󵄩󵄩󵄩 ≤ 𝐶 max 󵄩󵄩󵄩𝜑 (𝑡) − 𝜓 (𝑡)󵄩󵄩󵄩 ,
𝑡∈[−𝜏,0]
𝑛 = 1, 2, . . . ,
2𝑠
󵄩2
󵄩 󵄩2
󵄩󵄩
(𝑛)
(𝑛)
(𝑛)
󵄩󵄩𝜇𝑛+1 󵄩󵄩󵄩 − 𝑘󵄩󵄩󵄩𝜇𝑛 󵄩󵄩󵄩 − 2 ∑ 𝑑𝑗 Re ⟨𝑉𝑗 , 𝑃𝑗 − 𝑙𝑉𝑗 ⟩
𝑗=1
(22)
2𝑠+2 2𝑠+2
= − ∑ ∑ 𝑀𝑖𝑗 ⟨𝑟𝑖 , 𝑟𝑗 ⟩ ,
𝑖=1 𝑗=1
where
𝑟1 = 𝑤𝑛+1 ,
(𝑛)
,
𝑟𝑗 = 𝑄𝑗−2
(20)
(19)
Proof. It follows from a fairly straightforward (but tedious)
computation and (𝑘, 𝑙) algebraically stability that (compare
also [22, 23])
Definition 2. The TSRK method (7a)–(7c) is called asymptotically stable for (1) with (2)–(4) if lim𝑛 → ∞ ‖𝑤𝑛 ‖ = 0.
3.2. Numerical Stability of the Methods. Numerical stability
is an important feature of an effective numerical method. An
unstable numerical method may be consistent of high order,
yet arbitrarily small perturbations will eventually cause large
deviations from the true solution. In this section, we will
focus on the asymptotic stability of the TSRK method.
).
(𝑛−1)
𝑟𝑗 = 𝑄𝑗−𝑠−2
,
𝑟2 = 𝑤𝑛 ,
𝑗 = 3, 4, . . . , 𝑠 + 2,
(23)
𝑗 = 𝑠 + 3, 𝑠 + 4, . . . , 2𝑠 + 2.
when the following condition holds:
2 2
̃ ) ≤ 𝑙,
ℎ (𝛼 + 𝛽 + 𝜎 + 𝜎𝛾 𝑝
(21)
where 𝐶 depends only on the method, the parameters
𝛼, 𝛽, 𝜎, 𝛾, 𝜂, and 𝜏.
By means of (𝑘, 𝑙) algebraical stability of the method, we have
the following:
2𝑠
󵄩󵄩
󵄩2
󵄩 󵄩2
(𝑛)
(𝑛)
(𝑛)
󵄩󵄩𝜇𝑛+1 󵄩󵄩󵄩 ≤ 𝑘󵄩󵄩󵄩𝜇𝑛 󵄩󵄩󵄩 + 2 ∑ 𝑑𝑗 Re ⟨𝑉𝑗 , 𝑃𝑗 − 𝑙𝑉𝑗 ⟩ .
𝑗=1
(24)
Abstract and Applied Analysis
5
It follows from (2)–(4) that
Substituting (25) into (24), using (14) we get the following:
2 Re ⟨𝑊𝑗(𝑛) , 𝑄𝑗(𝑛) ⟩
󵄩2
󵄩󵄩
󵄩󵄩𝜇𝑛+1 󵄩󵄩󵄩
(𝑛)
(𝑛)
̃ )
= 2ℎ Re ⟨𝑊𝑗(𝑛) , 𝑓 (𝑡𝑛 + 𝑐𝑗 ℎ, 𝑌𝑗(𝑛) , 𝑌𝑗 , 𝑌
𝑗
2𝑠
󵄩2
󵄩
󵄩 󵄩2
≤ 󵄩󵄩󵄩𝜇𝑛 󵄩󵄩󵄩 + ∑ 𝑑𝑗 { (2ℎ𝛼 + ℎ𝛽 + ℎ𝜎 − 2𝑙) 󵄩󵄩󵄩󵄩𝑊𝑗(𝑛) 󵄩󵄩󵄩󵄩
(𝑛)
(𝑛)
̃ )⟩
−𝑓 (𝑡𝑛 + 𝑐𝑗 ℎ, 𝑍𝑗(𝑛) , 𝑍𝑗 , 𝑍
𝑗
=
2ℎ Re ⟨𝑊𝑗(𝑛) , 𝑓 (𝑡𝑛
−𝑓 (𝑡𝑛 +
+
𝑗=1
𝑚
󵄩
󵄩
󵄩2
󵄩
̃ 2 ∑ 󵄩󵄩󵄩𝑊𝑗(𝑛−𝑞) 󵄩󵄩󵄩 } .
+ℎ𝛽 󵄩󵄩󵄩󵄩𝑊𝑗(𝑛−𝑚) 󵄩󵄩󵄩󵄩 + ℎ𝜎𝛾2 𝑝
󵄩
󵄩
(𝑛)
̃ (𝑛) )
𝑐𝑗 ℎ, 𝑌𝑗(𝑛) , 𝑌𝑗 , 𝑌
𝑗
𝑞=0
(26)
(𝑛)
̃ (𝑛) )⟩
𝑐𝑗 ℎ, 𝑍𝑗(𝑛) , 𝑌𝑗 , 𝑌
𝑗
(𝑛)
By induction, we get the following:
(𝑛)
̃ )
+ 2ℎ Re ⟨𝑊𝑗(𝑛) , 𝑓 (𝑡𝑛 + 𝑐𝑗 ℎ, 𝑍𝑗(𝑛) , 𝑌𝑗 , 𝑌
𝑗
󵄩󵄩
󵄩2
󵄩󵄩𝜇𝑛+1 󵄩󵄩󵄩
󵄩 󵄩2
≤ 󵄩󵄩󵄩𝜇−1 󵄩󵄩󵄩
(𝑛)
(𝑛)
̃ )⟩
−𝑓 (𝑡𝑛 + 𝑐𝑗 ℎ, 𝑍𝑗(𝑛) , 𝑍𝑗 , 𝑍
𝑗
󵄩
󵄩2
󵄩 󵄩󵄩 (𝑛) 󵄩󵄩
󵄩
≤ 2ℎ𝛼󵄩󵄩󵄩󵄩𝑊𝑗(𝑛) 󵄩󵄩󵄩󵄩 + 2ℎ𝛽 󵄩󵄩󵄩󵄩𝑊𝑗(𝑛) 󵄩󵄩󵄩󵄩 ⋅ 󵄩󵄩󵄩𝑊𝑗 󵄩󵄩󵄩
󵄩
󵄩
󵄩󵄩 (𝑛) 󵄩󵄩 󵄩󵄩󵄩 ̃ (𝑛) 󵄩󵄩󵄩
+ 2ℎ𝜎 󵄩󵄩󵄩𝑊𝑗 󵄩󵄩󵄩 ⋅ 󵄩󵄩𝑊𝑗 󵄩󵄩
󵄩
󵄩
󵄩
󵄩2
󵄩2 󵄩󵄩 (𝑛) 󵄩󵄩2
󵄩
≤ 2ℎ𝛼󵄩󵄩󵄩󵄩𝑊𝑗(𝑛) 󵄩󵄩󵄩󵄩 + ℎ𝛽 (󵄩󵄩󵄩󵄩𝑊𝑗(𝑛) 󵄩󵄩󵄩󵄩 + 󵄩󵄩󵄩𝑊𝑗 󵄩󵄩󵄩 )
󵄩
󵄩
𝑛
2𝑠
󵄩2
󵄩
̃ 2 ) 󵄩󵄩󵄩𝑊𝑗(𝑖) 󵄩󵄩󵄩 }
+ ∑ ∑ 𝑑𝑗 {(2ℎ𝛼 + 2ℎ𝛽 + ℎ𝜎 − 2𝑙 + ℎ𝜎𝛾2 𝑝
󵄩
󵄩
𝑖=−1 𝑗=1
−2 2𝑠
󵄩2
󵄩
̃ 2 ) 󵄩󵄩󵄩𝑊𝑗(𝑖) 󵄩󵄩󵄩 }
+ ∑ ∑ 𝑑𝑗 {(ℎ𝛽 + ℎ𝜎𝛾2 𝑝
󵄩
󵄩
𝑖=−𝑚 𝑗=1
󵄩2 󵄩󵄩 ̃ (𝑛) 󵄩󵄩󵄩2
󵄩
+ ℎ𝜎 (󵄩󵄩󵄩󵄩𝑊𝑗(𝑛) 󵄩󵄩󵄩󵄩 + 󵄩󵄩󵄩𝑊
󵄩)
󵄩 𝑗 󵄩󵄩
𝑛
2𝑠
󵄩2
󵄩
󵄩 󵄩2
̃ 2 − 𝑙) 󵄩󵄩󵄩𝑊𝑗(𝑖) 󵄩󵄩󵄩 }
≤ 󵄩󵄩󵄩𝜇−1 󵄩󵄩󵄩 + 2 ∑ ∑ 𝑑𝑗 {ℎ (𝛼 + 𝛽 + 𝜎 + 𝜎𝛾2 𝑝
󵄩
󵄩
󵄩2
󵄩
= (2ℎ𝛼 + ℎ𝛽 + ℎ𝜎) 󵄩󵄩󵄩󵄩𝑊𝑗(𝑛) 󵄩󵄩󵄩󵄩
𝑖=−1 𝑗=1
󵄩󵄩 (𝑛) 󵄩󵄩2
󵄩󵄩 (𝑛) 󵄩󵄩2
̃ 󵄩󵄩
+ ℎ𝛽󵄩󵄩󵄩𝑊𝑗 󵄩󵄩󵄩 + ℎ𝜎󵄩󵄩󵄩𝑊
󵄩
󵄩
󵄩 𝑗 󵄩󵄩
(25)
̃ 𝑞 𝑔 (𝑡𝑛 + 𝑐𝑗 ℎ, 𝑡𝑛−𝑞 +
−ℎ ∑ 𝑝
𝑞=0
2
̃𝑞
+ ℎ𝜎 (𝑚 + 1) ℎ2 𝑝
𝑞=0
(𝑛−𝑞) 󵄩2
𝑐𝑗 ℎ, 𝑍𝑗 )󵄩󵄩󵄩󵄩
󵄩2
󵄩
≤ (2ℎ𝛼 + ℎ𝛽 + ℎ𝜎) 󵄩󵄩󵄩󵄩𝑊𝑗(𝑛) 󵄩󵄩󵄩󵄩
󵄩2
󵄩
+ ℎ𝛽󵄩󵄩󵄩󵄩𝑊𝑗(𝑛−𝑚) 󵄩󵄩󵄩󵄩 + ℎ𝜎 (𝑚 + 1) ℎ2 𝛾2
𝑚
󵄩2
󵄩
̃ 2𝑞 ⋅ 󵄩󵄩󵄩𝑊𝑗(𝑛−𝑞) 󵄩󵄩󵄩 .
×∑𝑝
󵄩
󵄩
𝑞=0
𝑗=1
󵄩
󵄩2
max 󵄩󵄩󵄩󵄩𝑊𝑗(𝑖) 󵄩󵄩󵄩󵄩
−𝑚−1≤𝑖≤−2
󵄩2
󵄩 󵄩2 󵄩
≤ 󵄩󵄩󵄩𝑤0 󵄩󵄩󵄩 + 󵄩󵄩󵄩𝑤−1 󵄩󵄩󵄩
󵄩󵄩
󵄩
2𝑠
2
̃ )
+ 𝑚ℎ ∑ 𝑑𝑗 (𝛽 + 𝜎𝛾2 𝑝
𝑗=1
󵄩
󵄩2
max 󵄩󵄩󵄩󵄩𝑊𝑗(𝑖) 󵄩󵄩󵄩󵄩
−𝑚−1≤𝑖≤−2
2𝑠
̃ 2 ) ∑ 𝑑𝑗 ) max 󵄩󵄩󵄩󵄩𝜑 (𝑡) − 𝜓 (𝑡)󵄩󵄩󵄩󵄩2 .
≤ (2 + 𝜏 (𝛽 + 𝜎𝛾2 𝑝
𝑡∈[−𝜏,0]
𝑗=1
(27)
𝑚
󵄩
(𝑛−𝑞)
× ∑ 󵄩󵄩󵄩󵄩𝑔 (𝑡𝑛 + 𝑐𝑗 ℎ, 𝑡𝑛−𝑞 + 𝑐𝑗 ℎ, 𝑌𝑗 )
−𝑔 (𝑡𝑛 + 𝑐𝑗 ℎ, 𝑡𝑛−𝑞 +
2𝑠
󵄩 󵄩2
̃2)
≤ 󵄩󵄩󵄩𝜇−1 󵄩󵄩󵄩 + 𝑚ℎ ∑ 𝑑𝑗 (𝛽 + 𝜎𝛾2 𝑝
󵄩󵄩2
󵄩
(𝑛−𝑞) 󵄩
𝑐𝑗 ℎ, 𝑍𝑗 )󵄩󵄩󵄩󵄩
󵄩2
󵄩2
󵄩
󵄩
≤ (2ℎ𝛼 + ℎ𝛽 + ℎ𝜎) 󵄩󵄩󵄩󵄩𝑊𝑗(𝑛) 󵄩󵄩󵄩󵄩 + ℎ𝛽󵄩󵄩󵄩󵄩𝑊𝑗(𝑛−𝑚) 󵄩󵄩󵄩󵄩
2𝑠
󵄩2
󵄩
̃ 2 ) 󵄩󵄩󵄩𝑊𝑗(𝑖) 󵄩󵄩󵄩 }
∑ 𝑑𝑗 {(ℎ𝛽 + ℎ𝜎𝛾2 𝑝
󵄩
󵄩
𝑖=−𝑚−1 𝑗=1
󵄩󵄩 (𝑛) 󵄩󵄩2
󵄩2
󵄩
= (2ℎ𝛼 + ℎ𝛽 + ℎ𝜎) 󵄩󵄩󵄩󵄩𝑊𝑗(𝑛) 󵄩󵄩󵄩󵄩 + ℎ𝛽󵄩󵄩󵄩𝑊𝑗 󵄩󵄩󵄩
󵄩
󵄩
󵄩󵄩 𝑚
󵄩󵄩
̃ 𝑞 𝑔 (𝑡𝑛 + 𝑐𝑗 ℎ, 𝑡𝑛−𝑞 + 𝑐𝑗 ℎ, 𝑌𝑗(𝑛−𝑞) )
+ ℎ𝜎 󵄩󵄩󵄩󵄩ℎ ∑ 𝑝
󵄩󵄩 𝑞=0
󵄩
𝑚
−2
+ ∑
Hence, ‖𝜇𝑛+1 ‖ ≤ 𝐶 max𝑡∈[−𝜏,0] ‖𝜑(𝑡) − 𝜓(𝑡)‖, where 𝐶 =
̃ 2 ) ∑2𝑠
√2 + 𝜏(𝛽 + 𝜎𝛾2 𝑝
𝑗=1 𝑑𝑗 .
The proof of Theorem 3 is completed.
Theorem 4. Assume that a TSRK method (7a)–(7c) is (𝑘, 𝑙)algebraically stable with 0 < 𝑘 < 1, then the TSRK method
(7a)–(7c) with (9), (12) and (14) is asymptotically stable for the
problem (1) with (2)–(4), when the following condition holds:
1
1
̃ 2 ) < 𝑙.
ℎ (𝛼 + 𝛽 + 𝜎 + 𝜎𝛾2 𝑝
2
2
(28)
6
Abstract and Applied Analysis
Proof. Let
𝑢 = (2𝛼 + 𝛽 + 𝜎) ℎ − 2𝑙,
(29)
̃2) ℎ }
{ (𝛽 + 𝜎𝛾2 𝑝
𝑘 = max {𝑘,
.
}
(−𝑢)1/𝑚
}
{
̃ 2 ) < 𝑙, we have 𝑢 <
Then when ℎ(𝛼 + 𝛽 + (1/2)𝜎 + (1/2)𝜎𝛾2 𝑝
̃ 2 )ℎ and 0 < 𝑘 < 1.
−(𝛽 + 𝜎𝛾2 𝑝
The application of Theorem 3 yields the following:
󵄩2
󵄩 󵄩2
󵄩󵄩
󵄩󵄩𝜇𝑛+1 󵄩󵄩󵄩 ≤ 𝑘󵄩󵄩󵄩𝜇𝑛 󵄩󵄩󵄩
2𝑠
󵄩2
󵄩
+ ∑ 𝑑𝑗 { (2ℎ𝛼 + ℎ𝛽 + ℎ𝜎 − 2𝑙) 󵄩󵄩󵄩󵄩𝑊𝑗(𝑛) 󵄩󵄩󵄩󵄩
𝑚
󵄩
󵄩
󵄩2
󵄩
̃ 2 ∑ 󵄩󵄩󵄩𝑊𝑗(𝑛−𝑞) 󵄩󵄩󵄩 }
+ℎ𝛽 󵄩󵄩󵄩󵄩𝑊𝑗(𝑛−𝑚) 󵄩󵄩󵄩󵄩 + ℎ𝜎𝛾2 𝑝
󵄩
󵄩
𝑞=0
󵄩 󵄩2
≤ 𝑘󵄩󵄩󵄩𝜇𝑛 󵄩󵄩󵄩
4. The Convergence of TSRK
Method for NDDEs
Τ
(𝑛) Τ
= (𝑌1
(𝑛) Τ
, . . . , 𝑌𝑠
Τ
𝑚
(𝑛−1) Τ
, 𝑌1
Τ
By induction, we get the following:
󵄩2
󵄩󵄩
󵄩󵄩𝜇𝑛+1 󵄩󵄩󵄩
Τ Τ
(𝑛)
(𝑛)
̃ )
𝐹 (𝑡𝑛 , 𝑌(𝑛) , 𝑌 , 𝑌
(𝑛)
(𝑛) Τ
(𝑛)
(𝑛) Τ
̃ ) ,...,
= (𝑓(𝑡𝑛 + 𝑐1 ℎ, 𝑌1(𝑛) , 𝑌1 , 𝑌
1
𝑛
󵄩󵄩𝜇−1 󵄩󵄩󵄩2 + ∑ 𝑘 𝑛−𝑚−𝑖
󵄩 󵄩
̃ ) ,
𝑓(𝑡𝑛 + 𝑐𝑠 ℎ, 𝑌𝑠(𝑛) , 𝑌𝑠 , 𝑌
𝑠
𝑖=−1
𝑗=1
+ ∑ ∑ 𝑑𝑗 𝑘
𝑖=−𝑚−1 𝑗=1
≤𝑘
󵄩2
󵄩
̃ 2 ) ⋅ 󵄩󵄩󵄩𝑊𝑗(𝑖) 󵄩󵄩󵄩
(ℎ𝛽 + ℎ𝜎𝛾2 𝑝
󵄩
󵄩
̃ )⋅
⋅ ∑ 𝑑𝑗 (𝛽 + 𝜎𝛾 𝑝
𝑗=1
𝑛+2
2
𝑗=1
Τ
(𝑛−1) Τ
) ) ∈ 𝐶2𝑠𝑁.
(𝑛)
(𝑛)
̃ ) + 𝐶22 𝜁(𝑛) .
𝜁(𝑛+1) = ℎ𝐶21 𝐹 (𝑡𝑛 , 𝑌(𝑛) , 𝑌 , 𝑌
󵄩
󵄩2
max 󵄩󵄩󵄩𝑊𝑗(𝑖) 󵄩󵄩󵄩󵄩
−𝑚−1≤𝑖≤−2󵄩
2𝑠
{ 𝑛+2
}
𝑛−𝑚+2
̃2) }
≤ {2𝑘
+𝑘
⋅ 𝜏⋅ ∑ 𝑑𝑗 (𝛽 + 𝜎𝛾2 𝑝
𝑗=1
{
}
󵄩2
󵄩󵄩
⋅ max 󵄩󵄩𝜑 (𝑡) − 𝜓 (𝑡)󵄩󵄩󵄩 .
𝑡∈[−𝜏,0]
(𝑛)
(𝑛)
𝑛−𝑚+2
󵄩2 󵄩 󵄩2
󵄩
(󵄩󵄩󵄩𝑤−1 󵄩󵄩󵄩 + 󵄩󵄩󵄩𝑤0 󵄩󵄩󵄩 ) + 𝑘
2𝑠
̃
,𝑌
𝑠
̃ ) + 𝐶12 𝜁(𝑛) ,
𝑌(𝑛) = ℎ𝐶11 𝐹 (𝑡𝑛 , 𝑌(𝑛) , 𝑌 , 𝑌
󵄩
󵄩2
max 󵄩󵄩󵄩𝑊(𝑖) 󵄩󵄩󵄩
−𝑚−1≤𝑖≤−2󵄩 𝑗 󵄩
̃ )⋅
⋅ 𝜏 ⋅ ∑ 𝑑𝑗 (𝛽 + 𝜎𝛾2 𝑝
(𝑛−1)
𝑓(𝑡𝑛−1 + 𝑐𝑠 ℎ, 𝑌𝑠(𝑛−1) , 𝑌𝑠
) ,...,
Thus, process (8a)–(8c) can be written in the more compact
form:
󵄩󵄩𝜇−1 󵄩󵄩󵄩2 + 𝑘 𝑛−𝑚+2 ⋅ 𝑚ℎ
󵄩 󵄩
2 2
̃
,𝑌
1
(35)
𝑛+2 󵄩
2𝑠
≤𝑘
(31)
(𝑛−1) Τ
(𝑛−1)
𝑓(𝑡𝑛−1 + 𝑐1 ℎ, 𝑌1(𝑛−1) , 𝑌1
󵄩2
󵄩
̃ ) ⋅ 󵄩󵄩󵄩𝑊𝑗(𝑖) 󵄩󵄩󵄩
× ∑ 𝑑𝑗 (𝑢𝑘 + ℎ𝛽 + ℎ𝜎𝛾 𝑝
󵄩
󵄩
2 2
𝑛−𝑚−𝑖
Τ
) ∈ 𝐶2𝑠𝑁,
Τ Τ
Τ
Τ
(30)
2𝑠
(𝑛−1) Τ
, . . . , 𝑌𝑠
𝜁(𝑛+1) = (𝑦(𝑛+1) , 𝑦(𝑛) ) ∈ 𝐶2𝑁,
𝑞=0
−2
Τ Τ
Τ
̃ (𝑛) = (𝑌
̃ (𝑛) , . . . , 𝑌
̃ (𝑛) , 𝑌
̃ (𝑛−1) , . . . , 𝑌
̃ (𝑛−1) ) ∈ 𝐶2𝑠𝑁,
𝑌
1
𝑠
1
𝑠
󵄩
󵄩2
̃ 2 ∑ 󵄩󵄩󵄩𝑊𝑗(𝑛−𝑞) 󵄩󵄩󵄩 } .
+ℎ𝜎𝛾2 𝑝
󵄩
󵄩
𝑚
Τ
𝑌(𝑛) = (𝑌1(𝑛) , . . . , 𝑌𝑠(𝑛) , 𝑌1(𝑛−1) , . . . , 𝑌𝑠(𝑛−1) ) ∈ 𝐶2𝑠𝑁,
(𝑛)
𝑗=1
2𝑠
The proof of the theorem is completed.
𝑌
2𝑠
󵄩2
󵄩
󵄩
󵄩
+ ∑ 𝑑𝑗 {𝑢󵄩󵄩󵄩󵄩𝑊𝑗(𝑛) 󵄩󵄩󵄩󵄩 + ℎ𝛽 󵄩󵄩󵄩󵄩𝑊𝑗(𝑛−𝑚) 󵄩󵄩󵄩󵄩
𝑛+2 󵄩
Because 𝜇𝑛 = (𝑤𝑛 Τ , 𝑤𝑛−1 Τ )Τ , we can get from (33) that
󵄩 󵄩
lim 󵄩󵄩𝑤 󵄩󵄩 = 0.
(34)
𝑛 → ∞ 󵄩 𝑛󵄩
4.1. Some Concepts. In order to study the convergence of the
method, we define the following:
𝑗=1
≤𝑘
The inequality together with the knowledge 0 < 𝑘 < 1 leads
to the following:
󵄩
󵄩
lim 󵄩󵄩𝜇 󵄩󵄩 = 0.
(33)
𝑛 → ∞ 󵄩 𝑛+1 󵄩
(32)
(36a)
(36b)
Definition 5. Method (7a)–(7c) with an approximation procedure (9) is said to be 𝐷 convergent of order 𝑝 if the global
error satisfies a bound of the form
󵄩
󵄩󵄩
󵄩󵄩𝐻 (𝑡𝑛 ) − 𝜁(𝑛) 󵄩󵄩󵄩
󵄩
󵄩
󵄩
󵄩
󵄩
󵄩
≤ 𝜌1 (𝑡𝑛 ) (󵄩󵄩󵄩󵄩𝐻 (𝑡0 ) − 𝜁(0) 󵄩󵄩󵄩󵄩 + max 󵄩󵄩󵄩󵄩𝑌 (𝑡𝑘 ) − 𝑌(𝑘) 󵄩󵄩󵄩󵄩 + ℎ𝑝 ) ,
𝑘≤0
𝑛 ≥ 1, ℎ ∈ (0, ℎ0 ] ,
(37)
Abstract and Applied Analysis
7
where 𝐻(𝑡) and 𝑌(𝑡) are defined by
𝐻 (𝑡) = (𝑦 (𝑡 + ℎ) , 𝑦 (𝑡)) ∈ 𝐶2𝑁,
𝑌 (𝑡) = (𝑦 (𝑡 + 𝑐1 ℎ) , . . . , 𝑦 (𝑡 + 𝑐𝑠 ℎ) , 𝑦 (𝑡 − ℎ + 𝑐1 ℎ) , . . . ,
𝑦 (𝑡 − ℎ + 𝑐𝑠 ℎ)) ∈ 𝐶2𝑠𝑁,
(38)
𝐿 𝑖 are dependent on the method, the bounds 𝑀𝑖 , 𝑁𝑖 , the
parameters 𝛼, 𝛽, 𝛾, 𝜎, and 𝜏.
First, we give a preliminary result which will later be used
̃ ∈ 𝐶2𝑠𝑁
̃ 𝑍, 𝑍, 𝑍
several times. To simplify, we denote 𝑌, 𝑌, 𝑌,
(𝑛)
(𝑛)
(𝑛)
(𝑛)
̃ , and 𝜁, 𝜛 ∈ 𝐶2𝑁 for 𝜁(𝑛+1) ,
̃ , 𝑍(𝑛) , 𝑍 𝑍
for 𝑌(𝑛) , 𝑌 , 𝑌
(𝑛+1)
(𝑛+1)
(𝑛+1) (𝑛)
̃ and 𝛿̃ by the
, where 𝜛
= (𝑧
, 𝑧 ). Define Δ
𝜛
following:
where 𝜌1 (𝑡) and ℎ0 depend on 𝑀𝑖 , 𝛼, 𝛽, 𝛾, 𝜎, and 𝜏.
̃ , (42a)
̃ = 𝑌 − 𝑍 − ℎ𝐶11 (𝐹 (𝑡, 𝑌, 𝑌, 𝑌)
̃ − 𝐹 (𝑡, 𝑍, 𝑍, 𝑍))
Δ
Definition 6. TSRK Method (7a)–(7c) is said to be diagonally
stable if there exist a 2𝑠 × 2𝑠 diagonal matrix 𝑄 > 0 such that
Τ
the matrix 𝑄𝐶11 + 𝐶11
𝑄 is positive definite.
̃ .
̃ − 𝐹 (𝑡, 𝑍, 𝑍, 𝑍))
𝛿̃ = 𝜁 − 𝜛 − ℎ𝐶21 (𝐹 (𝑡, 𝑌, 𝑌, 𝑌)
Remark 7. The concepts of algebraic stability and diagonal
stability of TSRK method are the generalizations of corresponding concepts of Runge-Kutta methods. Although it
is difficult to examine these conditions, many results have
been found, especially, there exist algebraically stable and
diagonally stable multistep formulas of arbitrarily high order
(cf. [24]).
Definition 8. TSRK Method (7a)–(7c) is said to have generalized stage order 𝑝 if 𝑝 is the largest integer which possesses
the following properties.
(42b)
Theorem 9. Suppose method (7a)–(7c) is diagonally stable,
then there exist constants ℎ1 , 𝑝2 , and 𝑝3 such that
󵄩
󵄩 ̃ ̃ 󵄩󵄩
󵄩 ̃ 󵄩󵄩
󵄩
󵄩
󵄩󵄩 + ℎ 󵄩󵄩󵄩Y − Z󵄩󵄩󵄩 + ℎ 󵄩󵄩󵄩Y
‖Y − Z‖ ≤ p2 (󵄩󵄩󵄩󵄩Δ
󵄩
󵄩
󵄩 − Z󵄩󵄩) ,
󵄩
ℎ ∈ (0, ℎ1 ] ,
(43a)
󵄩
󵄩 ̃ ̃ 󵄩󵄩
󵄩 ̃ 󵄩󵄩 󵄩󵄩̃󵄩󵄩
󵄩
󵄩
󵄩󵄩
󵄩
󵄩󵄩 + 󵄩󵄩𝛿󵄩󵄩 + ℎ 󵄩󵄩󵄩Y − Z󵄩󵄩󵄩 + ℎ 󵄩󵄩󵄩Y
󵄩󵄩𝜁 − 𝜛󵄩󵄩󵄩 ≤ p3 (󵄩󵄩󵄩󵄩Δ
󵄩
󵄩 󵄩 󵄩
󵄩 − Z󵄩󵄩) ,
󵄩
ℎ ∈ (0, ℎ1 ] .
(43b)
For any given problem (1) and ℎ ∈ [0, ℎ0 ], there exists an
abstract function 𝐻ℎ (𝑡),
Proof. Since the method (7a)–(7c) is diagonally stable, there
exists a positive definite diagonal matrix 𝑄 such that the
Τ
𝑄 is positive definite. Therefore, the
matrix 𝐸 = 𝑄𝐶11 + 𝐶11
matrix 𝐶11 is obviously nonsingular and there exists a 𝑙 > 0
depends only on the method such that the matrix
𝐻ℎ (𝑡) = (𝐻1ℎ (𝑡) , 𝐻2ℎ (𝑡)) ∈ 𝐶2𝑁
−Τ
𝐸𝑙 = 𝐶11
𝐸𝐶11 − 2𝑙𝑄
(39)
such that
󵄩󵄩
󵄩
󵄩󵄩𝐻 (𝑡) = 𝐻ℎ (𝑡)󵄩󵄩󵄩 ≤ 𝑝1 ℎ𝑝 ,
󵄩
󵄩
󵄩󵄩 ℎ 󵄩󵄩
󵄩󵄩 ℎ 󵄩󵄩
𝑝+1
󵄩󵄩Δ (𝑡)󵄩󵄩 ≤ 𝑝1 ℎ ,
󵄩󵄩𝛿 (𝑡)󵄩󵄩 ≤ 𝑝1 ℎ𝑝+1 ,
󵄩
󵄩
󵄩
󵄩
is also positive definite.
Define
(40)
𝑊 = 𝑌 − 𝑍,
ℎ
ℎ
𝑌 (𝑡) = ℎ𝐶11 𝑌 (𝑡) + 𝐶12 𝐻 (𝑡) + Δ (𝑡) ,
(41a)
𝐻ℎ (𝑡 + ℎ) = ℎ𝐶21 𝑌󸀠 (𝑡) + 𝐶22 𝐻ℎ (𝑡) + 𝛿ℎ (𝑡) .
(41b)
The function 𝑌 (𝑡) is defined by the following:
󸀠
󸀠
𝑌 (𝑡) = (𝑦 (𝑡 + 𝑐1 ℎ) , . . . , 𝑦 (𝑡 + 𝑐𝑠 ℎ) , 𝑦 (𝑡 − ℎ + 𝑐1 ℎ) , . . . ,
󸀠
2𝑠𝑁
𝑦 (𝑡 − ℎ + 𝑐𝑠 ℎ)) ∈ 𝐶
̃
̃=𝑌
̃ − 𝑍,
𝑊
̃ − 𝐹 (𝑡, 𝑍, 𝑍, 𝑌))
̃ ,
𝐾2 = ℎ (𝐹 (𝑡, 𝑍, 𝑌, 𝑌)
̃ ;
̃ − 𝐹 (𝑡, 𝑍, 𝑍, 𝑍))
𝐾3 = ℎ (𝐹 (𝑡, 𝑍, 𝑍, 𝑌)
(45a)
(45b)
(45c)
(45d)
then
󸀠
󸀠
𝑊 = 𝑌 − 𝑍,
̃ − 𝐹 (𝑡, 𝑍, 𝑌, 𝑌))
̃ ,
𝐾1 = ℎ (𝐹 (𝑡, 𝑌, 𝑌, 𝑌)
where the maximum stepsize ℎ0 > 0 and the constant 𝑝1
depend only on the method and the bounds 𝑀𝑖 , Δℎ (𝑡) and
𝛿ℎ (𝑡), they are defined by the following equations:
󸀠
(44)
.
(41c)
ℎ
Particularly, when 𝐻(𝑡) = 𝐻 (𝑡), generalized stage order is
called stage order.
4.2. 𝐷-Convergence and Proofs. In this section, we focus on
the error analysis of TSRK method for (1). For the sake of
simplicity, we always assume that all constants ℎ𝑖 , 𝛾𝑖 , 𝑑𝑖 , and
̃ = 𝑊 − 𝐶11 (𝐾1 + 𝐾2 + 𝐾3 ) ,
Δ
(46a)
𝛿̃ = 𝜁 − 𝜛 − 𝐶21 (𝐾1 + 𝐾2 + 𝐾3 ) .
(46b)
Using (2)–(4), (44), (46a), and (46b), we have for ℎ𝛼 ≤ 𝑙,
0 ≤ ⟨𝑊, 2𝑙𝑄𝑊⟩ + 2Re⟨𝐾1 , −𝑄𝑊⟩
−1
= − ⟨𝑊, 𝐸𝑙 𝑊⟩ + 2Re ⟨𝑊, 𝑄 (𝐶11
𝑊 − 𝐾1 )⟩
−1 ̃
Δ + 𝐾2 + 𝐾3 )⟩
≤ − 𝜆 1 ‖𝑊‖2 + 2Re ⟨𝑊, 𝑄 (𝐶11
󵄩󵄩 ̃ 󵄩󵄩
󵄩 −1 󵄩󵄩
󵄩󵄩
≤ − 𝜆 𝑙 ‖𝑊‖2 + 2 󵄩󵄩󵄩󵄩𝑄𝐶11
󵄩󵄩󵄩 ‖𝑊‖ ⋅ 󵄩󵄩󵄩Δ
󵄩
󵄩󵄩 󵄩󵄩
󵄩 ̃ 󵄩󵄩
󵄩󵄩 ,
+ 2ℎ𝛽 ‖𝑄‖ ⋅ ‖𝑊‖ ⋅ 󵄩󵄩󵄩𝑊󵄩󵄩󵄩 + 2ℎ𝜎 ‖𝑄‖ ⋅ ‖𝑊‖ ⋅ 󵄩󵄩󵄩󵄩𝑊
󵄩
(47)
8
Abstract and Applied Analysis
where 𝜆 l is the minimum eigenvalue of 𝐸𝑙 . Therefore,
󵄩
󵄩 ̃ ̃󵄩󵄩
󵄩 ̃ 󵄩󵄩
󵄩
󵄩
󵄩󵄩 + ℎ 󵄩󵄩󵄩𝑌 − 𝑍󵄩󵄩󵄩 + ℎ 󵄩󵄩󵄩𝑌
‖𝑌 − 𝑍‖ ≤ 𝑝2 (󵄩󵄩󵄩󵄩Δ
󵄩
󵄩
󵄩 − 𝑍󵄩󵄩) ,
󵄩
ℎ ∈ (0, ℎ1 ] ,
(48)
where 𝑝4 = ‖𝐷‖ ⋅ max{(2𝛼 + 𝛽 + 𝜎), 𝛽, 𝜎}, ‖ ⋅ ‖ is a norm on
𝐶2𝑠𝑁 defined by the following:
1/2
‖𝑈‖𝐺 = ⟨𝑈, 𝐺𝑈⟩
= ( ∑ 𝑔𝑖𝑗 ⟨𝑢𝑖 , 𝑢𝑗 ⟩)
2
󵄩 −1 󵄩󵄩
󵄩󵄩 , 𝛽 ‖𝑄‖ , 𝜎 ‖𝑄‖} ,
max {󵄩󵄩󵄩󵄩𝑄𝐶11
󵄩
𝜆1
(49a)
𝛼 ≤ 0,
{1
ℎ1 = {
𝑙
min (1, ) 𝛼 > 0.
𝛼
{
(49b)
𝑈 = (𝑢1 , 𝑢2 , . . . , 𝑢2𝑠 ) ∈ 𝐶2𝑠𝑁,
(𝑛)
(𝑛)
(𝑛)
With algebraic stability, the matrix
(50)
Τ
Τ
Τ
𝐺𝐶22
𝐶12
𝐷 − 𝐶22
𝐺𝐶21
𝐺 − 𝐶22
] (56)
Τ
Τ
Τ
𝐷𝐶12 − 𝐶21 𝐺𝐶22 𝐷𝐶11 + 𝐶11 𝐷 − 𝐶21
𝐺𝐶21
⟨𝑢(𝑛+1) , 𝐺𝑢(𝑛+1) ⟩ − ⟨𝑢(𝑛) , 𝐺𝑢(𝑛) ⟩ − 2Re ⟨𝑊(𝑛) , 𝐷𝐾(𝑛) ⟩
= − ⟨⟨𝑢(𝑛) , 𝐾(𝑛) ⟩, 𝑀 ⟨𝑢(𝑛) , 𝐾(𝑛) ⟩⟩ ≤ 0.
(57)
Using (2)–(4), we further obtain the following:
(51a)
(51b)
⟨𝑢(𝑛+1) , 𝐺𝑢(𝑛+1) ⟩
≤ ⟨𝑢(𝑛) , 𝐺𝑢(𝑛) ⟩ + 2Re ⟨𝑊(𝑛) , 𝐷𝐾(𝑛) ⟩
≤ ⟨𝑢(𝑛) , 𝐺𝑢(𝑛) ⟩ + 2Re ⟨𝑊(𝑛) , 𝐷𝐾1(𝑛) ⟩
+ 2Re ⟨𝑊(𝑛) , 𝐷𝐾2(𝑛) + 𝐷𝐾3(𝑛) ⟩
𝑍(𝑛) = (𝑍1(𝑛) , . . . , 𝑍𝑠(𝑛) , 𝑍1(𝑛−1) , . . . , 𝑍𝑠(𝑛−1) ) ∈ 𝐶2𝑠𝑁,
(𝑛)
𝑀=[
is nonnegative definite. As in [25], we have
where
𝑍
(𝑛)
(55)
Consider the compact form of (11a)–(11c):
̃ ) + 𝐶22 𝜛(𝑛) ,
𝜛(𝑛+1) = ℎ𝐶21 𝐹 (𝑡𝑛 , 𝑍(𝑛) , 𝑍 , 𝑍
(𝑛)
𝑢(𝑛+1) = 𝐶21 𝐾(𝑛) + 𝐶22 𝑢(𝑛) .
−1
where 𝑝3 = max{1, (1 + 𝑝2 )‖𝐶21 𝐶11
‖}, which completes the
proof of Theorem 9.
(𝑛)
(𝑛)
𝑊(𝑛) = 𝐶11 𝐾(𝑛) + 𝐶12 𝑢(𝑛) ,
ℎ ∈ (0, ℎ1 ] ,
(𝑛)
𝑢𝑖 ∈ 𝐶𝑁.
̃ )) ,
̃ ) − 𝐹 (𝑡𝑛 , 𝑍(𝑛) , 𝑍 , 𝑍
𝐾(𝑛) = ℎ (𝐹 (𝑡𝑛 , 𝑌(𝑛) , 𝑌 , 𝑌
󵄩
󵄩󵄩
󵄩󵄩𝜁 − 𝜛󵄩󵄩󵄩 =
̃ ) + 𝐶12 𝜛(𝑛) ,
𝑍(𝑛) = ℎ𝐶11 𝐹 (𝑡𝑛 , 𝑍(𝑛) , 𝑍 , 𝑍
(54)
Proof. Define 𝑢(𝑛) = 𝜁(𝑛) − 𝜛(𝑛) , we get from (45a)–(45d) that
From (43a), (49a), and (49b), it follows that
󵄩󵄩̃
̃ 󵄩󵄩󵄩󵄩
󵄩󵄩𝛿 + 𝐶21 𝐶11 −1 (𝑊 − Δ)
󵄩
󵄩
󵄩󵄩 󵄩󵄩 󵄩󵄩 ̃ 󵄩󵄩
󵄩󵄩̃󵄩󵄩
󵄩󵄩
−1 󵄩
󵄩
≤ 󵄩󵄩󵄩𝛿󵄩󵄩󵄩 + 𝑝1 ℎ 󵄩󵄩󵄩𝐶21 𝐶11 󵄩󵄩󵄩 (󵄩󵄩󵄩𝑊󵄩󵄩󵄩 + 󵄩󵄩󵄩𝑊󵄩󵄩󵄩)
󵄩󵄩
󵄩
−1 󵄩
󵄩󵄩 ⋅ 󵄩󵄩󵄩Δ
+ (1 + 𝑝1 ) 󵄩󵄩󵄩󵄩𝐶21 𝐶11
󵄩󵄩 󵄩󵄩 ̃ 󵄩󵄩󵄩
󵄩
󵄩 ̃ ̃󵄩󵄩
󵄩 ̃ 󵄩󵄩 󵄩󵄩̃󵄩󵄩
󵄩
󵄩
󵄩󵄩 + 󵄩󵄩𝛿󵄩󵄩 + ℎ 󵄩󵄩󵄩𝑌 − 𝑍󵄩󵄩󵄩 + ℎ 󵄩󵄩󵄩𝑌
≤ 𝑝3 (󵄩󵄩󵄩󵄩Δ
󵄩
󵄩 󵄩 󵄩
󵄩 − 𝑍󵄩󵄩) ,
󵄩
,
𝑖,𝑗=1
where
𝑝2 =
1/2
2𝑠
(𝑛)
(𝑛)
(𝑛−1)
= (𝑍1 , . . . , 𝑍𝑠 , 𝑍1
(𝑛−1)
, . . . , 𝑍𝑠
≤ ⟨𝑢(𝑛) , 𝐺𝑢(𝑛) ⟩ + 2ℎ𝛼 ⟨𝑊(𝑛) , 𝐷𝑊(𝑛) ⟩
) ∈ 𝐶2𝑠𝑁,
̃(𝑛) , . . . , 𝑍
̃(𝑛) , 𝑍
̃(𝑛−1) , . . . , 𝑍
̃(𝑛−1) ) ∈ 𝐶2𝑠𝑁,
̃(𝑛) = (𝑍
𝑍
1
𝑠
1
𝑠
(52)
𝜛(𝑛+1) = (𝑧(𝑛+1) , 𝑧(𝑛) ) ∈ 𝐶2𝑁.
Theorem 10. Suppose the method (7a)–(7c) is algebraically
stable for the matrices 𝐺 and 𝐷, then for (36a), (36b), (51a),
and (51b), one has the following:
󵄩2
󵄩󵄩 (𝑛+1)
󵄩󵄩𝜁
− 𝜛(𝑛+1) 󵄩󵄩󵄩󵄩𝐺
󵄩
󵄩2
󵄩
≤ 󵄩󵄩󵄩󵄩𝜁(𝑛) − 𝜛(𝑛) 󵄩󵄩󵄩󵄩𝐺
󵄩2 󵄩
󵄩2
󵄩
+ 𝑝4 ℎ (󵄩󵄩󵄩󵄩𝑌(𝑛) − 𝑍(𝑛) 󵄩󵄩󵄩󵄩 + 󵄩󵄩󵄩󵄩𝑌(𝑛−𝑚) − 𝑍(𝑛−𝑚) 󵄩󵄩󵄩󵄩
󵄩󵄩 (𝑛)
󵄩2
̃(𝑛) 󵄩󵄩󵄩󵄩 ) ,
̃ −𝑍
+󵄩󵄩󵄩󵄩𝑌
󵄩󵄩
󵄩
(𝑛) 󵄩
󵄩
󵄩 󵄩󵄩
󵄩
+ 2ℎ𝛽 󵄩󵄩󵄩󵄩𝐷1/2 𝑊(𝑛) 󵄩󵄩󵄩󵄩 ⋅ 󵄩󵄩󵄩𝐷1/2 𝑊 󵄩󵄩󵄩
󵄩
󵄩
󵄩󵄩 1/2 (𝑛) 󵄩󵄩 󵄩󵄩󵄩 1/2 ̃ (𝑛) 󵄩󵄩󵄩
+ 2ℎ𝜎 󵄩󵄩󵄩𝐷 𝑊 󵄩󵄩󵄩 ⋅ 󵄩󵄩𝐷 𝑊 󵄩󵄩
󵄩
󵄩
≤ ⟨𝑢(𝑛) , 𝐺𝑢(𝑛) ⟩ + 2ℎ𝛼 ⟨𝑊(𝑛) , 𝐷𝑊(𝑛) ⟩
󵄩󵄩 (𝑛) 󵄩󵄩2
󵄩2
󵄩
+ ℎ𝛽 (‖𝐷‖ ⋅ 󵄩󵄩󵄩󵄩𝑊(𝑛) 󵄩󵄩󵄩󵄩 + ‖𝐷‖ ⋅ 󵄩󵄩󵄩𝑊 󵄩󵄩󵄩 )
󵄩
󵄩
󵄩󵄩 (𝑛) 󵄩󵄩2
󵄩2
󵄩
̃ 󵄩󵄩 )
+ ℎ𝜎 (‖𝐷‖ ⋅ 󵄩󵄩󵄩󵄩𝑊(𝑛) 󵄩󵄩󵄩󵄩 + ‖𝐷‖ ⋅ 󵄩󵄩󵄩𝑊
󵄩󵄩
󵄩
(53)
󵄩2
󵄩
≤ ⟨𝑢(𝑛) , 𝐺𝑢(𝑛) ⟩ + (2ℎ𝛼 + ℎ𝛽 + ℎ𝜎) ‖𝐷‖ ⋅ 󵄩󵄩󵄩󵄩𝑊(𝑛) 󵄩󵄩󵄩󵄩
󵄩󵄩 (𝑛) 󵄩󵄩2
󵄩2
󵄩
̃ 󵄩󵄩 ) ,
+ ‖𝐷‖ (ℎ𝛽󵄩󵄩󵄩󵄩𝑊(𝑛−𝑚) 󵄩󵄩󵄩󵄩 + ℎ𝜎󵄩󵄩󵄩𝑊
󵄩󵄩
󵄩
which gives (53). The proof is completed.
(58)
Abstract and Applied Analysis
9
In the following, we assume that the method (7a)–(7c) has
generalized stage order 𝑝; that is, there exists a function 𝐻ℎ (𝑡)
̂ (𝑛) and 𝑦
̂ (𝑛+1) by
such that (40) holds. For any > 0, we define 𝑌
the following:
Proof. A combination of (36a) and (41a) leads to the following:
𝑌(𝑘) − 𝑌 (𝑡𝑘 )
(𝑛)
̂̃ ) + 𝐶 𝐻ℎ (𝑡 ) ,
̂ (𝑛) = ℎ𝐶11 𝐹 (𝑡𝑛 , 𝑌
̂ (𝑛) , 𝑌 (𝑡𝑛 − 𝜏) , 𝑌
𝑌
12
𝑛
(𝑘)
(62)
(59a)
̃ (𝑡𝑘 )) ]
−𝐹 (𝑡𝑘 , 𝑌 (𝑡𝑘 ) , 𝑌 (𝑡𝑘 ) , 𝑌
(𝑛)
̂̃ ) + 𝐶 𝐻ℎ (𝑡 ) ,
̂ (𝑛) , 𝑌 (𝑡𝑛 − 𝜏) , 𝑌
̂ (𝑛+1) = ℎ𝐶21 𝐹 (𝑡𝑛 , 𝑌
𝑦
22
𝑛
+ 𝐶12 (𝜁(𝑘) − 𝐻ℎ (𝑡𝑘 )) − Δℎ (𝑡) .
(59b)
where
(𝑘)
̃ )
= ℎ𝐶11 [𝐹 (𝑡𝑘 , 𝑌(𝑘) , 𝑌 , 𝑌
It follows from Theorem 9 that
̂̃
𝑌
(𝑛)
= (∫
𝑡𝑛 +𝑐1 ℎ
𝑔 (𝑡, 𝜉, 𝐻1ℎ (𝜉) 𝑑𝜉 , . . . ,
𝑡𝑛 −𝜏+𝑐1 ℎ
∫
𝑡𝑛 +𝑐𝑠 ℎ
𝑡𝑛 −𝜏+𝑐𝑠 ℎ
∫
𝑔 (𝑡, 𝜉, 𝐻1ℎ (𝜉) 𝑑𝜉 , . . . ,
󵄩󵄩 (𝑘)
󵄩
󵄩
󵄩
̃ −𝑌
̃ (𝑡𝑘 )󵄩󵄩󵄩
≤ 𝑝2 (ℎ 󵄩󵄩󵄩󵄩𝑌(𝑘−𝑚) − 𝑌 (𝑡𝑘 − 𝜏)󵄩󵄩󵄩󵄩 + ℎ 󵄩󵄩󵄩𝑌
󵄩󵄩
󵄩
𝑔 (𝑡, 𝜉, 𝐻2ℎ (𝜉) 𝑑𝜉 , . . . ,
󵄩
󵄩
+ 󵄩󵄩󵄩󵄩𝐶12 (𝜁(𝑘) − 𝐻ℎ (𝑡𝑘 )) − Δℎ (𝑡)󵄩󵄩󵄩󵄩 ) .
𝑡𝑛 +𝑐1 ℎ
𝑡𝑛 −𝜏+𝑐1 ℎ
∫
𝑡𝑛 +𝑐𝑠 ℎ
(60)
𝑡𝑛 +𝑐1 ℎ
𝑔 (𝑡, 𝜉, 𝐻1ℎ
𝑡𝑛 −𝜏+𝑐1 ℎ
∫
𝑡𝑛 +𝑐𝑠 ℎ
𝑡𝑛 −𝜏+𝑐𝑠 ℎ
∫
𝑡𝑛 −𝜏−ℎ+𝑐1 ℎ
∫
(𝜉) 𝑑𝜉 , . . . ,
𝑔 (𝑡, 𝜉, 𝐻1ℎ (𝜉) 𝑑𝜉 , . . . ,
𝑡𝑛 −ℎ+𝑐1 ℎ
𝑡𝑛 −ℎ+𝑐𝑠 ℎ
𝑡𝑛 −ℎ−𝜏+𝑐𝑠 ℎ
󵄩󵄩 (𝑘)
󵄩
󵄩󵄩𝑌
̃ −𝑌
̃ (𝑡𝑘 )󵄩󵄩󵄩
󵄩󵄩
󵄩󵄩
󵄩󵄩
𝑚
󵄩󵄩
̃ 𝑔 (𝑡𝑘 + 𝑐1 ℎ, 𝑡𝑘−𝑞 + 𝑐1 ℎ, 𝑌1(𝑘−𝑞) ) , . . . ,
≤ 󵄩󵄩󵄩󵄩(ℎ ∑ 𝑝
󵄩󵄩 𝑞=0 𝑞
󵄩
𝑚
̃ 𝑞 𝑔 (𝑡𝑘 + 𝑐𝑠 ℎ, 𝑡𝑘−𝑞 + 𝑐𝑠 ℎ, 𝑌𝑠(𝑘−𝑞) ) ,
ℎ∑𝑝
𝑔 (𝑡, 𝜉, 𝐻1ℎ (𝜉) 𝑑𝜉 , . . . ,
𝑞=0
𝑚
𝑔 (𝑡, 𝜉, 𝐻1ℎ (𝜉) 𝑑𝜉 ) .
),...,
𝑞=0
𝑚
̃ 𝑞 𝑔 (𝑡𝑘−1 + 𝑐𝑠 ℎ, 𝑡𝑘−𝑞−1 + 𝑐𝑠 ℎ, 𝑌𝑠(𝑘−𝑞−1) ))
ℎ ∑𝑝
𝑞=0
𝑛
󵄩󵄩 (𝑘)
󵄩󵄩2
∑ 󵄩󵄩󵄩𝑌 − 𝑌(𝑡𝑘 − 𝜏)󵄩󵄩󵄩
󵄩
󵄩
𝑘=1
− (∫
𝑡𝑛 +𝑐1 ℎ
𝑡𝑛 −𝜏+𝑐1 ℎ
𝑛
󵄩
󵄩2
= ∑ 󵄩󵄩󵄩󵄩𝑌(𝑘−𝑚) − 𝑌(𝑡𝑘 − 𝜏)󵄩󵄩󵄩󵄩
∫
𝑘=1
𝑛
󵄩
󵄩2
≤ 𝑝5 ( ∑ 󵄩󵄩󵄩󵄩𝜁(𝑘) − 𝐻ℎ (𝑡𝑘 )󵄩󵄩󵄩󵄩 + 𝐶1 ℎ(𝑝+1) + 𝐶2 ℎ2(]+1)
(61)
𝑘=1
𝑘=−𝑚+1
(𝑘−𝑞−1)
̃ 𝑞 𝑔 (𝑡𝑘−1 + 𝑐1 ℎ, 𝑡𝑘−𝑞−1 + 𝑐1 ℎ, 𝑌1
ℎ∑𝑝
Theorem 11. Suppose the method (7a)–(7c) is diagonally stable
and its generalized stage order is 𝑝, then there exist constants
𝑝5 and ℎ2 such that
0
󵄩
󵄩2
+𝐶3 ∑ 󵄩󵄩󵄩󵄩𝑌(𝑘) − 𝑌(𝑡𝑘 )󵄩󵄩󵄩󵄩 ) ,
(63)
A combination of (4) and (9) gives the following:
𝑔 (𝑡, 𝜉, 𝐻2ℎ (𝜉) 𝑑𝜉 )
𝑡𝑛 −𝜏+𝑐𝑠 ℎ
= (∫
󵄩󵄩 (𝑘)
󵄩
󵄩󵄩𝑌 − 𝑌 (𝑡𝑘 )󵄩󵄩󵄩
󵄩
󵄩
ℎ ∈ (0, ℎ2 ] ,
where 𝑝5 , 𝐶1 , 𝐶2 , and 𝐶3 depend on 𝑝2 , 𝑠, 𝑑1 , ℎ1 , and
‖𝐶12 ‖.
𝑡𝑛 +𝑐𝑠 ℎ
𝑡𝑛 −𝜏+𝑐𝑠 ℎ
∫
𝑡𝑛 +𝑐1 ℎ
𝑡𝑛 −𝜏+𝑐1 ℎ
(64)
𝑔 (𝑡, 𝜉, 𝑦 (𝜉) 𝑑𝜉 , . . . ,
𝑔 (𝑡, 𝜉, 𝑦 (𝜉) 𝑑𝜉 , . . . ,
𝑔 (𝑡, 𝜉, 𝑦 (𝜉 − ℎ) 𝑑𝜉 , . . . ,
󵄩󵄩
󵄩󵄩
𝑔 (𝑡, 𝜉, 𝑦 (𝜉 − ℎ) 𝑑𝜉 ) 󵄩󵄩󵄩󵄩
∫
󵄩󵄩
𝑡𝑛 −𝜏+𝑐𝑠 ℎ
󵄩
𝑡𝑛 +𝑐𝑠 ℎ
̃ ℎ] ,
≤ 2𝑠𝑑
1
10
Abstract and Applied Analysis
̃ and ] depend only on the compound quadrature
where 𝑑
1
(CQ) formula (9), which on substitution into (63) gives the
following:
𝑛
󵄩
󵄩2
∑ 󵄩󵄩󵄩󵄩𝑌(𝑘) − 𝑌(𝑡𝑘 )󵄩󵄩󵄩󵄩
Using Theorem 9, we have the following:
󵄩
󵄩󵄩 (𝑛)
󵄩󵄩𝑌 − 𝑌
̂ (𝑛) 󵄩󵄩󵄩
󵄩󵄩
󵄩󵄩
󵄩
󵄩 󵄩 󵄩
≤ 𝑝2 [ 󵄩󵄩󵄩𝐶12 󵄩󵄩󵄩 ⋅ 󵄩󵄩󵄩󵄩𝜁(𝑛) − 𝐻ℎ (𝑡𝑛 )󵄩󵄩󵄩󵄩
𝑘=1
𝑛
󵄩
󵄩2
≤ 3 (𝑝12 ℎ2 ∑ 󵄩󵄩󵄩󵄩𝑌(𝑘−𝑚) − 𝑌(𝑡𝑘 − 𝜏)󵄩󵄩󵄩󵄩
(67)
󵄩󵄩 (𝑛)
󵄩
󵄩
󵄩
󵄩̃
̂̃ (𝑛) 󵄩󵄩󵄩] ,
+ℎ 󵄩󵄩󵄩󵄩𝑌(𝑛−𝑚) − 𝑌 (𝑡𝑛−𝑚 )󵄩󵄩󵄩󵄩 + ℎ 󵄩󵄩󵄩𝑌
−𝑌
󵄩󵄩
󵄩󵄩
󵄩󵄩
𝑘=1
𝑛
󵄩
󵄩2
󵄩 󵄩2
+ 𝑝22 󵄩󵄩󵄩𝐶12 󵄩󵄩󵄩 ∑ 󵄩󵄩󵄩󵄩𝜁(𝑘) − 𝐻ℎ (𝑡𝑘 )󵄩󵄩󵄩󵄩
which on substitution into (66) gives the following:
𝑘=1
󵄩2
󵄩
+𝑝12 ⋅ 󵄩󵄩󵄩󵄩Δℎ (𝑡𝑘 )󵄩󵄩󵄩󵄩 +
󵄩󵄩 (𝑛+1)
󵄩(2)
󵄩󵄩𝜁
̂ (𝑛+1) 󵄩󵄩󵄩󵄩𝐺
−𝑦
󵄩
̃ 2 ℎ2]+2 ) ,
𝑝22 4𝑠2 𝑑
1
𝑛
󵄩
󵄩2
∑ 󵄩󵄩󵄩󵄩𝑌(𝑘−𝑚) − 𝑌(𝑡𝑘 − 𝜏)󵄩󵄩󵄩󵄩
≤ (1 +
(65)
𝑛
󵄩
󵄩2
= ∑ 󵄩󵄩󵄩󵄩𝑌(𝑘−𝑚) − 𝑌(𝑡𝑘 − 𝑚ℎ)󵄩󵄩󵄩󵄩
̃ 2 ℎ2] ,
+ 𝑝4 (1 + 3ℎ2 𝑝22 ) ℎ2𝑠𝑑
𝑘=1
𝑛−𝑚
󵄩
󵄩2
= ∑ 󵄩󵄩󵄩󵄩𝑌(𝑖) − 𝑌(𝑡𝑖 )󵄩󵄩󵄩󵄩
where 𝜆 2 is the minimum characteristic value of 𝐺. On the
other hand,
𝑖=1−𝑚
󵄩󵄩 (𝑛)
󵄩2
󵄩󵄩𝜁 − 𝐻ℎ (𝑡𝑛 )󵄩󵄩󵄩
󵄩
󵄩𝐺
𝑛
󵄩
󵄩2
≤ ∑ 󵄩󵄩󵄩󵄩𝑌(𝑘) − 𝑌(𝑡𝑘 )󵄩󵄩󵄩󵄩
𝑘=1
󵄩2 󵄩 (𝑛)
󵄩
󵄩2
̂ − 𝐻ℎ (𝑡𝑛 )󵄩󵄩󵄩󵄩𝐺
̂ (𝑛) 󵄩󵄩󵄩󵄩𝐺 + 󵄩󵄩󵄩󵄩𝑦
≤ 󵄩󵄩󵄩󵄩𝜁(𝑛) − 𝑦
0
󵄩
󵄩2
+ ∑ 󵄩󵄩󵄩󵄩𝑌(𝑘) − 𝑌(𝑡𝑘 )󵄩󵄩󵄩󵄩 .
󵄩 󵄩 (𝑛)
󵄩
󵄩
̂ − 𝐻ℎ (𝑡𝑛 )󵄩󵄩󵄩󵄩𝐺
̂ (𝑛) 󵄩󵄩󵄩󵄩𝐺 ⋅ 󵄩󵄩󵄩󵄩𝑦
+ 2󵄩󵄩󵄩󵄩𝜁(𝑛) − 𝑦
𝑘=1−𝑚
Therefore, there exist 𝑝5 and ℎ2 such that (61) holds. The proof
of Theorem 11 is completed.
Theorem 12. Suppose method (7a)–(7c) is algebraically stable
and diagonally stable and its generalized stage order is 𝑝. Then
the method with quadrature formula (9) is 𝐷-convergent of
order at least min{𝑝, ]}.
Proof. A combination of (36b), (59b), and (53) leads to the
following:
󵄩2
󵄩󵄩 (𝑛+1)
󵄩󵄩𝜁
̂ (𝑛+1) 󵄩󵄩󵄩󵄩𝐺
−𝑦
󵄩
(69)
󵄩2
󵄩
̂ (𝑛) 󵄩󵄩󵄩󵄩𝐺
≤ (1 + ℎ) 󵄩󵄩󵄩󵄩𝜁(𝑛) − 𝑦
1 󵄩 (𝑛)
󵄩2
̂ − 𝐻ℎ (𝑡𝑛 )󵄩󵄩󵄩󵄩𝐺.
+ (1 + ) 󵄩󵄩󵄩󵄩𝑦
ℎ
In view of (41a)–(41c) and (59a) and (59b), the application of
Theorem 9 leads to the following:
󵄩󵄩
󵄩󵄩 ℎ
󵄩󵄩 󵄩󵄩 ℎ
󵄩󵄩
󵄩󵄩 (𝑛)
ℎ
󵄩
󵄩
󵄩 󵄩
󵄩
󵄩󵄩𝑦
󵄩 ̂ − 𝐻 (𝑡𝑛 )󵄩󵄩 ≤ 𝑝3 (󵄩󵄩Δ (𝑡𝑛 )󵄩󵄩 + 󵄩󵄩𝛿 (𝑡𝑛 )󵄩󵄩) ,
(70)
ℎ ∈ (0, ℎ0 ]
󵄩2
󵄩
̂ (𝑛) 󵄩󵄩󵄩󵄩𝐺 + 𝑝4 ℎ
≤ 󵄩󵄩󵄩󵄩𝜁(𝑛) − 𝑦
󵄩󵄩 (𝑛)
󵄩2
󵄩̃
̂̃ (𝑛) 󵄩󵄩󵄩 ] .
+󵄩󵄩󵄩𝑌
−𝑌
󵄩󵄩
󵄩󵄩
󵄩󵄩
(68)
󵄩
󵄩2
+ 𝑝4 (1 + 3ℎ2 𝑝22 ) ℎ󵄩󵄩󵄩󵄩𝑌(𝑛−𝑚) − 𝑌(𝑡𝑛−𝑚 )󵄩󵄩󵄩󵄩
𝑘=1
󵄩2
󵄩󵄩
2
̂ (𝑛) 󵄩󵄩󵄩 + 󵄩󵄩󵄩󵄩𝑌(𝑛−𝑚) − 𝑌(𝑡𝑛−𝑚 )󵄩󵄩󵄩󵄩
× [󵄩󵄩󵄩𝑌(𝑛) − 𝑌
󵄩󵄩 󵄩
󵄩
󵄩
󵄩 󵄩2
3ℎ𝑝4 𝑝22 󵄩󵄩󵄩𝐶12 󵄩󵄩󵄩
󵄩
󵄩2
) ⋅ 󵄩󵄩󵄩󵄩𝜁(𝑛) − 𝐻ℎ (𝑡𝑛 )󵄩󵄩󵄩󵄩𝐺
𝜆2
(66)
which gives
󵄩󵄩 (𝑛)
󵄩󵄩2
󵄩󵄩 󵄩󵄩 ℎ
󵄩󵄩 2
ℎ
2 󵄩
󵄩󵄩 ℎ
󵄩󵄩𝑦
󵄩
󵄩 󵄩
󵄩
󵄩 ̂ − 𝐻 (𝑡𝑛 )󵄩󵄩𝐺 ≤ 𝜆 1 𝑝3 (󵄩󵄩Δ (𝑡𝑛 )󵄩󵄩 + 󵄩󵄩𝛿 (𝑡𝑛 )󵄩󵄩) ,
ℎ ∈ (0, ℎ0 ] ,
(71)
Abstract and Applied Analysis
11
where 𝜆 1 denotes the maximum eigenvalue of the matrix 𝐺.
A combination of (40) and (66)–(71) leads to the following:
3
2.5
󵄩2
󵄩󵄩 (𝑛)
󵄩󵄩𝜁 − 𝐻ℎ (𝑡𝑛 )󵄩󵄩󵄩
󵄩𝐺
󵄩
2
󵄩
󵄩2
≤ (1 + ℎ) [𝛾1 󵄩󵄩󵄩󵄩𝜁(𝑛−1) − 𝐻ℎ (𝑡𝑛−1 )󵄩󵄩󵄩󵄩𝐺
󵄩
󵄩2
+ 𝛾2 󵄩󵄩󵄩󵄩𝑌(𝑛−𝑚−1) − 𝑌 (𝑡𝑛=𝑚−1 )󵄩󵄩󵄩󵄩
2]+1
+𝛾3 ℎ
1.5
(72)
1
]
1
+ 4 (1 + ) 𝜆 1 𝑝32 𝑝12 ℎ2𝑝+2 ,
ℎ
0.5
ℎ ∈ (0, ℎ3 ] ,
0
where
−1
−0.5
0
0.5
1
1.5
2
2.5
3
3.5
Figure 1: The numerical solution of (79a) and (79b) for (77) with
ℎ = 0.1.
ℎ3 = min {ℎ0 , ℎ1 , ℎ2 } ≤ 1,
󵄩 󵄩2
3ℎ𝑝4 𝑝22 󵄩󵄩󵄩𝐶12 󵄩󵄩󵄩
𝛾1 = 1 +
,
𝜆2
𝛾2 = 𝑝4 ℎ (1 +
(73)
3𝑝22 ) ℎ2 ,
Table 1: Convergence order of two-step Runge-Kutta methods for
system (77).
𝑚
Convergence order
8
4.0022
16
3.9982
32
3.9964
64
3.9960
̃ 2 𝑝 (1 + 3ℎ2 𝑝2 ) .
𝛾3 = 2𝑠𝑑
4
2
Considering Theorem 11, we further obtain the following:
𝑛
× ∑ (1 + ℎ)𝑖−1 𝛾1 𝑖−1 }
󵄩2
󵄩󵄩 (𝑛)
󵄩󵄩𝜁 − 𝐻ℎ (𝑡𝑛 )󵄩󵄩󵄩
󵄩𝐺
󵄩
󵄩
󵄩2
≤ (1 + ℎ)𝑛 𝛾1𝑛 󵄩󵄩󵄩󵄩𝜁(0) − 𝐻ℎ (𝑡0 )󵄩󵄩󵄩󵄩𝐺 + (1 + ℎ) 𝛾2
𝑛
󵄩
󵄩2
× ∑ (1 + ℎ)𝑖−1 𝛾1𝑖−1 󵄩󵄩󵄩󵄩𝑌(𝑛−𝑚−𝑖) − 𝑌(𝑡𝑛=𝑚−𝑖 )󵄩󵄩󵄩󵄩𝐺
𝑖=1
1
+ [(1 + ℎ) 𝛾3 ℎ2]+1 + 4 (1 + ) 𝜆 1 𝑝32 𝑝12 ℎ2𝑝+2 ]
ℎ
𝑖=1
𝑛
× exp [(1 + ℎ) 𝛾2 ∑ (1 + ℎ)𝑖−1 𝛾1𝑖−1 ] ,
𝑖=1
(75)
(74)
1
+ [(1 + ℎ) 𝛾3 ℎ2]+1 + 4 (1 + ) 𝜆 1 𝑝32 𝑝12 ℎ2𝑝+2 ]
ℎ
𝑛
× ∑ (1 + ℎ)𝑖−1 𝛾1 𝑖−1 .
𝑖=1
Using discrete Bellman inequality, we have the following:
where ℎ ∈ (0, ℎ0 ], and ℎ0 = min{ℎ0 , ℎ1 , ℎ2 , ℎ3 } ≤ 1.
Considering ‖𝐻(𝑡) − 𝐻ℎ (𝑡𝑛 )‖ ≤ 𝑝0 ℎ𝑝 , we obtain the
following:
󵄩󵄩 (𝑛)
󵄩
󵄩󵄩𝜁 − 𝐻 (𝑡𝑛 )󵄩󵄩󵄩 ≤
󵄩
󵄩
󵄩󵄩 (𝑛)
󵄩
󵄩󵄩𝜁 − 𝐻ℎ (𝑡𝑛 )󵄩󵄩󵄩
󵄩
󵄩
󵄩
󵄩
+ 󵄩󵄩󵄩󵄩𝐻ℎ (𝑡𝑛 ) − 𝐻 (𝑡𝑛 )󵄩󵄩󵄩󵄩
󵄩2
󵄩󵄩 (𝑛)
󵄩󵄩𝜁 − 𝐻ℎ (𝑡𝑛 )󵄩󵄩󵄩
󵄩𝐺
󵄩
󵄩
󵄩2
≤ {(1 + ℎ)𝑛 𝛾1𝑛 󵄩󵄩󵄩󵄩𝜁(0) − 𝐻ℎ (𝑡0 )󵄩󵄩󵄩󵄩𝐺 + (1 + ℎ) 𝛾2
𝑛
0
𝑖=1
𝑗=1−𝑚
󵄩
󵄩2
× ∑ (1 + ℎ)𝑖−1 𝛾1𝑖−1 ∑ 󵄩󵄩󵄩󵄩𝑌(𝑗) − 𝑌(𝑡𝑗 )󵄩󵄩󵄩󵄩𝐺
(76)
󵄩
󵄩
≤ 󵄩󵄩󵄩󵄩𝜁(𝑛) − 𝐻ℎ (𝑡𝑛 )󵄩󵄩󵄩󵄩 + 𝑝0 ℎ𝑝 .
Considering (75) and (76), we can easily conclude that
method (7a)–(7c) with quadrature formula (9) is 𝐷convergent of order at least > min{𝑝, ]}. The proof is completed.
12
Abstract and Applied Analysis
𝑦󸀠 (𝑡) = − (6 + sin 𝑡) 𝑦 (𝑡) + 𝑦 (𝑡 −
3
𝑡
2.5
−∫
𝑡−𝜋/4
1.5
−
𝜋
≤𝑡≤0
4
𝑡 ≥ 0,
(77)
and its perturbed problem
1
𝑧󸀠 (𝑡) = − (6 + sin 𝑡) 𝑧 (𝑡) + 𝑧 (𝑡 −
0.5
−∫
𝑡
𝑡−𝜋/4
0
−1
sin (V) 𝑦 (V) 𝑑V + 5 exp (cos 𝑡) ,
𝑦 (𝑡) = exp (cos 𝑡) ,
2
𝜋
)
4
−0.5
0
0.5
1
1.5
2
2.5
3
Figure 2: The numerical solution of (79a) and (79b) for (78) with
ℎ = 0.1.
5. Numerical Experiments
Consider the following nonlinear Volterra delay integro-differential equations:
𝑎11
𝜋
≤ 𝑡 ≤ 0.
4
𝑎22
0.87165188291653
−0.08663699023763
0.50361252124048
𝑏1
𝑏2
𝑏̃1
0.95532987568936
0.79063681672548
2√15 − 7
𝑐1
𝑐2
1.59379439197950
𝜃
1
0.44316674917114
𝜃
1
(𝑛)
𝑚/2
̃ (𝑛) = ℎ [sin (𝑡𝑛 + 𝑐𝑗 ℎ)𝑌(𝑛) + 4 ∑ sin (𝑡𝑛−2𝑞+1 + 𝑐𝑗 ℎ) 𝑌(𝑛−2𝑞+1)
𝑌
𝑗
𝑗
𝑗
3
𝑞=1
+ 2 ∑ sin (𝑡𝑛−2𝑞 +
𝑡 ≥ 0,
(78)
One may check that this system has the solution 𝑦(𝑡) =
exp(cos 𝑡) and 𝑎 = −4, 𝛽 = 1/2, 𝜎 = 1/2, 𝛾 = 1. With
Theorem 9, we conclude that system (77) satisfies stability
and convergence properties.
Apply the two-step Runge-Kutta method induced by the
GL method in [12]
𝑎21
̃ is computed by the compound Simpson
to (77) and (78), 𝑌
𝑗
rule with an even integer 𝑚 ≥ 4 formula:
𝑞=1
−
𝑎12
0.47790690818421
(𝑚−2)/2
sin (V) 𝑧 (V) 𝑑V + 5 exp (cos 𝑡) ,
𝑧 (𝑡) = 2,
3.5
𝜋
)
4
𝑏̃2
8 − 2√15
(79a)
Using the condition (28), we know this method is stable; the
convergence order is shown in Table 1.
For Figures 1 and 2, it is obvious that the corresponding
method for VDIDEs is stable and convergent, and the convergent order is min{4, 4}.
6. Conclusions
(𝑛−2𝑞)
𝑐𝑗 ℎ) 𝑌𝑗
+ sin (𝑡𝑛−𝑚 + 𝑐𝑗 ℎ) 𝑌𝑗(𝑛−𝑚) ] .
(79b)
This paper is devoted to the stability and convergence
analysis of the two-step Runge-Kutta (TSRK) methods with
compound quadrature formula for the numerical solution
for a nonlinear Volterra delay integro-differential equations.
Nonlinear stability and 𝐷-convergence are introduced and
proved.
Abstract and Applied Analysis
We believe that the results presented in this paper can
be extended to other general DIDEs and NDIDEs. However,
it is difficult to extend the results presented in this paper to
more general delay differential equations; the discussion will
be discussed later.
Acknowledgments
This work was supported by the Education Foundation of
Heilongjiang Province of China (12523039) and the Doctor Foundation of Heilongjiang Institute of Technology
(2012BJ27).
References
[1] M. Shakourifar and M. Dehghan, “On the numerical solution
of nonlinear systems of Volterra integro-differential equations
with delay arguments,” Computing, vol. 82, no. 4, pp. 241–260,
2008.
[2] S. F. Li, “𝐵-convergence properties of multistep Runge-Kutta
methods,” Mathematics of Computation, vol. 62, no. 206, pp.
565–575, 1994.
[3] K. Burrage and J. C. Butcher, “Nonlinear stability of a general
class of differential equation methods,” BIT Numerical Mathematics, vol. 20, no. 2, pp. 185–203, 1980.
[4] C. Huang, S. Li, H. Fu, and G. Chen, “Nonlinear stability of
general linear methods for delay differential equations,” BIT
Numerical Mathematics, vol. 42, no. 2, pp. 380–392, 2002.
[5] K. Burrage, “High order algebraically stable multistep RungeKutta methods,” SIAM Journal on Numerical Analysis, vol. 24,
no. 1, pp. 106–115, 1987.
[6] S. Li, “Stability and 𝐵-convergence properties of multistep
Runge-Kutta methods,” Mathematics of Computation, vol. 69,
no. 232, pp. 1481–1504, 2000.
[7] P. Linz, Analytical and Numerical Methods for Volterra Equations, SIAM Studies in Applied Mathematics, Society for Industrial and Applied Mathematics, Philadelphia, Pa, USA, 1985.
[8] S. F. Li, “Nonlinear stability of general linear methods,” Journal
of Computational Mathematics, vol. 9, no. 2, pp. 97–104, 1991.
[9] C. T. H. Baker and N. J. Ford, “Convergence of linear multistep
methods for a class of delay-integro-differential equations,”
International series of Numerical Mathematics, vol. 86, pp. 47–
59, 1988.
[10] C. T. H. Baker and N. J. Ford, “Asymptotic error expansions for
linear multistep methods for a class of delay integro-differential
equations,” Bulletin de la Société Mathématique de Grèce, vol. 31,
pp. 5–18, 1990.
[11] C. T. H. Baker and N. J. Ford, “Stability properties of a scheme
for the approximate solution of a delay-integro-differential
equation,” Applied Numerical Mathematics, vol. 9, no. 3–5, pp.
357–370, 1992.
[12] C. Zhang and S. Vandewalle, “Stability analysis of RungeKutta methods for nonlinear Volterra delay-integro-differential
equations,” IMA Journal of Numerical Analysis, vol. 24, no. 2, pp.
193–214, 2004.
[13] M. De la Sen and N. Luo, “On the uniform exponential
stability of a wide class of linear time-delay systems,” Journal
of Mathematical Analysis and Applications, vol. 289, no. 2, pp.
456–476, 2004.
13
[14] M. De la Sen, “Stability of impulsive time-varying systems and
compactness of the operators mapping the input space into the
state and output spaces,” Journal of Mathematical Analysis and
Applications, vol. 321, no. 2, pp. 621–650, 2006.
[15] C. T. H. Baker and A. Tang, “Stability analysis of continuous
implicit Runge-Kutta methods for Volterra integro-differential
systems with unbounded delays,” Applied Numerical Mathematics, vol. 24, no. 2-3, pp. 153–173, 1997.
[16] C. Zhang and S. Vandewalle, “General linear methods for
Volterra integro-differential equations with memory,” SIAM
Journal on Scientific Computing, vol. 27, no. 6, pp. 2010–2031,
2006.
[17] J. X. Kuang, J. X. Xiang, and H. J. Tian, “The asymptotic stability
of one-parameter methods for neutral differential equations,”
BIT Numerical Mathematics, vol. 34, no. 3, pp. 400–408, 1994.
[18] J. K. Hale and S. M. Verduyn Lunel, Introduction to FunctionalDifferential Equations, Springer, Berlin, Germany, 1993.
[19] Z. Chengjian and Z. Shuzi, “Nonlinear stability and Dconvergence of Runge-Kutta methods for delay differential
equations,” Journal of Computational and Applied Mathematics,
vol. 85, no. 2, pp. 225–237, 1997.
[20] B.-Y. Guo, Z.-Q. Wang, H.-J. Tian, and L.-L. Wang, “Integration
processes of ordinary differential equations based on LaguerreRadau interpolations,” Mathematics of Computation, vol. 77, no.
261, pp. 181–199, 2008.
[21] B.-Y. Guo and X.-Y. Zhang, “A new generalized Laguerre
spectral approximation and its applications,” Journal of Computational and Applied Mathematics, vol. 181, no. 2, pp. 342–363,
2005.
[22] R. Frank, J. Schneid, and C. W. Ueberhuber, “The concept of 𝐵convergence,” SIAM Journal on Numerical Analysis, vol. 18, no.
5, pp. 753–780, 1981.
[23] T. Koto, “Stability of Runge-Kutta methods for delay integrodifferential equations,” Journal of Computational and Applied
Mathematics, vol. 145, no. 2, pp. 483–492, 2002.
[24] C. M. Huang, “Algebraic stability of hybrid methods,” Chinese
Journal of Numerical Mathematics and Applications, vol. 17, pp.
67–74, 1995.
[25] S. F. Li, Theory of Computational Methods for Stiff Differential
Equations, Hunan Science and Technology Publisher, Changsha, China, 1997.
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International Journal of
Differential Equations
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International Journal of
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Mathematical Physics
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Journal of
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