R E S E A R C H
Open Access
Dissipativity of Runge-Kutta methods for a
class of nonlinear
functional-integro-differential equations
Qing Liao and Liping Wen
**Correspondence:
[email protected] School of Mathematics and Computational Science, Xiangtan University, Xiangtan, Hunan 411105, P.R. China
Abstract
This paper is concerned with the dissipativity of Runge-Kutta methods for a class of nonlinear functional-integro-differential equations (FIDEs). The dissipativity results of Runge-Kutta methods for the FIDEs are given. It is shown under a suitable condition that an algebraically stable Runge-Kutta method is dissipative when applied to the FIDEs. Numerical examples are given to illustrate the correctness of our theoretical results.
Keywords: functional-integro-differential equation; Runge-Kutta method; dissipativity; algebraic stability; dynamical systems
1 Introduction
Among various properties of dynamical systems, dissipativity is one of important char-acteristics. A dissipative dynamical system is characterized by possessing a bounded ab-sorbing set that all trajectories enter in a finite time and thereafter remain inside []. In the study of numerical methods for these systems, one natural wish is for the numerical solution to preserve the dissipativity of the analytic solution.
Over the past few decades, the dissipativity of the analytic solution and numerical meth-ods for some special class dynamical systems of VFDEs have been studied widely. One can refer to the following works and corresponding authors: [–] for ordinary differ-ential equations (ODEs), [–] for delay differdiffer-ential equations (DDEs), and [–] for other kinds of Volterra functional differential equations, such as delay integro-differential equations (DIDEs), neutral delay differential equations (NDDEs), neutral delay integro-differential equations (NDIDEs) and so on.
In this paper, we investigate numerical dissipativity of a class of nonlinear functional-integro-differential equations (FIDEs) (see (.) in the next section). In and , Zhang and Qin studied the stability of Runge-Kutta methods [] and one-leg methods [] for this kind of problems, respectively. Recently, we also studied the dissipativity of systems (.) and of one-leg methods for FIDEs (.) []. In addition we do not find more dissipativity results for this kind of nonlinear FIDEs. The aim of this paper is to investigate the dissipativity of Runge-Kutta methods for (.).
Runge-Kutta methods are deduced. In Section , some numerical experiments are given to illustrate the theoretical results which we stated in previous sections.
2 The descriptions of problem class and numerical methods
LetCdbe ad-dimensional complex Euclidean space with the inner product·,·and the
corresponding norm · . For any nonnegative diagonal matrixB=diag(b,b, . . . ,bs), we
define a pseudo inner product onCds:= (Cd)sby
Y,ZB= s
j=
bjYj,Zj, Y= (Y,Y, . . . ,Ys)∈Cds,Z= (Z,Z, . . . ,Zs)∈Cds,
and the corresponding pseudo norm onCdsby
YB=
Y,YB.
It is obvious that whenBis positive definite, they are the inner product and the norm onCds, respectively.
Consider nonlinear functional integro-differential equations (FIDEs) of the form (cf. [, ])
⎧ ⎨ ⎩
d dt[x(t) –
t
t–τg(t,ξ,x(ξ))dξ] =f(t,x(t),x(t–τ)), t∈[t, +∞),
x(t) =ϕ(t), t–τ≤t≤t, (.)
whereτ > is a given constant delay, the functionsf : [t, +∞)×Cd×Cd→Cd,g:D×
Cd→Cd, andϕ: [t–τ,t]→Cdare assumed to be continuous so that system (.) has
a unique solutionx(t), andf andgsatisfy also the conditions
Ref(t,u,v),u–w ≤γ+αu+βv+ηw,
t≥t,u,v,w∈Cd (.)
and
g(t,ξ,u)≤λu, (t,ξ)∈D,u∈Cd, (.)
whereγ,α,β,η,λare given real constants andγ, –α,β,ηare nonnegative, andλ> with λτ< ,
D:=(t,ξ) :t∈[t, +∞),ξ∈[t–τ,t]
.
In order to investigate the numerical dissipativity of (.), we assume further thatf sat-isfies the condition: for any constantM> , there existsL> which is only dependent on Msuch thatf(t,u,v) ≤Lholds for anyt≥t,u ≤Mandv ≤M.
Definition .(cf. []) Problem (.) in FIDEs is said to be dissipative inCdif there exists
such that for any given continuous initial functionϕ: [t–τ,t]→Cdwithϕ(t) contained
infor allt∈[t–τ,t], the corresponding solutionx(t) of the problem is contained in Bfor allt≥t∗. HereBis called an absorbing set inCd.
In our recent paper [], we studied the dissipativity of (.) and gave the following re-sults.
Theorem . Suppose that x(t)is a solution of problem(.)where f and g satisfy(.) withα< and(.),respectively,and there exists constant <δ< such that
– λτ
β+ (η–α)λτ
|α| ≤δ. (.)
Then,
(i) for anyt≥t,we have
x(t)
≤
– λτ –γ ( –δ)α+
– λτ – λτeμτ¯ φe
–μ¯(t–t),
whereφ=supt–τ≤ξ≤tϕ(ξ),andμ¯> is defined as
¯ μ=inf
t≥t
μ(t) :μ(t) +α+β+ (η–α)λτ e
μ(t)τ
– λτeμ(t)τ =
;
here and later,the symbolsγ,α,β,η,λare given by(.)and(.); (ii) for any givenε> ,there existst∗=t∗(φ,ε)such that
x(t)≤ – λτ
–γ
( –δ)α+ε, t≥t
∗.
Hence system(.)is dissipative with an absorbing set
B=B
,
– λτ
–γ ( –δ)α +ε
.
Remark . In [] and [], the authors studied the stability of Runge-Kutta methods and one-leg methods for FIDEs (.) on a limited closed interval [,T], but the mono-tonicity condition
Ref(t,u,v) –f(t,u,v),u–u– (w–w) ≤αu–u+βv–v+ηw–w,
The aim of this paper is to investigate whether the Runge-Kutta methods for system (.) preserve the dissipativity of the system itself.
It is well known that an s-stage Runge-Kutta method for ODEs can be expressed as
c A
bT=
c a a · · · as
c a a · · · as
..
. · · · · cs as as · · · ass
b b · · · bs
, (.)
whereA= (aij)∈Rs×s,b= (b,b, . . . ,bs)T∈Rsandc= (c,c, . . . ,cs)T∈Rswith ≤ci≤
(i= , , . . . ,s) andsj=bj= .
The following algebraic stability concept is the basis for studying the dissipativity of Runge-Kutta methods.
Definition .(see [, ]) Runge-Kutta method (.) is said to be algebraically stable if
B=diag(b,b, . . . ,bs) and M=BA+ATB–bbT
are nonnegative definite.
Let the step sizeh= τ
m with some positive integermandtn=t+nh. An adaptation of
method (.) for solving problem (.) leads to (see [])
⎧ ⎨ ⎩
Xi(n)–Zi(n)=xn–zn+h
s
j=aijf(tn+cjh,X(jn),X
(n–m)
j ), i= , , . . . ,s,
xn+–zn+=xn–zn+h
s
j=bjf(tn+cjh,Xj(n),X
(n–m)
j ),
(.)
wherexn,X(in)denote approximations tox(tn),x(tn+cih) andznandZi(n)approximations
toz(tn) andz(tn+cih), respectively. Here and later, we put that
z(t) =
t t–τ
gt,ξ,x(ξ)dξ. (.)
In addition,
⎧ ⎨ ⎩
xn=ϕ(tn), n≤,
Xi(n)=ϕ(tn+cih), tn+cih≤t.
(.)
As to the computation of integral termszn,Zi(n), we apply the compound quadrature
formulas
zn=h m
i=
vig(tn,tn–i,xn–i), (.)
Z(jn)=h
m
i= vig
tn+cjh,tn–i+cjh,X(jn–i)
where the quadrature formulas (.) and (.) can be derived from a uniform repeated rule (cf. [, , ]). For the numerical dissipativity analysis, we assume (.) or (.) to satisfy the following condition:
h
(m+ )m
i=
|vi|<v withmh=τand a positive constantv. (.)
Definition . Method (.) with a quadrature formula is said to be dissipative if, when-ever the method is applied with a step sizehto a dynamical system of the form (.) subject to (.)-(.), there exists a constantrsuch that, for any initial functionϕ(t), there exists n(ϕ,¯ h),ϕ¯=supt–τ≤t≤tϕ(t)such that
xn ≤r, n>n (.)
holds.
3 Dissipativity of Runge-Kutta methods
In this section we focus on the dissipativity analysis of algebraically stable Runge-Kutta methods with respect to nonlinear FIDEs.
Theorem . Assume that Rung-Kutta method(.)is algebraically stable and bj> for
j= , , . . . ,s,and that problem(.)satisfies(.)and(.)with(.)andα+β+ηvλ< . Then method(.)with(.)-(.)and(.)for FIDEs(.)is dissipative.
Proof For simplicity, we let
ti(n)=tn+cih, Qi=hf
ti(n),Xi(n),Xi(n–m), i= , , . . . ,s. It is well known (see, for example, []) that
xn+–zn+–xn–zn– s
i= biRe
Xi(n)–Zi(n),Qi = – s
i=
s
j=
mijQi,Qj, (.)
wheremij=biaij+bjaji–bibj.
By means of algebraic stability of the method, (.) leads to
xn+–zn+≤ xn–zn+ s
i= biRe
Xi(n)–Z(in),Qi. (.)
Using conditions (.) and (.) , then (.) gives
xn+–zn+≤ xn–zn+ h s
i= bi
γ+αXi(n)+βXi(n–m)+ηZ(in). (.)
We let
Hence by induction, from (.) we can obtain that
Making the square of the both sides and using (.) and the Cauchy-Schwarz inequality, we get
Similarly, from (.), (.) and (.) we can also obtain
Zi(n)≤hλ
Hence it can be deduced that
Therefore, substituting (.) into (.) shows
xn–zn≤hnγ+ h
α+β+ηvλ
n–
j=
X(j)B
+( +τ λ)+τβ+ηvλ max –τ≤ξ≤
ϕ(ξ). (.)
Whenγ= , it follows from (.) andα+β+ηνλ< that
lim
n→∞X
(n)
B= ,
which shows that for anyε> , there existsn=n(ϕ,¯ h) > such that
Xj(n)<ε, Xj(n–m)<ε, j= , . . . ,s,n≥n (.)
and
Z(n)
j <νλε, j= , . . . ,s,n≥n. (.)
On the other hand, from (.) we have
xn–zn=
X(in)–Z
(n)
i –h s
j= aijf
tj(n),Xj(n),Xj(n–m)
≤Xi(n)+Z(in)+h
s
j= |aij|f
tj(n),X(jn),Xj(n–m), i= , , . . . ,s. (.)
Therefore, we can obtain that
xn–zn ≤hL s
j=
|aij|+ ( +νλ)ε, n≥n, (.)
where
L= sup
u≤ε
v≤ε
f(t,u,v), t∈[, +∞),u,v∈Cd.
Whenγ > , using techniques similar to those presented in [], we can conclude that there existr> and a positive integern(ϕ,¯ h) such that
xn–zn ≤r forn≥n, (.)
where
r=
+τβ+ηνλR+ (m+ )hγ,
n=[( +τ λ)
+τ(β+ηνλ)]ϕ¯
with
The next thing to do in the proof is estimatingxn. Because of the fact that
In the following, we consider two cases. First, when μ+θ ϕ≥ϕ, we can obtain by induction that
xn+j ≤μ
j
k=
θk+θj+ϕ, j= , , , . . . . (.)
In fact, (.) implies (.) satisfied forj= . It is easy to see that
μ
j
k=
θk+θj+ϕ=μ
j–
k=
θk+θj(μ+θ ϕ)
≥μ
j–
k=
θk+θjϕ, j≥. (.)
If (.) holds forj<l, wherelis a positive integer, then it follows from (.) and (.) that
xn+l ≤μ+θ
μ
l–
k=
θk+θlϕ
=μ
l
k=
θk+θl+ϕ,
which shows that (.) holds for anyj≥.
Second, whenμ+θ ϕ<ϕ, then forl= , , . . . , it can be given by induction that
xn+ml+j ≤μ
l
k=
θk+θl+ϕ for anyj∈ {, , . . . ,m– }. (.)
In order to prove this conclusion, we first consider the case ofl= .
As a matter of fact, whenl= , (.) implies that (.) holds forj= . If here (.) holds forj<q<m– , then from (.) we have
xn+q ≤μ+θmax{μ+θ ϕ,ϕ} ≤μ+θ ϕ,
which shows (.) holds forl= .
Suppose that (.) holds forl<p, wherepis a positive integer. Whenj= , (.) reads
xn+pm ≤μ+θ max
≤k≤mxn+pm–k
≤μ+θ
μ
p–
k=
θk+θpϕ
=μ
p
k=
If it holds forj<q<m– that
where we have used that
μ
This completes the proof of Theorem ..
Remark . It is well known that thesstage Gauss, Radau IA, Radau IIA and Lobatto IIIC Runge-Kutta methods are all algebraically stable [], then from Theorem . they can preserve the dissipativity of the system when applied to FIDEs (.).
4 Numerical experiments
As an example, we consider the following two-dimensional system:
For this system, we choose
α= –
, β=
, η=
, λ= π, δ=
, γ =
( –a)+ ( +b), τ= π ,
which ensures all the conditions of Theorem . hold. System (.) is dissipative and pos-sesses an absorbing setB=B(, ( –a)+ ( +b)+ε) for any givenε> .
In order to solve system (.), we use the third order Radau IIA method where
c A
bT=
–
. (.)
Method (.) is algebraically stable and order . We letτ =mhwith a given positive integermand apply the composite Simpson’s rule to approach the integral terms
zn=
tn
tn–τ
gtn,ξ,x(ξ)
dξ and Zi(n)=
tn+cih
tn+cih–τ
gtn+cih,ξ,x(ξ)
dξ.
Here we can letv=in (.) and haveα+β+ηvλ< . According to Theorem ., the numerical solution is dissipative.
Now we let the step sizeh= .π/ and consider different initial functions fort∈ [π
, ] as follows:
(I) y(t) =sin(t)et,y(t) = t; (II) y(t) =cos(t),y(t) = sin(t); (III) y(t) = sin(t),y(t) =cos(t),
respectively. The numerical results are shown in Figures , , , , and .
These numerical examples prove that problem (.) is dissipative. Therefore, the numer-ical examples illustrate the correctness of our theoretnumer-ical results.
Figure 2 The numerical solution of (4.1) with initial function (I) anda= 2,b= 2 in the interval [76π, 10π].
Figure 3 The numerical solution of (4.1) with initial function (II) anda= 3,b= 2 in the interval [1, 10π].
Figure 5 The numerical solution of (4.1) with initial function (III) anda= 3,b= 4 in the interval [1, 10π].
Figure 6 The numerical solution of (4.1) with initial function (III) anda= 3,b= 4 in the interval [4π
3, 10π].
Acknowledgements
This work is supported by the National Natural Science Foundation of China (11371302, 11571291).
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
All authors contributed equally to the writing of this paper. All authors read and approved the final manuscript.
Publisher’s Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Received: 31 October 2016 Accepted: 4 May 2017
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![Figure 1 The numerical solution of (4.1) with initial function (I) and a = 2, b = 2 in the interval [0,10π].](https://thumb-us.123doks.com/thumbv2/123dok_us/964900.1118386/11.595.119.478.531.709/figure-numerical-solution-initial-function-i-interval-p.webp)
![Figure 4 The numerical solution of (4.1) with initial function (II) and a = 3, b = 2 in the interval[ 56π ,10π].](https://thumb-us.123doks.com/thumbv2/123dok_us/964900.1118386/12.595.117.479.79.712/figure-numerical-solution-initial-function-ii-interval-p.webp)
![Figure 6 The numerical solution of (4.1) with initial function (III) and a = 3, b = 4 in the interval[ 43π ,10π].](https://thumb-us.123doks.com/thumbv2/123dok_us/964900.1118386/13.595.117.485.79.489/figure-numerical-solution-initial-function-iii-interval-p.webp)
