Journal of Computational and Applied Mathematics 228 (2009) 524–537

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Journal of Computational and AppliedMathematics

journal homepage: www.elsevier.com/locate/cam

Recent advances in the numerical analysis of Volterra functionaldifferential equations with variable delaysHermann BrunnerDepartment of Mathematics and Statistics, Memorial University of Newfoundland, St. John’s, NL, Canada A1C 5S7

a r t i c l e i n f o

Article history:Received 28 August 2006Received in revised form 2 March 2007

MSC:65R2034K0634K28

Keywords:Volterra functional integro-differentialequationsVanishing delaysProportional delaysPantograph equationCollocation solutionsOptimal order of superconvergence

a b s t r a c t

The numerical analysis of Volterra functional integro-differential equations with vanishingdelays has to overcome a number of challenges that are not encountered when solving‘classical’ delay differential equations with non-vanishing delays. In this paper I shalldescribe recent results in the analysis of optimal (global and local) superconvergenceorders in collocation methods for such evolutionary problems. Following a brief surveyof results for equations containing Volterra integral operators with non-vanishing delays,the discussion will focus on pantograph-type Volterra integro-differential equations with(linear and nonlinear) vanishing delays. The paper concludes with a section on openproblems; these include the asymptotic stability of collocation solutions uh on uniformmeshes for pantograph-type functional equations, and the analysis of collocation methodsfor pantograph-type functional equations with advanced arguments.

© 2009 Published by Elsevier B.V.

1. Introduction

The aim of this paper is to describe recent and new results on the superconvergence properties of collocation solutionsto the initial-value problem for the Volterra functional integro-differential equation (VFIDE)

u′(t) = a(t)u(t)+ b(t)u(θ(t))+ g(t)+ (Vu)(t)+ (Vθu)(t), t ∈ I := [t0, T] (t0 ≥ 0). (1.1)

The Volterra integral operators V : C(I)→ C(I) and Vθ : C(Iθ)→ C(I) (with Iθ := [θ(t0), θ(T)]) in (1.1) are given by

(Vf )(t) :=∫ t

t0

K0(t, s)f (s)ds, t ∈ I, (1.2)

and

(Vθf )(t) :=∫ θ(t)

t0

K1(t, s)f (s)ds, t ∈ I; (1.3)

they correspond to kernels K0 and K1 that are continuous on the domains

D := {(t, s) : t0 ≤ s ≤ t, t ∈ I} and Dθ := {(t, s) : θ(t0) ≤ s ≤ θ(t), t ∈ I},

respectively. In the particular case where K1(t, s) = −K0(t, s) on Dθ the VFIDE (1.1) reduces to

u′(t) = a(t)u(t)+ b(t)u(θ(t))+ g(t)+ (Wθu)(t), t ∈ I, (1.4)

E-mail address: hermann@math.mun.ca.

0377-0427/$ – see front matter © 2009 Published by Elsevier B.V.doi:10.1016/j.cam.2008.03.024

H. Brunner / Journal of Computational and Applied Mathematics 228 (2009) 524–537 525

corresponding to the Volterra integral operator Wθ : C(Iθ)→ C(I) defined by

(Wθf )(t) :=∫ t

θ(t)K(t, s)f (s)ds, t ∈ I, (1.5)

whose kernel K is continuous on

D̄θ := {(t, s) : θ(t) ≤ s ≤ t, t ∈ I}.

The delay (or: lag) function θ is of the form θ(t) := t − τ(t), t ∈ I, where the delay τ = τ(t) is non-negative on I. We willassume that θ satisfies either the conditions

(DN1) τ(t) ≥ τ0 > 0 for all t ∈ I (non-vanishing delay),(DN2) θ is strictly increasing on I,(DN3) θ ∈ Cd(I) for some d ≥ 0;

or

(DV1) τ(t0) = 0, τ(t) > 0 for t ∈ (t0, T] (vanishing delay),(DV2) θ(t) ≤ q1t on I for some q1 ∈ (0, 1), and mint∈I θ

′(t) =: q0 > 0 for some q0 ≤ q1,(DV3) θ ∈ Cd(I) for some d ≥ 1.

Accordingly, the VFIDE (1.1) is complemented by an initial condition that is either

u(t) = φ(t), t ∈ [θ(t0), t0], (1.6)

(if the delay satisfies condition (DN1)), where φ is a given (continuous) initial function, or

u(t0) = u0 (1.7)

(if the delay is such that condition (DV1) holds).The approximate solution uh will be sought by collocation in some finite-dimensional function space (for example the

space of continuous piecewise polynomials with respect to a given mesh Ih of I). If the set of collocation points in I is denotedby Xh, the collocation equation corresponding to (1.1) may be written in continuous form, namely

u′h(t) = a(t)uh(t)+ b(t)uh(θ(t))+ g(t)− δh(t)+ (Vuh)(t)+ (Vθuh)(t), t ∈ I, (1.8)

with initial values

uh(t) := φ(t) if t ∈ [θ(t0), t0] (1.9)

(non-vanishing delay case), or

uh(t0) = u0 (1.10)

(vanishing delay case), respectively. The defect (or residual) δh(t) has the property δh(t) = 0 whenever t ∈ Xh, by the definitionof collocation. Hence, the collocation error eh := u− uh solves the functional equation

e′h(t) = a(t)eh(t)+ b(t)eh(θ(t))+ δh(t)+ (Veh)(t)+ (Vθeh)(t), t ∈ I, (1.11)

where the initial conditions are, respectively,

eh(t) = 0, t ∈ [θ(t0), t0] (1.12)

or

eh(t0) = 0. (1.13)

In the analysis of superconvergence (on I, or on Ih) of uh, the representation of the solution eh of (1.11) plays a key role. Thisrepresentation depends crucially on the nature of the delay: for non-vanishing delays it is given by a standard ‘variation-of-constants formula’ (or, more precisely, by a ‘resolvent representation’; cf. (2.7)). However, in the case of a vanishing delaysatisfying (DV1)–(DV3), such a resolvent representation does not exist, as will be shown in Section 2.2 (Theorem 2.3 andCorollary 2.5), and this will imply that the classical (local) superconvergence results are in general no longer valid.

In Section 2 we discuss and juxtapose these solution representations and then use them, in Sections 3 and 4, to obtainoptimal local superconvergence results, with the focus being on problems with vanishing proportional delays. Section 5contains an extension of these superconvergence results to higher-order VFIDEs with vanishing delays. Finally, in Section 6we describe a number of open problems, ranging from the question on asymptotic stability of collocation solutions forpantograph-type VFIDEs to that on the numerical analysis of pantograph-type VFIDEs with advanced arguments.

526 H. Brunner / Journal of Computational and Applied Mathematics 228 (2009) 524–537

2. Representation of the collocation error

2.1. VFIDEs with non-vanishing delays

If the delay τ(t) in θ(t) = t − τ(t) does not vanish on I and satisfies (DN2) and (DN3), it induces the so-called primarydiscontinuity points (or breaking points) {ξµ} at which the regularity of the solution of (1.1) may be lower than that of thegiven data (cf. [8, Section 3.1.3]). These points are given by the recursion

θ(ξµ) = ξµ−1, µ = 1, 2, . . . (ξ0 = t0), (2.1)

and they satisfy ξµ − ξµ−1 ≥ τ0 (µ ≥ 1), owing to (DN1). For arbitrarily smooth functions in (1.1) and (1.6), the (unique)solution u of (1.1) and (1.6) is in general not smooth on I: its regularity at t = ξµ is only Cµ (µ = 0, 1, . . . ), while itis arbitrarily smooth in each interval (ξµ, ξµ+1]. This is in complete analogy to the regularity theory for delay differentialequations with non-vanishing delays; compare, e.g., [8]. (However, note the possible ‘supersmoothing’ of the solution forincreasing µ in ξµ when b(t) ≡ 0 and V = 0; cf.[18].)

The initial-value problem for (1.1) is equivalent to a sequence of ‘local’ initial-value problems on the intervals I(µ):=

[ξµ, ξµ+1] (µ ≥ 0):

u′(t) = a(t)u(t)+ gµ(t)+∫ t

ξµ

K0(t, s)u(s)ds, t ∈ I(µ), (2.2)

with

gµ(t) := g(t)+ b(t)u(θ(t))+∫ ξµ

t0

K0(t, s)u(s)ds+ (Vθu)(t) (µ ≥ 1), (2.3)

and, for µ = 0,

g0(t) := g(t)+ b(t)φ(θ(t))+ (Vθφ)(t), t ∈ I(0). (2.4)

Eq. (2.2) is a standard Volterra integro-differential equation on I(µ), with non-homogeneous term gµ(t) depending on thislocal interval. For given initial value u(ξµ), its solution is

u(t) = r0(t, ξµ)u(ξµ)+∫ t

ξµ

r0(t, s)gµ(s)ds, t ∈ I(µ); (2.5)

the resolvent kernel r0 is defined by the solution of the resolvent equation

∂r0(t, s)

∂s= −r0(t, s)a(s)−

∫ t

sr0(t, v)K0(v, s)dv, ξµ ≤ s ≤ t ≤ ξµ+1, (2.6)

subject to the initial condition r0(t, t) = 1 for t ∈ I(µ) (see, e.g., [30] or [12, Section 3.1]). Using a simple induction argumenton (2.5), together with the expressions (2.4) and (2.3), we readily derive the following representation theorem (see also[11, pp. 69–70]).

Theorem 2.1. Let the given functions a, b, g, K0, K1, φ in (1.1) be continuous on their respective domains, and assume that θis subject to (DN1)–(DN3). Then on I(µ) (µ ≥ 1) the unique solution u ∈ C1(I) of the initial-value problem (1.1) and (1.6) can bewritten in the form

u(t) = r0(t, ξµ)u(ξµ)+∫ t

ξµ

r0(t, s)g(s)ds+ Fµ(t)+ Φµ(t), (2.7)

with

Fµ(t) :=µ−1∑ν=1ρµ,ν(t)u(ξν)+

∫ ξ1

ξ0

rµ,0(t, s)g0(s)ds+µ−1∑ν=1

∫ ξν+1

ξν

rµ,ν(t, s)g(s)ds,

Φµ(t) :=∫ θµ(t)

ξ0

qµ,0(t, s)g0(s)ds+µ−1∑ν=1

∫ θµ−ν(t)

ξν

qµ,ν(t, s)g(s)ds.

On the first interval I(0) the solution representation reduces to

u(t) = r0(t, t0)u(t0)+∫ t

t0

r1(t, s)g0(s)ds, (2.8)

where u(t0) = φ(t0). The functions ρµ,ν, rµ,ν, and qµ,ν depend on a, b, K0, K1, r0 and θ and are continuous on their respectivedomains; r0 = r0(t, s) denotes the resolvent kernel for K0 = K0(t, s) defined by the resolvent equation (2.6).

H. Brunner / Journal of Computational and Applied Mathematics 228 (2009) 524–537 527

The following result is obtained from Theorem 2.1 by replacing u and g by eh and δh, respectively.

Corollary 2.2. The unique solution eh of the initial-value problem (1.11) and (1.12) governing the collocation error on the intervalI(µ) is given by

eh(t) = r0(t, ξµ)eh(ξµ)+∫ t

ξµ

r0(t, s)δh(s)ds+ F̃µ(t)+ Φ̃µ(t) (µ ≥ 1), (2.9)

with

F̃µ(t) :=µ−1∑ν=1ρµ,ν(t)eh(ξν)+

∫ ξ1

ξ0

rµ,0(t, s)δh(s)ds+µ−1∑ν=1

∫ ξν+1

ξν

rµ,ν(t, s)δh(s)ds,

and

Φ̃µ(t) :=∫ θµ(t)

ξ0

qµ,0(t, s)δh(s)ds+µ−1∑ν=1

∫ θµ−ν(t)

ξν

qµ,ν(t, s)δh(s)ds.

On the first interval I(0) the representation (2.9) reduces to the classical resolvent representation

eh(t) = r0(t, t0)eh(t0)+∫ t

t0

r0(t, s)δh(s)ds, (2.10)

where eh(t0) = 0.

2.2. VFIDEs with vanishing delays

The delay differential equation corresponding to (1.1) with a(t) ≡ 0 and V = Vθ = 0,

u′(t) = a(t)u(t)+ b(t)u(θ(t))+ g(t), t ∈ [0, T], (2.11)

where θ(t) = qt (0 < q < 1), is known as the (variable coefficient) pantograph equation. The constant coefficient versionwas first studied, in the context of the modelling of the wave motion of an overhead wire induced by the pantograph ofan electric locomotive moving at a constant speed, in [61,29], and analyzed in detail in, e.g., [45,36]. (Compare also [65,2] and its references, for the physical framework of pantograph dynamics.) Thus, we will refer to the VFIDE (1.1) withθ(t) = qt (0 < q < 1, t ∈ [0, T]) as a pantograph-type VFIDE.

The survey paper [36], as well as the papers [38,41,28] conveys much insight into the properties of solutions to (2.11)and its nonlinear counterparts.

Consider now the general initial-value problem (1.1) and (1.7), with the delay function θ being subject to the conditions(DV1)–(DV3). It is readily verified that (1.1) is equivalent to the delay integral equation

u(t) = g0(t)+∫ t

0H0(t, s)u(s)ds+

∫ θ(t)

0H1(t, s)u(s)ds, t ∈ I, (2.12)

where we have set

g0(t) := u(t0)+∫ t

0g(s)ds, (2.13)

H0(t, s) := a(s)+∫ t

sK0(v, s)dv, (2.14)

H1(t, s) := b(θ−1(s))θ′(θ−1(s))+∫ t

θ−1(s)K1(v, s)dv. (2.15)

Theorem 2.3. If the given functions in the pantograph-type VFIDE (1.1) are continuous on their respective domains, and if thedelay function θ is subject to (DV1)–(DV3), then its unique solution u ∈ C1(I) has the representation

u(t) = g0(t)+∫ t

t0

R0(t, s)g0(s)ds+∞∑k=1

∫ θk(t)

0H1,k(t, s)g0(s)ds+M(t), t ∈ I, (2.16)

where

M(t) :=∞∑k=1

∫ θk(t)

0H(0,1)

k (t, s)g0(s)ds, (2.17)

528 H. Brunner / Journal of Computational and Applied Mathematics 228 (2009) 524–537

and θ0(t) := t. The kernels H0,k(t, s) and H1,k(t, s) are the iterated kernels associated with the kernels H0(t, s) and H1(t, s) definedin (2.14) and (2.15), g0 is given by (2.13), and

R0(t, s) :=∫ t

0

∞∑k=1

H0,k(t, s)g0(s)ds, (t, s) ∈ D, (2.18)

is the resolvent kernel corresponding to the kernel H0(t, s). The infinite series in (2.16)–(2.18) converge absolutely and uniformlyon I, and the continuous functions H(0,1)

k (t, s) (k ≥ 1) in (2.17) all vanish identically when H0(t, s) ≡ 0 or H1(t, s) ≡ 0.

Proof. The representation (2.16) follows readily by applying standard Picard iteration to the integral Eq. (2.12). (We notethat the representation (2.16) corresponding toH0(t, s) ≡ 0 is already implicitly contained in [1]; see also [24].) The followinglemma (an obvious extension of Dirichlet’s formula regarding the change in the order of integration in integrals with variablelimits of integration) plays a key role in the proof. �

Lemma 2.4. Let k and ` be given non-negative integers, and assume that the delay function θ is subject to the hypotheses(DV1)–(DV3). Then for any continuous function φ on D(k+`)

θ := {(t, s) : 0 ≤ s ≤ θk+`(t), t ∈ I},

∫ θk(t)

0

(∫ θ`(s)

0φ(s, v)dv

)ds =

∫ θk+`(t)

0

(∫ θk(t)

θ−`(v)φ(s, v)ds

)dv, (t, s) ∈ D(k+`)

θ . (2.19)

The result of Theorem 2.3 yields the desired representation for the collocation error eh := u− uh.

Corollary 2.5. The unique solution eh of the initial-value problem (1.11) and (1.13) for the collocation error is given by

eh(t) = dh(t)+∫ t

0R0(t, s)dh(s)ds+

∞∑k=1

∫ θk(t)

0H1,k(t, s)dh(s)ds+ M̃(t), t ∈ I, (2.20)

where

M̃(t) :=∞∑k=0

∫ θk(t)

0H(0,1)

k (t, s)dh(s)ds

and θ0(t) := t. We have set

dh(t) :=∫ t

0δh(s)ds. (2.21)

If a(t) ≡ 0 and V = 0 in (1.11), then the representation (2.20) assumes the form

eh(t) = dh(t)+∞∑k=1

∫ θk(t)

0H1,k(t, s)dh(s)ds, t ∈ I. (2.22)

3. Collocation methods in piecewise polynomial spaces

3.1. VFIDEs with non-vanishing delays

Since the non-vanishing delay τ = τ(t) in the delay function θ(t) = t − τ(t) in (1.1) induces the primary discontinuitypoints ZM = {ξµ : µ = 0, 1, . . . ,M; ξµ = t0}, the mesh Ih underlying a discretization method for such VFIDEs will have tocontain the set ZM if the method is to attain its classical order; that is, Ih will be a constrained mesh of the form

Ih :=M⋃µ=0

I(µ)h , with I(µ)

h := {t(µ)n : ξµ = t(µ)

0 < t(µ)1 < · · · < t(µ)

Nµ = ξµ+1}, (3.1)

where the Nµ are given positive integers. We will call Ih the global mesh, while the I(µ)h will be referred to as the underlying

local meshes. Here, we have assumed without loss of generality that T in I = [t0, T] is such that ξM+1 = T for some M ≥ 1. Wewill use the notation

σ(µ)n := [t

(µ)n , t(µ)

n+1], h(µ)n := t(µ)

n+1 − t(µ)n , h(µ)

:= max(n)

h(µ)n , h := max

(µ)h(µ).

Since this paper is mainly concerned with the analysis of the attainable orders of (global and local) superconvergence incollocation solutions for (1.1), we will assume from now on that Nµ = N for all µ.

H. Brunner / Journal of Computational and Applied Mathematics 228 (2009) 524–537 529

In the analysis of optimal superconvergence properties for collocation solutions to (1.1) with non-vanishing delays, themesh Ih has to possess a further property (see also [4] and [8, Section 6.3] in the case of DDEs): we will say that the globalmesh Ih (on I := [t0, T]) is θ-invariant if it is constrained and maps any local mesh I(µ)

h onto the previous one; that is,

θ(I(µ)h ) = I(µ−1)

h (3.2)

holds for allµ = 1, . . . ,M. We observe that θ-invariance implies Nµ = N for allµ ≥ 0. Moreover, if the mesh Ih is θ-invariantthen

t ∈ I(µ)h H⇒ θµ−ν(t) ∈ I(ν)h (ν = 0, 1, . . . ,µ; µ = 1, . . . ,M). (3.3)

The collocation space used to approximate the solution of (1.1) is the piecewise polynomial space

S(0)m (Ih) := {v ∈ C(I) : v|

σ(µ)n∈ πm (0 ≤ n < N; 0 ≤ µ ≤ M)}. (3.4)

Here, πk denotes the space of (real) polynomials of degree not exceeding k. The dimension of this linear space is

dim S(0)m (Ih) = (M + 1)Nm+ 1.

Accordingly, we choose the set Xµ of collocation points as

Xh :=

M⋃µ=0

X(µ)h , (3.5)

based on the M + 1 local sets

X(µ)h := {t

(µ)n,i := t(µ)

n + cih(µ)n : 0 < c1 < · · · < cm ≤ 1 (0 ≤ n ≤ N − 1)}

of cardinality Nm.For sufficiently regular solutions (i.e. for Cm-data) and arbitrarily chosen collocation parameters {ci} we have the global

convergence estimates

‖u(ν)− u(ν)

h ‖∞ ≤ Cνhm (ν = 0, 1). (3.6)

Since the delay τ in the VFIDE (1.1) does not vanish on I, a gain of one can be achieved in the global order of convergenceof uh by a judicious choice of the {ci}. More precisely, it is known (see [9,11]) that for Cm+1-data and collocation parameterssatisfying the orthogonality condition

J0 :=∫ 1

0

m∏i=1

(s− ci)ds = 0, (3.7)

the collocation solution uh ∈ S(0)m (Ih) is globally superconvergent on I:

‖u− uh‖∞ ≤ Chm+1, (3.8)

provided the mesh Ih is θ-invariant.The optimal (local) superconvergence order on Ih of the collocation solution uh ∈ S(0)

m (Ih), with θ-invariant mesh Ih, alsocoincides with the one for first-order ordinary (or delay) differential equations. We summarize the results in the followingtheorem (cf. [4] for DDEs, and [9,11] for general VFIDEs).

Theorem 3.1. Assume:

(a) The given functions a, b, g, K0, K1 and φ in (1.1) and (1.6) are in Cm+κ on their respective domains, for some κ with1 ≤ κ ≤ m, as in (d) below.

(b) The delay function θ is subject to the hypotheses (DN1)–(DN3) in Section 1, with d ≥ m+ κ+ 1 in (DN3).(c) uh ∈ S(0)

m (Ih) is the collocation solution, with θ-invariant mesh Ih, for the initial-value problem for the VFIDE (1.1).(d) The collocation parameters {ci} defining the collocation points in (3.5) are such that the orthogonality conditions

Jν :=∫ 1

0sν

m∏i=1

(s− ci)ds = 0, ν = 0, 1, . . . , κ− 1, (3.9)

hold, with Jκ 6= 0,

530 H. Brunner / Journal of Computational and Applied Mathematics 228 (2009) 524–537

Then the collocation error eh := u− uh satisfies, for all sufficiently small h > 0,

maxt∈Ih|eh(t)| ≤ Chm+κ (3.10)

where the constant C depends on the {ci} but not on h. In particular, if the {ci} are the m Gauss points in (0, 1) (i.e. if κ = m), thenwe obtain the optimal local order estimate

maxt∈Ih|eh(t)| ≤ Ch2m, (3.11)

while for u′h we only attain the error order |e′h(tn)| = O(hm).If in (d) we have cm = 1 and κ = m− 1 (the {ci} are then the Radau II points), there holds

maxt∈Ih\{t0}

|e(ν)h (t)| ≤ Cνh2m−1, ν = 0, 1.

We shall see in the following section that while the global superconvergence result (3.8) remains valid for vanishingdelays, this is no longer true for the optimal local superconvergence estimate (3.11) when m ≥ 3.

4. FVIDEs with vanishing delays

If the delay function θ in (1.1) satisfies the conditions (DV1)–(DV3) and is smooth, then the analytical solution of (1.1)with smooth data a, b, g, K0, K1 is smooth on the entire interval I, in sharp contrast to solutions to such functional equationswith non-vanishing delays. An important example of a vanishing delay (which motivated the subsequent analysis) is givenby the proportional delay,

θ(t) = qt = t − (1− q)t =: t − τ(t), 0 < q < 1, t ∈ [0, T]. (4.1)

We have already encountered this (linear) vanishing delay in the pantograph Eq. (2.11).Thus, since the analytical solution to the general pantograph-type VFIDE (1.1) with smooth data is smooth on the entire

interval [0, T], in contrast to solutions to such functional equations with non-vanishing delays, the (unconstrained) meshunderlying the approximating piecewise polynomial space will be chosen as

Ih := {tn : 0 = t0 < t1 < · · · < tN = T}, (4.2)

and we set hn := tn+1 − tn, h := max{hn : 0 ≤ n ≤ N − 1}. Our focus will be on superconvergence for uniform meshes Ih (butsee Remark 4.3 at the end of Section 4 on optimal superconvergence on geometric meshes).

For given Ih and m ≥ 1, the collocation solution uh to (1.1) will be an element of

S(0)m (Ih) := {v ∈ C(I) : v|[tn,tn+1] ∈ πm (0 ≤ n ≤ N − 1)} (4.3)

(with πm denoting the space of (real) polynomials of degree not exceeding m); it is determined by the collocation equation

u′h(t) = a(t)uh(t)+ b(t)uh(θ(t))+ g(t)+ (Vuh)(t)+ (Vθuh)(t), t ∈ Xh; uh(t0) = u0. (4.4)

Here,

Xh := {tn + cihn : 0 < c1 < · · · < cm ≤ 1 (0 ≤ n ≤ N − 1)} (4.5)

is the set of collocation points corresponding to given collocation parameters {ci}.A standard discrete Gronwall argument can be used to show that for arbitrarily chosen collocation parameters {ci}, the

collocation error tends to zero uniformly, as h→ 0. If the given functions in (1.1) have continuous derivatives of order m ontheir respective domains, then we obtain the basic error estimate (cf. [13, Ch. 5])

‖u(ν)− u(ν)

h ‖∞ ≤ Cνhm (ν = 0, 1), (4.6)

where the constants Cν depend on {ci} and on q, but not on h. Moreover, the uniform estimate for the defect δh induced bythe collocation solution uh ∈ S(0)

m (Ih) (recall (1.8)),

‖δh‖∞ ≤ D0hm, (4.7)

where m cannot be replaced by m+ 1, holds for any set {ci} of collocation parameters.In the papers [10,64,42,60] the authors studied the attainable order of continuous piecewise polynomial collocation

solutions for pantograph-type differential and integral equations at the first interior point t = t1 = h of a uniform mesh.The results gave a first indication that the classical optimal local superconvergence results of ordinary differential equations(related to certain Padé approximants to the exact solution) might no longer hold for such functional equations. (See also [74]for an extension to pantograph differential equations with two proportional delays.)

The analysis of optimal global and local superconvergence properties of the collocation solution uh ∈ S(0)m (Ih) on I and Ih takes

as its starting point the error representation (2.20) given in Corollary 2.5. We note that the optimal global superconvergence

H. Brunner / Journal of Computational and Applied Mathematics 228 (2009) 524–537 531

estimate (3.8) remains valid, provided the collocation parameters {ci} satisfy the orthogonality condition (3.7) (comparealso [11, Section 5]). This can be verified by employing techniques closely related to those that will be described in the proofof Theorem 4.1.

The optimal order of local superconvergence of uh on a uniform mesh Ih is described in the following theorem.

Theorem 4.1. Assume:(a) The given functions in the pantograph-type VFIDE (1.1) satisfy a, b, g ∈ Cm+κ(I), and K0 ∈ Cm+κ(D), K1 ∈ Cm+κ(Dθ), for

some κ with 1 ≤ κ ≤ m;(b) the delay function θ is subject to the hypotheses (DV1)–(DV3), with d ≥ m+ κ (κ ≥ 1);(c) uh ∈ S(0)

m (Ih), with uniform mesh Ih, is the collocation solution to (1.1) corresponding to collocation parameters {ci} satisfyingthe orthogonality conditions (3.9).

Then for all sufficiently small h > 0 the optimal local superconvergence order of uh is given by

max{|u(t)− uh(t)| : t ∈ Ih} ≤ C(q1)

{h2m if m = 1, 2,

hm+2 if m ≥ 3,(4.8)

regardless of the choice of κ. In particular, these estimates are optimal for the Gauss points {ci}: the exponent m + 2 cannot, ingeneral, be replaced by m+ 3.

Proof. For ease of exposition, and without loss of essential ingredients, we will consider the ‘pure delay’ case correspondingto a(t) ≡ 0 and V = Vθ in (1.1), for which the error Eq. (1.11) has the form

e′h(t) = b(t)eh(θ(t))+ δh(t)+ (Vθeh(t)), t ∈ [0, T].

Let t = tn = nh (n = 1, . . . ,N) be a given mesh point and define

qk,n := bθk(tn)/hc, γk,n := θ

k(tn)/h− qk,n, k∗n(q) := max{k : qk,n ≥ 1}. (4.9)

Thus, the representation (2.22) for the solution eh of the above error equation at tn can be written as

eh(tn) = dh(tn)+ SIn + SIIn, n = 1, . . . ,N, (4.10)

with

dh(t) :=∫ t

0δh(s)ds, t ∈ [0, T],

SIn :=k∗n(q)∑k=1

∫ tqk,n

0H1,k(tn, s)

(∫ s

0δh(v)dv

)ds

= hk∗n(q)∑k=1

qk,n−1∑`=0

H1,k(tn, t` + sh)

(h`−1∑ν=0

∫ 1

0δh(tν + vh)dv+ h

∫ s

0δh(t` + vh)dv

)ds, (4.11)

and

SIIn := h∞∑k=1

∫ γk,n

0H1,k(tn, tqk,n + sh)

(∫ tqk,n+sh

0δh(v)dv

)ds

= h∞∑k=1

∫ γk,n

0H1,k(tn, tqk,n + sh)

hqk,n−1∑`=0

∫ 1

0δh(t` + vh)dv+ h

∫ s

0δh(tqk,n + vh)dv

ds. (4.12)

Since the defect δh is piecewise Cm+κ on I (by (1.8) and assumption (a)), we may write

dh(tn) = hn−1∑`=0

∫ 1

0δh(s)ds = h

n−1∑`=0

{m∑j=1

wjδh(t` + cjh)+ E(`)n,j

},

where {wj} and E(`)n,j denote, respectively, the weights of the interpolatory m-point quadrature formula with abscissas {cj}

and the corresponding quadrature error. By the orthogonality conditions (3.9) these quadrature formulas have degree ofprecision m+ κ− 1 ≥ m, and hence – since δh vanishes at the collocation points – we are led to the estimate

|dh(tn)| ≤ hn−1∑`=0|E(`)

n,j | ≤ Qmhm+κ· Nh = QmTh

m+κi (n = 1, . . . ,N). (4.13)

Using the above quadrature argument and the estimate (4.7), we obtain

h`−1∑ν=0

∣∣∣∣∣∫ 1

0δh(tν + vh)dv

∣∣∣∣∣+ h

∣∣∣∣∫ s

0δh(t` + vh)dv

∣∣∣∣ ≤ TQmhm+κ+ h · D0h

m=: D̃0h

m+1.

532 H. Brunner / Journal of Computational and Applied Mathematics 228 (2009) 524–537

Thus, we are led to

|SIn| ≤ h · D̃0hm+1·

k∗n(q)∑k=1

qk,n−1∑`=0

∫ 1

0|H1,k(tn, t` + sh)|ds. � (4.14)

Lemma 4.2. Assume that the delay function θ satisfies the hypotheses (DV1)–(DV3), and define

β := max{|b(θ−1(t))|θ′(θ−1(t)) : t ∈ I}, K̄1 := max{|K1(t, s)| : (t, s) ∈ Dθ}.

The iterated kernels H1,k(t, s) corresponding to the kernel H1(t, s) in (2.15) are determined recursively by

H1,k+1(t, s) :=∫ θ(t)

θ−k(s)H1(t, v)H1,k(v, s)dv

=

∫ θk(t)

θ−1(s)H1,k(t, v)H1(v, s)dv, (t, s) ∈ D(k+1)

θ (k ≥ 1),

with H1,1(t, s) := H1(t, s); they can be bounded by

|H1,k(t, s)| ≤(β+ K̄1T)k

(k− 1)!q(k−1)(k−2)/21 [q1t − q−(k−1)

1 s]k−1, (t, s) ∈ D(k)θ (k ≥ 1). (4.15)

The proof of this result can be found in [15].If we employ the above result in (4.14), we obtain the estimate

|SIn| ≤ D̃0hm+2

k∗n(q)∑k=1

(β+ K̄1T)kTk−1

(k− 1)!q(k−1)(3k−4)/21 =: C0(q1)h

m+2. (4.16)

Similarly, (4.12) together with (4.7) and Lemma 4.2 allows us to derive the estimate

|SIIn | ≤ C1(q1)hm+2 (1 ≤ n ≤ N). (4.17)

Hence, collecting the above estimates for the three terms in (4.1), we see that the desired optimal superconvergenceestimate,

|eh(tn)| ≤ (C0(q1)+ C1(q1))hm+2=: C(q1)h

m+2, n = 1, . . . ,N,

is valid for any m ≥ 2 and any q1 ∈ (0, 1); as the above analysis shows, the power hm+2 cannot be replaced by hm+3, exceptin trivial cases. The verification of the optimal local superconvergence order 2m when m = 1 is left to the reader.

The above result answers the question regarding the optimal order of local superconvergence on uniform meshes for thepantograph delay differential equation (2.11):

Corollary 4.3. The collocation solution uh ∈ S(0)m (Ih) (m ≥ 2) for the pantograph delay differential equation (2.11), with uniform

mesh Ih and collocation points based on the Gauss points {ci}, has the optimal local superconvergence order (on Ih) described by

max{|eh(t)| : t ∈ Ih} ≤ C(q)hp∗ , (4.18)

where

p∗ ={

2m if m = 1, 2,m+ 2 if m ≥ 3.

Remark 4.1. The local superconvergence result (4.8) is not valid for iterated collocation solutions (corresponding to uh ∈

S(−1)m−1(Ih)) to pantograph-type Volterra integral equations of the form

u(t) = g(t)+ (Vu)(t)+ (Vθu)(t), t ∈ [0, T] :

if θ(t) = qt (0 < q < 1) and if collocation is at the Gauss points, the optimal local order p∗ = m+2 (m ≥ 3) is attained only ifq = 1/2 and m is even [14]; otherwise we have only p∗ = m+ 1 (which coincides with the order of global superconvergenceon I).

Remark 4.2. Collocation for the pantograph equation (2.11) in S(0)m (Ih) is equivalent to a class of continuous implicit m-stage

Runge–Kutta methods. On the other hand, there are (explicit and implicit) CIRK methods that do not correspond to collocationmethods. To the author’s knowledge, the general (optimal) order theory for such CIRK methods on uniform meshes has notyet been investigated. (A brief introduction to CIRK methods for linear and nonlinear pantograph DDEs can be found in [8,pp. 164–171] and, especially, [31, pp. 66–69].)

H. Brunner / Journal of Computational and Applied Mathematics 228 (2009) 524–537 533

Remark 4.3. It has been shown in [5] that on the so-called quasi-geometric meshes Ih the collocation solution uh ∈ S(0)m (Ih) for

the generalized pantograph delay differential equation (2.11) and corresponding to the Gauss points exhibits the optimallocal superconvergence order 2m at the mesh points. This result has been extended to pantograph-type VFIDEs in [6].Compare also [16] and [15, Section 6] for related optimal superconvergence results on geometric meshes.

Quasi-geometric meshes were first introduced in [54,55] and in [7] to derive asymptotic stability results for Runge–Kuttasolutions for (2.11). Compare also Section 6.2 for additional details regarding the use of such non-uniform meshes.

5. Higher-order pantograph-type integro-differential equations

In [3] Bélair studied various properties of solutions for the general second-order DDEs with proportional delays,

u′′(z)+n∑

j=1aj(z)u

′(pjz)+m∑

k=1bk(z)u(qkz) = 0,

where the complex constants pj and qk satisfy 0 < |pj|, |qk| < 1. Particular attention is given to the DDE u′′ + b(z)u(qz) =0, 0 < q < 1. For remarks on the theory of more general higher-order pantograph-type functional differential equationsthe reader is referred to [36, pp. 26–31] and [26]; see also [63,59].

The superconvergence analysis of Section 4 can be extended to higher-order pantograph-type VFIDEs. To illustrate this,we consider the second-order initial-value problem

u′′(t) = a(t)u(t)+ b(t)u(θ(t))+ g(t)+ (Vu)(t)+ (Vθu)(t), t ∈ I := [0, T], (5.1)

u(0) = u0, u′(0) = v0, (5.2)

with delay function θ subject to the hypotheses (DV1)-(DV3) of Section 1.It is easily verified that (5.1) and (5.2) are equivalent to the pantograph-type Volterra integral equation

u(t) = g0(t)+∫ t

0G0(t, s)u(s)ds+ g(t)+

∫ θ(t)

0G1(t, s)u(s)ds, t ∈ I, (5.3)

where now g0 stands for

g0(t) := u(0)+ u′(0)t +∫ t

0(t − s)g(s)ds. (5.4)

The kernels in (5.3) are given by

G0(t, s) :=∫ t

sH0(v, s)dv, (t, s) ∈ D, (5.5)

and

G1(t, s) :=∫ t

θ−1(s)H1(v, s)dv, (t, s) ∈ Dθ; (5.6)

the kernels H0(t, s) and H1(t, s) are the ones introduced in (2.14) and (2.15).The VFIDE (5.1) will be solved by collocation in the C1 piecewise polynomial space

S(1)m+1(Ih) := {v ∈ C1(I) : v|[tn,tn+1] ∈ πm+1 (0 ≤ n ≤ N − 1)};

its dimension is Nm+2, and the underlying mesh Ih is assumed to be uniform. Hence, we employ again the collocation pointsXh defined in (4.5), with hn = h = T/N. In analogy to (1.11) the collocation error eh := u − uh is the unique solution of theinitial-value problem

e′′h(t) = a(t)eh(t)+ b(t)eh(θ(t))+ δh(t)+ (Veh)(t)+ (Vθeh)(t), t ∈ I, (5.7)eh(0) = e′h(0) = 0,

where the defect δh satisfies δh(t) = 0 for all t ∈ Xh.It seems that the only investigation of the attainable order of collocation solutions to (5.1) (at t = t1 = h) is that in [72];

it extends some of the results of [10]. However, the optimal global or local superconvergence results on I and Ih given belowappear to be new.

The analysis in Section 2.2 suggests that eh admits a representation very similar to the one stated in Corollary 2.5, sincethat representation is based on the solution (2.16) of the Volterra functional integral equation (2.12), whose role is nowassumed by the VFIE (5.3), and where the previous iterated kernels H0,k(t, s) and H1,k(t, s) are to be replaced by the iteratedkernels G0,k(t, s) and G1,k(t, s) (generated by recurrence relations like those in Lemma 4.2) corresponding to the kernelsdefined in (5.5) and (5.6); the expression for dh(t) in the proof of Theorem 4.1 is now given by

dh(t) :=∫ t

0(t − s)δh(s)ds, t ∈ I. (5.8)

534 H. Brunner / Journal of Computational and Applied Mathematics 228 (2009) 524–537

In the case where a(t) ≡ 0 and V = 0 in (5.7) the error representation is (recall also (2.22))

eh(t) = dh(t)+∞∑k=1

∫ θk

0G1,k(t, s)dh(s)ds, t ∈ [0, T], (5.9)

where dh is now given by (5.8). All this implies that the proof of the local superconvergence results in Theorem 4.1 is readilyadapted to the present situation; in particular, the integral

∫ s0 δh(v)dv in (4.11) now becomes, by (5.4),

∫ s0(s− v)δh(v)dv. The

estimates (4.5) in Lemma 4.2 form the basis for the analogous estimates for the iterated kernels G1,k(t, s) corresponding tothe kernel G1(t, s) in the functional integral Eq. (5.3). Hence, the local superconvergence results of Theorem 4.1 carry over tothe collocation solution uh ∈ S(1)

m+1(Ih), with uniform mesh Ih, for (5.7). For example, if m ≥ 2 and collocation is at the Gausspoints, we obtain the optimal order estimate

max{|eh(t)| : t ∈ Ih} ≤ C(q1)hm+2 (m ≥ 2), (5.10)

(with q1 defined in condition (DV2)), provided the given functions in the VFIDE (5.1) satisfy the assumptions (a) and (b) ofTheorem 4.1, with κ ≥ 1.

It is clear from the above observations that the optimal local superconvergence result (5.10) also holds for collocationsolutions uh ∈ S(d)

m+d, with d := k − 1, to any kth-order analogue of the pantograph-type VFIDE (5.1). In this case, theconvolution integral in (5.8) becomes

dh(t) :=1

(k− 1)!

∫ t

0(t − s)k−1δh(s)ds (k ≥ 2),

and (5.5) and (5.6) are to be replaced by the appropriate (k− 2)-fold integrals of G0(t, s) and G1(t, s). The upper bounds forthe iterated kernels G1,k(t, s) in (5.9) associated with the kernel G1(t, s) follow readily from an obvious adaptation of theresult in Lemma 4.2.

Note however that – to the best of my knowledge – the superconvergence analysis of collocation solutions to (first- andhigher-order) pantograph-type DDEs and VFIDEs with multiple proportional delays remains open.

6. Open problems

6.1. Dynamics of error constants C(q)

It is well known (see for example [29,36,39,57]) that the numerical approximation of solutions to pantograph DDEs withq = 1− ε (0 < ε � 1) is not trivial (the reader not familiar with this phenomenon is encouraged to solve the initial-valueproblem for (6.2), with a = 0 and q = 0.99, on [0, 1000]; compare also [29, pp. 299–306] and [57, p. 310] on numericalresults for similar ‘difficult’ pantograph equations).

As far as the local superconvergence analysis of piecewise polynomial collocation solutions is concerned, it is crucial tounderstand the ‘dynamics’ (magnitude, limit), as q1 → 1−, of the error constants C(q1) occurring in the order estimatesof Theorem 4.1 and Corollary 4.3 (where q1 = q; cf. (DV2)), and in (5.10). This problem remains open. (I would like toacknowledge the discussion with Professor P.G. Ciarlet [25] on this topic.)

6.2. Asymptotic behaviour of collocation solutions

The analysis of the asymptotic behaviour of collocation solutions on uniform meshes for the pantograph-type Volterrafunctional integro-differential equation (VFIDE)

u′(t) = au(t)+ bu(qt)+∫ t

0k0(t − s)u(s)ds+

∫ qt

0k1(t − s)u(s)ds, t ≥ 0 (0 < q < 1), (6.1)

remains elusive. It has been shown in [45,44,36,54,41,40,57] that for the constant coefficient pantograph equation (i.e. for(6.1) with k0 = k1 = 0),

u′(t) = au(t)+ bu(qt), t ≥ 0 (0 < q < 1), (6.2)

the solution has the asymptotic property u(t)→ 0, as t→∞, if the coefficients a and b satisfy

Re(a) < 0 and |b| < |a|; (6.3)

for q = 1/2 these conditions are also necessary for asymptotic stability (cf. [37]). Related results on the asymptotic behaviourof solutions to the pantograph DDEs can be found in [38,26,23], and in [53] for partial (reaction–diffusion) DDEs withproportional delay.

No asymptotic stability results seem to be known for the general VFIDE (6.1). In particular, the problem of characterizingthe (continuous) kernels k0 and k1 for which, under the conditions (6.3), the solutions of (6.1) are asymptotically stable isopen.

H. Brunner / Journal of Computational and Applied Mathematics 228 (2009) 524–537 535

The same is essentially true for the asymptotic stability of numerical solutions to the pantograph DDE (6.2) basedon uniform meshes. While there are stability results for the trapezoidal method and for the θ-method and its one-legversion (see, e.g., [19–21,36,56,48–50,39,57,62,51] – and especially the illuminating discussions in [37,40] – comparealso the conjecture in [43] and its partial solution in [47]), and for collocation solutions uh ∈ S(0)

1 when q = 1/2 (orq = 1/ν, ν ∈ N, ν ≥ 3) [22], the asymptotic stability analysis of general collocation solutions in S(0)

m (Ih) (m ≥ 1) to(6.2) with arbitrary q ∈ (0, 1) remains open if the underlying mesh Ih is uniform.

The stability (and contractivity) of numerical solutions (including Runge–Kutta and collocation solutions) for VFIDEs withnon-vanishing delays is studied in [17,73,35,71]. For VFIDEs with vanishing delays analogous results for uniform meshesremain to be found.

We conclude this section by observing that, in contrast to the case of uniform meshes, there exists considerable work onthe asymptotic behaviour of numerical solutions for pantograph-type DDEs and VFIDEs on quasi-geometric meshes. Suchmeshes were introduced in [55, p. 180] and [7, pp. 281–282] (see also [8, pp. 168–171]). It is shown in these papersthat most of the classical numerical stability (and contractivity) results for ODEs carry over to VFIDEs with vanishingproportional delays, provided the underlying mesh is quasi-geometric. θ-methods are considered in [50,46,32,73,31], whilethe asymptotic properties of more general Runge–Kutta solutions are investigated in [46,67,69,70,68,35,52]. The recentpaper [31] presents a general, elegant new stability framework for pantograph DDEs. (The reader may also wish to consult[27] where a different approach to the analysis of numerical asymptotic stability for pantograph-type DDEs is described.)The papers [68,31] contain extensive lists of references on the development of these numerical asymptotic analyses.

6.3. Linear multistep methods for pantograph-type VFIDEs

The integrated form of (6.1) is, at t = tn = nh (n ≥ 1),

u(tn) = u(0)+

∫ tn

0h0(tn − s)u(s)ds+

∫ qtn

0h1(tn, s)u(s)ds, t ≥ 0, (6.4)

where the new kernels are

h0(t − s) := a+∫ t

sk0(v− s)dv, h1(t, s) := b/q+

∫ t

q−1sk1(v− s)dv

(cf. (2.14) and (2.15)). In [19] the authors discuss linear multistep methods for the pantograph DDE (6.2) when q assumesthe special value of q = 1/2. (Note that in this case we have qtn = tn/2 if n is even, and qtn = tbn/2c + h/2.)

While the boundedness and asymptotic stability of linear multistep (and one-leg) methods for pantograph DDEs havebeen considered in [70] for non-uniform (i.e. geometric) meshes, the systematic construction of such methods and theanalysis of their optimal convergence on uniform meshes remain to be studied. A possible approach could be based on anextension of the theory of convolution quadratures (compare the paper [58] for background and additional references) todeal with terms of the form∫ qtn

0h1(tn, s)u(s)ds =

∫ tqn

0h1(tn, s)u(s)ds+ h

∫ γn

0h1(tn, s)u(s)ds n ≥ 1,

in (6.4), where qn := bqnc and γn := qn− qn ∈ [0, 1).

6.4. Pantograph-type VFIDEs with advanced arguments

If the parameter q in the pantograph Eq. (6.2) satisfies q > 1, then the corresponding initial-value problem is ill-posed.A detailed analysis of such advanced differential equations can be found in [45] (Theorem 2) and in [44]. This ill-posednesswill cause difficulties in the numerical solution of such functional differential equations; a simple illustration of this is givenin [37] (pp. 137–138) for the trapezoidal method. The general numerical analysis of, e.g., piecewise polynomial collocationmethods for pantograph-type DDEs (and VFIDEs) with advanced arguments remains to be carried out.

This is an important challenge for numerical analysts since functional differential equations with advanced proportionalarguments arise in practical applications: second-order pantograph-type DDEs with q > 1 arise for example in themathematical modelling of the growth of a population of cells where the individual cells grow, such that each mother celldivides into q > 1 daughter cells of the same size. See [33,34,66] for the details; the theory of such functional differentialequations is discussed in [59].

Acknowledgements

This paper is based on two invited lectures presented at the Workshop on Innovative Methods for Solving Evolution Problemswith Memory (Capri, June 19–21, 2006). The research was supported in part by the Natural Sciences and Engineering ResearchCouncil (NSERC) of Canada. The author also gratefully acknowledges the financial support extended to him by the Universitàdegli Studi di Napoli “Federico II” during a visit in June 2006.

536 H. Brunner / Journal of Computational and Applied Mathematics 228 (2009) 524–537

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