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Collocation methods for nonlinear convolution Volterra integral equations with multiple proportional delays
Applied Mathematics and Computation 218 (2012) 10848–10860
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Collocation methods for nonlinear convolution Volterra integral equations with multiple proportional delays Kai Zhang ⇑, Jie Li, Haiming Song Department of Mathematics, Jilin University, Changchun, Jilin 130023, PR China
a r t i c l e
i n f o
a b s t r a c t In this paper, we apply the collocation methods to a class of nonlinear convolution Volterra integral equations with multiple proportional delays (NCVIEMPDs). We shall present the existence, uniqueness and regularity properties of analytic solution for this type equation, and then analyze the convergence and superconvergence properties of the collocation solution. The numerical results verify our theoretical analysis. Crown Copyright Ó 2012 Published by Elsevier Inc. All rights reserved.
Keywords: Collocation methods Nonlinear convolution Volterra integral equation Multiple proportional delays
1. Introduction Modeling many problems of physics, economics, stochastics, and other disciplines leads to nonlinear convolution Volterra integral equations with multiple proportional delays, where the rate of change of the process is not only determined by its present state but also by a certain past state. These are usually difficult to solve analytically and in many cases the solutions must be approximated. Numerical methods based on finite difference methods, Runge–Kutta methods, discontinuous Galerkin methods and spectral methods etc., have also been developed for various nonlinear Volterra integral equations and we refer to [1,3,5,7–9,11,17], and references therein for details about the rich literature. In [15], Ma and Brunner derived a posteriori error estimates for nonlinear Volterra integro-differential equations (NVIDEs) possessing the nonstandard memory term, and studied the discontinuous Galerkin (DG) method for NVIDEs with fully discretized memory terms. The assumption of nonstandard memory term in [15] is that the derivatives of nonstandard memory term are bounded. Ma also used high order collocation methods for Black Scholes equation in economics (c.f. [16]) under the frame of previous work. Guan et al. relaxed the assumption on nonstandard memory term to Lipschitz continuity (c.f. [11]), but the Lipschitz function is constrained to small enough in some integral sense, which is little strict to extend. In [6,12], some Volterra integral equations with nonlinear convolution are also studied, and Brunner also gives some convergence and superconvergence results for Volterra functional equations with multiple proportional delays (c.f. [4]). To the best of our knowledge, there are few works about convergence of collocation methods for nonlinear convolution Volterra integral equations with multiple proportional delays, which means the memory is extended from nonlinear term to nonlinear convolution term. Motivated by the work in [15,11], in this paper, we shall study the collocation method for nonlinear convolution Volterra integral equations with multiple delay (or: lag) functions hk ¼ hk ðtÞ; k ¼ 1; 2; . . . ; p of the form
uðtÞ ¼ f ðtÞ þ
p X
V hk u ðtÞ;
t 2 I :¼ ½0; T;
k¼1
where p is some positive integer. The Volterra integral operators V hk ðk ¼ 1; 2; . . . ; pÞ : CðIÞ ! CðIÞ are defined by
V hk u ðtÞ :¼
Z
hk ðtÞ
K k ðt; sÞG1 ðuðt sÞÞG2 ðuðsÞÞds;
0
⇑ Corresponding author. E-mail addresses: [email protected] (K. Zhang), [email protected] (J. Li), [email protected] (H. Song). 0096-3003/$ - see front matter Crown Copyright Ó 2012 Published by Elsevier Inc. All rights reserved. http://dx.doi.org/10.1016/j.amc.2012.04.045
ð1Þ
K. Zhang et al. / Applied Mathematics and Computation 218 (2012) 10848–10860
10849
where f ; G1 ; G2 and K k are given smooth functions. The delay functions hk ðtÞ; k ¼ 1; 2; . . . ; p are assumed to have the following properties: (1) hk ð0Þ ¼ 0, and hk is strictly increasing on I; k t on I for some q k 2 ð0; 1Þ; (2) hk ðtÞ 6 q (3) hk 2 C mk ðIÞ for some integer mk P 0. An important special case is the linear vanishing delay or proportional delay, i.e. hk ðtÞ ¼ qk t ¼ t ð1 qk Þt :¼ t sk ðtÞ with 0 < qk < 1, which are known as the pantograph delay functions (c.f. [17]). We shall concern on the corresponding NCVIEMPDs given by
uðtÞ ¼ f ðtÞ þ
p X
V qk u ðtÞ :¼ f ðtÞ þ LðG1 ðuÞ; G2 ðuÞÞðtÞ;
t 2 I;
ð2Þ
k¼1
where L is an operator which is given by
LðG1 ðuÞ; G2 ðuÞÞðtÞ ¼
p Z X k¼1
qk t
K k ðt; sÞG1 ðuðt sÞÞG2 ðuðsÞÞds :¼ Lu;
t2I
0
Rq t and V qk u ðtÞ :¼ 0 k K k ðt; sÞG1 ðuðt sÞÞG2 ðuðsÞÞds, k ¼ 1; 2; . . . ; p. Here, the nonstandard memory terms G1 ðuðt sÞÞG2 ðuðsÞÞ in each V qk u can be different, we assume they are uniform just for simplification in the rest of this paper. To the best of our knowledge, there exists few work on collocation method for NCVIEMPDs of form (2). In order to gain some insight approaches for nonlinear convolution Volterra integral equations, we present a study of piecewise polynomial collocation solutions for (2). There are several challenges for these NCVIEMPDs: the nonlinear convolution memory term is more tough than nonstandard memory term, and admits polynomial growth; the convergence and superconvergence proofs of integral equation are more difficult than corresponding integro-differential equations; the situations for the multiple proportional delays V qk u in (2) are more complex than single proportional delay. The rest of this paper is organized as follows: In Section 2, the existence, uniqueness and regularity of the analytic solution to (2) is presented. Section 3 is devoted to construct the collocation schemes. The conditions for the uniqueness of numerical schemes, convergence and superconvergence results of collocation scheme are shown in Section 4, and in Section 5, we give some numerical experiments to verify our theoretic results. The last section is devoted to some conclusion remarks and future work. 2. Existence, uniqueness and regularity of the analytic solution In order to show the existence, uniqueness and regularity of the analytic solution, we first introduce weighted norm space and some notations. Suppose that the space CðIÞ is endowed with exponentially weighted norm
kzkr ¼ maxjert zðtÞj; t2I
r P 0; 8z 2 CðIÞ
and kv k1 :¼ max jv ðtÞj. Then there exists positive number t2I
jðrÞ such that
jðrÞkzk1 6 kzkr 6 kzk1 ; z 2 CðIÞ: We assume the operators L and Gi in (2) satisfy following assumptions throughout this paper. Assumption A: There exist positive parameters L; M; N, l and kðrÞ such that, for i ¼ 1; 2 following inequalities hold
kGi ðu1 Þ Gi ðu2 Þkr 6 Mku1 u2 kr ; kGi ðu1 Þ Gi ðu2 Þk1 6 Mku1 u2 k1 ; kGi ðuÞk1 6 Lkukl1 ; kLðv 1 ; v 2 Þkr 6 Nkv 1 kr kv 2 kr ; kLðv 1 ; v 2 Þkr 6 kðrÞ minfkv 1 k1 kv 2 kr ; kv 1 kr kv 2 k1 g with
lim kðrÞ ¼ 0:
r!1
Define the linear operator T : L1 ðIÞ ! L1 ðIÞ by
T ðuÞðtÞ :¼ f ðtÞ þ LðG1 ðuÞ; G2 ðuÞÞðtÞ;
t 2 I;
then the Eq. (2) can be rewritten as
u ¼ T ðuÞ:
ð3Þ
By using the weighted norm technique (c.f. [13,18]), we can have following theorem about the existence and uniqueness of (2) or (3). Theorem 2.1. Assume that the given functions and operators in (2) satisfy (i) f 2 CðIÞ and K k 2 CðDqk Þ with Dqk ¼ fðt; sÞ : 0 6 s 6 qk tg; k ¼ 1; 2; . . . ; p ; (ii) L and Gi ði ¼ 1; 2Þ under the Assumption A.
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K. Zhang et al. / Applied Mathematics and Computation 218 (2012) 10848–10860
Then the NCVIEMPDs (2) has a unique solution u 2 CðIÞ.
Proof. Let Bq;r ðf Þ be a ball given by
Bq;r ðf Þ :¼ fu 2 CðIÞ : ku f kr 6 qg;
ðq > 0;
r P 0; f 2 CðIÞÞ
and we shall prove the conclusion by Banach fixed point theorem in Bq;r ðf Þ. We first show that the operator T : Bq;r ðf Þ ! Bq;r ðf Þ is a contraction mapping with q is small enough and enough. For any v ; v 1 and v 2 2 Bq;r ðf Þ,
ð4Þ
r is large
kT v f kr ¼ kLðG1 ðv Þ G1 ðf Þ þ G1 ðf Þ; G2 ðv Þ G2 ðf Þ þ G2 ðf ÞÞkr 6 kLðG1 ðv Þ G1 ðf Þ; G2 ðv Þ G2 ðf ÞÞkr þ kLðG1 ðv Þ G1 ðf Þ; G2 ðf ÞÞkr þ kLðG1 ðf Þ; G2 ðv Þ G2 ðf ÞÞkr þ kLðG1 ðf Þ; G2 ðf ÞÞkr 6 NM2 kv f k2r þ kðrÞMkG2 ðf Þk1 kv f kr þ kðrÞMkG1 ðf Þk1 kv f kr þ kðrÞkG1 ðf Þk1 kG2 ðf Þk1 6 NM2 q2 þ kðrÞMðkG2 ðf Þk1 þ kG1 ðf Þk1 Þq þ kðrÞkG1 ðf Þk1 kG2 ðf Þk1 and
kT v 1 T v 2 kr ¼ kLðG1 ðv 1 Þ; G2 ðv 1 ÞÞ LðG1 ðv 2 Þ; G2 ðv 2 ÞÞkr 6 kLðG1 ðv 1 Þ G1 ðv 2 Þ; G2 ðv 1 Þ G2 ðf Þ þ G2 ðf ÞÞkr þ kLðG1 ðv 2 Þ G1 ðf Þ þ G1 ðf Þ; G2 ðv 1 Þ G2 ðv 2 ÞÞkr 6 Mkv 1 v 2 kr NMkv 1 f kr þ kðrÞkG2 ðf Þk1 þ Mkv 1 v 2 kr NMkv 2 f kr þ kðrÞkG1 ðf Þk1 h i 6 2NM2 q þ kðrÞM kG2 ðf Þk1 þ kG1 ðf Þk1 kv 1 v 2 kr : From the continuity of Gi and f, we know that kG1 ðf Þk1 and kG2 ðf Þk1 are bounded. Since limr!1 kðrÞ ¼ 0, there exist sufficiently large numbers r1 and small q1 such that when r P r1 and q 6 q1 ; kT v f kr 6 q; kT v 1 T v 2 kr 6 12 kv 1 v 2 kr hold, which imply T is a contraction mapping in Bq;r ðf Þ. Therefore, Eq. (2) has a unique solution in Bq;r ðf Þ. Next, we show the uniqueness of the solution of (2) in the whole space CðIÞ. Suppose that u1 ; u2 2 CðIÞ are two solutions of (2), then
kui f kr ¼ kLðG1 ðui Þ; G2 ðui ÞÞkr 6 kðrÞkG1 ðui Þk1 kG2 ðui Þk1 ;
i ¼ 1; 2:
Since kG1 ðu1 Þk1 and kG2 ðu2 Þk1 are bounded, and limr!1 kðrÞ ¼ 0, we know that
lim kui f kr ¼ 0;
i ¼ 1; 2:
r!1
Thus for any q 6 q1 , there exists a sufficiently large number r2 such that when r P r2 , we have that kui f kr 6 q for i ¼ 1; 2, which implies u1 ; u2 2 Bq;r ðf Þ. Together with T is a contraction mapping in Bq;r ðf Þ, we obtain the uniqueness. h In order to analyze the regularity of the solution of the Eq. (2), we introduce some preliminary lemmas as follows. Lemma 2.1. [3]Consider the multiple delays functional equation
uðtÞ ¼ f ðtÞ þ
p Z X k¼1
qk t
K k ðt; sÞuðt sÞds;
t 2 I:
ð5Þ
0
If f 2 CðIÞ and K k 2 CðDqk Þ, then (5) has a unique solution u 2 CðIÞ. Lemma 2.2. For g 2 C 1 ðRÞ, the following identity holds
gðtÞ gðsÞ ¼ ðt sÞ
Z
1
g 0 ðs þ ðt sÞhÞdh;
8t; s 2 R:
0
Now, we are in the stage of showing the regularity result about the solution of (2). Theorem 2.2. Assume that the given functions in (2) satisfy f 2 C m ðIÞ; Gi 2 C m ðRÞ and K k 2 C m ðDqk Þ; k ¼ 1; 2; . . . ; p, for some integer v P 1, then the NCVIEMPDs (2) has a unique solution u 2 C m ðIÞ. Proof. By Theorem 2.1, we have already known that u 2 CðIÞ. For f 2 C 1 ðIÞ; Gi 2 C 1 ðRÞ and K k 2 C 1 Dqk , differentiate both sides of the Eq. (2) formally leading to
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K. Zhang et al. / Applied Mathematics and Computation 218 (2012) 10848–10860 p X qk K k ðt; qk tÞG1 ðuðt qk tÞÞG2 ðuðqk tÞÞ
u0 ðtÞ ¼ f 0 ðtÞ þ
k¼1
þ
p Z X
qk t
0
k¼1
@K k ðt; sÞG1 ðuðt sÞÞG2 ðuðsÞÞ þ K k ðt; sÞu0 ðt sÞG01 ðuðt sÞÞG2 ðuðsÞÞ ds: @t
Since the existence of a differentiable solution to the Eq. (2) is still unknown, consider the equation
e ðtÞ ¼ f 0 ðtÞ þ u
p X qk K k ðt; qk tÞG1 ðuðt qk tÞÞG2 ðuðqk tÞÞ k¼1
þ
p Z X k¼1
qk t 0
@K k e ðt sÞG01 ðuðt sÞÞG2 ðuðsÞÞ ds: ðt; sÞG1 ðuðt sÞÞG2 ðuðsÞÞ þ K k ðt; sÞ u @t
ð6Þ
e 2 CðIÞ. Lemma 2.1 implies that the above equation has a unique solution u e ðtÞ equals to u0 ðtÞ, that is to show limh!0 u e ðtÞ uðtþhÞuðtÞ Next, we shall prove u ¼ 0. For the simplification, we introduce h notations
x½g; v ; s ¼ sup gðtÞ t2I;0
;
v ðt þ sÞ v ðtÞ s
f ðt þ hÞ f ðtÞ A1 ¼ f 0 ðtÞ ; h " # Z p X 1 qk ðtþhÞ A2 ¼ qk K k ðt; qk tÞG1 ðuðt qk tÞÞG2 ðuðqk tÞÞ K k ðt þ h; sÞG1 ðuðt þ h sÞÞG2 ðuðsÞÞds ; h qk t k¼1 Z p Z qk t X @K k 1 qk t ðK k ðt þ h; sÞ K k ðt; sÞÞG1 ðuðt þ h sÞÞG2 ðuðsÞÞds ; A3 ¼ ðt; sÞG1 ðuðt sÞÞG2 ðuðsÞÞds h 0 @t 0 k¼1 Z Z p q t X k 1 qk t e ðt sÞG01 ðuðt sÞÞG2 ðuðsÞÞds K k ðt; sÞ u K k ðt; sÞG2 ðuðsÞÞðG1 ðuðt þ h sÞÞ G1 ðuðt sÞÞÞds : A4 ¼ h 0 0 k¼1 Then for any t; t þ h 2 ½0; d with sufficiently small d > 0 and h – 0, using (2) and (6), we obtain
uðt þ hÞ uðtÞ u e ðtÞ 6 jA1 j þ jA2 j þ jA3 j þ jA4 j: h
ð7Þ
Noting that
G1 ðuðt þ h sÞÞ G1 ðuðt sÞÞ ¼ ðuðt þ h sÞ uðt sÞÞ
Z
1
0
G01 ðuðt sÞ þ hðuðt þ h sÞ uðt sÞÞÞdh
and K k ; Gi ; G01 ; u are continuous, we can choose a positive number d such that
jA4 j 6
1 x½ ue ; u; h 2
and then taking sup-norm on both sides of (7), we have
1 x½ ue ; u; h 6 jA1 j þ jA2 j þ jA3 j: 2 e ðtÞ ¼ u0 ðtÞ for any t 2 ½0; d. Since f 2 C 1 ðIÞ and u 2 CðIÞ, we know that limh!0 jAi j ¼ 0ði ¼ 1; 2; 3Þ, which imply u e ðtÞ ¼ u0 ðtÞ holds on ½0; ld for some positive integer l, then for t; t þ h 2 ½ld; ðl þ 1Þd, we have Assume u
e ðtÞ u
uðt þ hÞ uðtÞ e4; ¼ A1 þ A 2 þ A 3 þ A h
where
e4 ¼ A
"Z p X k¼1
þ
0
tld
e ðt sÞG01 ðuðt sÞÞG2 ðuðsÞÞds K k ðt; sÞ u
p Z qk t X k¼1
tld
1 h
K k ðt; sÞu0 ðt sÞG01 ðuðt sÞÞG2 ðuðsÞÞds
Z
#
tld
K k ðt; sÞG2 ðuðsÞÞðG1 ðuðt þ h sÞÞ G1 ðuðt sÞÞÞds
0
1 h
Z
qk t
K k ðt; sÞG2 ðuðsÞÞðG1 ðuðt þ h sÞÞ G1 ðuðt sÞÞÞds :
tld
e ðtÞ ¼ u0 ðtÞ on ½ld; ðl þ 1Þd, therefore (2) has a solution u 2 C 1 ðIÞ by Using the same arguments on interval ½0; d, it follows that u mathematical induction. For integer v P 2, by the similar arguments, we can obtain the conclusion. h
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K. Zhang et al. / Applied Mathematics and Computation 218 (2012) 10848–10860
3. Collocation methods Let Ih ¼ ftn ¼ nh; n ¼ 0; . . . ; Ng with tN ¼ Nh ¼ T be a given uniform mesh on I and set hn :¼ ðt n ; t nþ1 ; n ¼ 0; . . . ; N 1. We shall be concerned with the collocation solution uh lying in the piecewise polynomial space ð1Þ
Sm1 ðIh Þ :¼ fv : v jhn 2 P m1 ; 0 6 n 6 N 1g;
ð8Þ
where P m1 ðm P 1Þ denotes the set of polynomials of degree not exceeding m 1. The dimension of the space equals Nm, and it is natural to choose the set of collocation points to be
ð1Þ Sm1 ðIh Þ
X h :¼ ftn;i ¼ tn þ ci h : 0 < c1 < < cm 6 1; n ¼ 0; . . . ; N 1g; since its cardinality is also Nm. Here, fci gm i¼1 is a given set of collocation parameters in ð0; 1. Since
qk tn;i ¼ qk ðt n þ ci hÞ ¼ qk ðnh þ ci hÞ ¼ qk ðn þ ci Þh; let
ðkÞ ðkÞ ðkÞ ðkÞ ðkÞ qk tn;i :¼ qn;i þ cn;i h ¼ qn;i h þ cn;i h ¼ tqðkÞ þ cn;i h 2 tqðkÞ ; tqðkÞ þ1 ; n;i
ðkÞ
ðkÞ
n;i
n;i
ðkÞ
with qn;i ¼ bqk ðn þ ci Þc, cn;i ¼ qk ðn þ ci Þ qn;i 2 ð0; 1. Here for any x 2 R; bxc is the greatest integer not exceeding x. ð1Þ We are looking for uh 2 Sm1 ðIh Þ satisfying the collocation equation
uh ðtÞ ¼ f ðtÞ þ
p X
V qk uh ðtÞ;
t 2 Xh :
ð9Þ
k¼1
Setting U n;j ¼ uh ðt n þ cj hÞðj ¼ 1; . . . ; mÞ, we can express unh (the restriction of uh on interval hn ) by interpolation
uh jhn ¼ unh ðtÞ ¼ uh ðtn þ shÞ ¼
m X Lj ðsÞU n;j ;
0 < s 6 1;
ð10Þ
j¼1
with Lagrange interpolation polynomials
Lj ðsÞ ¼
m Y s ck ; c ck k¼1;k–j j
0 < s 6 1; j ¼ 1; . . . ; m:
ð11Þ
Therefore, the global collocation solution uh on I is given by
uh ðtÞ ¼
N1 X
vn ðtÞunh ðtÞ;
n¼0
where vn ðtÞ is the characteristic function on hn . For each collocation point t n;i 2 X h , the collocation Eq. (9) becomes
uh ðt n;i Þ ¼ f ðt n;i Þ þ
p X
V qk uh ðtn;i Þ;
ð12Þ
k¼1
where
V qk uh ðt n;i Þ ¼ P1 þ P2 þ P 3 : ðkÞ
Z
qn;i 1
¼h
X
ðkÞ qn;i 1
K k t n;i ; t l þ sh G1
X Z
0
m X
! Lp ðci sÞU nl;p G2
p¼1
Z
1
K k ðtn;i ; tl þ shÞG1
ci
l¼0
þh
0
l¼0
þh
ci
cðkÞ n;i
! ! m X Lq ðsÞU l;q ds q¼1
! ! ! m m X X Lp ð1 þ ci sÞU nl1;p G2 Lq ðsÞU l;q ds p¼1
K k ðt n;i ; t qðkÞ n;i
q¼1
! ! m X þ shÞG1 uh tnqðkÞ þ ðci sÞh G2 Lq ðsÞU qðkÞ ;q ds : n;i
n;i
q¼1 ðkÞ
There are two cases for P 3 in ðV qk uh Þðtn;i Þ corresponding to tnqðkÞ þ ðci sÞh, s 2 ð0; cn;i : n;i
ðkÞ
(i) if ci 6 cn;i , then
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K. Zhang et al. / Applied Mathematics and Computation 218 (2012) 10848–10860
ðV qk uh Þðt n;i Þ ¼ P 1 þ P2 þ h þh
Z ci
(ii) if ci > c
ðkÞ n;i ,
cðkÞ n;i
Z
ci
! ! ! m m X X K k t n;i ; tqðkÞ þ sh G1 Lp ðci sÞU nqðkÞ ;p G2 Lq ðsÞU qðkÞ ;q ds n;i
0
n;i
p¼1
q¼1
n;i
! ! ! m m X X K k t n;i ; t qðkÞ þ sh G1 Lp ð1 þ ci sÞU nqðkÞ 1;p G2 Lq ðsÞU qðkÞ ;q ds ; n;i
n;i
p¼1
q¼1
n;i
then
ðV qk uh Þðt n;i Þ ¼ P 1 þ P2 þ h
Z
ðkÞ cn;i
! ! ! m m X X K k t n;i ; t qðkÞ þ sh G1 Lp ðci sÞU nqðkÞ ;p G2 Lq ðsÞU qðkÞ ;q ds : n;i
0
n;i
p¼1
q¼1
n;i
4. Theoretic results for the collocation solution on uniform mesh I h In this section, we shall present the existence and uniqueness of the collocation solution. Using projection operators, we propose a theorem about convergence of collocation method (12). The last part of this section is devoted to some results on superconvergence of collocation method for NCVIEMPDs. 4.1. The existence and uniqueness of the collocation solution For the simplification, we introduce notations
U n ¼ ðU n;1 ; . . . ; U n;m ÞT ;
F n ¼ ðf ðt n þ c1 hÞ; . . . ; f ðt n þ cm hÞÞT ;
n ¼ 0; 1; . . . ; N 1:
Then the existence and uniqueness of (12) is given by following theorem. Theorem 4.1. Assume that the given functions and operators in (2) satisfy (i) f 2 CðIÞ; Gi 2 CðRÞ and K k 2 CðDqk Þ, k ¼ 1; 2; . . . ; p; (ii) Gi under Assumption A. the Eq. (12) defines a unique collocation solution > 0, for any uniform mesh I with h < h, Then there exists a constant h h ð1Þ uh 2 Sm1 for all qk 2 ð0; 1Þ.
Proof. We use mathematical induction method with respect to index n to prove this result. For n ¼ 0, let u ¼ ðu1 ; . . . ; um ÞT . Define the operator K : Rm ! Rm as follows
ðKuÞi ¼ h
p Z X
cðkÞ 0;i
K k ðci h; shÞG1
0
k¼1
! ! m m X X Lp ðci sÞup G2 Lq ðsÞuq ds; p¼1
q¼1
then the Eq. (12) can be rewritten as
ðI KÞU 0 ¼ F 0 ;
ð13Þ
where I denotes the identity operator. Setting Br ð0Þ ¼ fu 2 R : kuk1 6 rg with kuk1 ¼ maxi jui j, then for /; u 2 Br ð0Þ, we have m
(Z ðkÞ ! " ! !# p c0;i m m m X X X X ðK/Þ ðKuÞ 6 h K k ðci h; shÞG1 Lp ðci sÞ/p G2 Lq ðsÞ/q G2 Lq ðsÞuq ds i i 0 p¼1 q¼1 q¼1 k¼1 Z ðkÞ " ! !# ! ) c0;i m m m X X X þ K k ðci h; shÞ G1 Lp ðci sÞ/p G1 Lp ðci sÞup G2 Lq ðsÞuq ds 0 p¼1 p¼1 q¼1 6 hk/ uk1 nR
p n X
o KMðmLÞ1þl Lrl þ KMðmLÞ1þl Lrl ;
k¼1
o where K ¼ maxt2I jK k ðt; sÞjds ; L ¼ maxj kLj k1 and L is the constant defined in Assumption A. For u 2 Br ð0Þ, we have t 0
! ! p Z cðkÞ p m m X X X X 0;i jðKuÞi j 6 h K k ðci h; shÞG1 Lp ðci sÞup G2 Lq ðsÞuq ds 6 h KðmLÞ2l L2 r 2l : 0 p¼1 q¼1 k¼1 k¼1
10854
K. Zhang et al. / Applied Mathematics and Computation 218 (2012) 10848–10860
~ such that for any h 2 ð0; hÞ, ~ we have Then there exists a positive number h
1 k/ uk1 2
kK/ Kuk1 6
and kKuk1 6 r:
Therefore, we know that K : Br ð0Þ ! Br ð0Þ is a contraction map. Hence the operator I K has a bounded inverse, which implies (12) has a unique bounded solution U 0 . Assume (12) has a unique bounded solution on n 6 l 1 for some positive integer l, we want to prove that it is also true for n ¼ l. Noting that when n – 0 and s 2 ð0; qk tn;i Þ, the points tn;i s and s can not belong to ðtn ; tnþ1 simultaneously. For n ¼ l, define the operator K on Rm as follows
ðKU l Þi ¼ f ðtl;i Þ þ
p Z X k¼1
qk t l;i
K k ðt l;i ; sÞG1 ðuh ðt l;i sÞÞG2 ðuh ðsÞÞds:
0
For /; u 2 Rm , we get
" ! !# ! m m m X X X jðK/Þi ðKuÞi j 6 h Lp ðci sÞ/p G1 Lp ðci sÞup Lq ðsÞU 0;q ds G2 K k ðtl;i ; shÞ G1 0 p¼1 p¼1 q¼1 k¼1 Z ðkÞ ! " ! !# ) cn;i m m m X X X þ K k tl;i ; tqðkÞ þ sh G1 Lp ðci sÞU 0;p G2 Lq ðsÞ/q G2 Lq ðsÞuq ds 0 l;i p¼1 q¼1 q¼1 (Z p X
ci
6 hk/ uk1
p n o X KMðmLÞ1þl LkU 0 kl1 þ KMðmLÞ1þl LkU 0 kl1 : k¼1
such that for any h 2 ð0; hÞ ð0; hÞ, ~ we have Then there exists a positive number h
kK/ Kuk1 6
1 k/ uk1 : 2
Hence K is a contraction map on Rm for n ¼ l, which implies there exists a unique solution U l in Rm as n ¼ l. This completes the proof. h 4.2. The convergence results for collocation solution ð1Þ
We now analyze the convergence order of the collocation solution uh 2 Sm1 ðIh Þ for the NCVIEMPDs (2) by using projection operators. ð1Þ
Theorem 4.2. Let uh 2 Sm1 ðIh Þ is the collocation solution defined in (12). Assume that the given functions and operators in (2) satisfy (i) f 2 CðIÞ; Gi 2 CðRÞ and K k 2 CðDqk Þ, k ¼ 1; 2; . . . ; p; (ii) Gi under Assumption A. Then for all sufficiently small h > 0, we have
ku uh k1 6 C kðI P h Þf k1 þ kðI P h ÞLuk1 ;
ð14Þ
where P h is the Lagrange interpolation operator corresponding to the collocation parameters fci g, and the constant C is independent on h. Furthermore, if the exact solution u 2 W m;1 ðIÞ, we obtain m
ku uh k1 6 Ch kukm;1 ; where kv km;1
j :¼ max sup d v ðtÞ . dt j 06j6m
t2I
Proof. The operator formulations for NCVIEMPDs (2) and its collocation Eq. (12) are given by
u ¼ f þ Lu; uh ¼ P h f þ P h Luh :
Based on the solvability of the NCVIEMPDs and its collocation equation, we obtain
ð15Þ
K. Zhang et al. / Applied Mathematics and Computation 218 (2012) 10848–10860
(
10855
u ¼ ðI LÞ1 f ; uh ¼ ðI P h LÞ1 P h f :
The error between u and uh can be expressed by
eh :¼ u uh ¼ ðI LÞ1 f ðI P h LÞ1 P h f ¼ ðI LÞ1 P h f ðI P h LÞ1 P h f þ ðI LÞ1 ðf P h f Þ ¼ ðI P h LÞ1 ðK P h LÞðI LÞ1 P h f þ ðI LÞ1 ðf P h f Þ ¼ ðI P h LÞ1 ðL P h LÞðI LÞ1 ðP h f f Þ þ ðI P h LÞ1 ðK P h LÞðI LÞ1 f þ ðI LÞ1 ðI P h Þf ¼ ðI P h LÞ1 ðL P h LÞðI LÞ1 ðP h I Þf þ ðI P h LÞ1 ðI P h ÞLu þ ðI LÞ1 ðI P h Þf ; which implies
ku uh k1 6 C kðI P h Þf k1 þ kðI P h ÞLuk1 : If u 2 W m;1 , from the error estimates of the interpolation operator P h , we know that m
m
kðI P h Þf k1 6 Ch kf km;1 6 Ch kukm;1 ; m
m
kðI P h ÞLuk1 6 Ch kLukm;1 6 Ch kukm;1 ; which leads to m
ku uh k1 6 Ch kukm;1 : h 4.3. The superconvergence results for collocation solution In the rest of this section, we discuss the superconvergence of collocation method for NCVIEMPDs. Define the iterated collocation solution uith associated with uh by
uith ðtÞ ¼ f ðtÞ þ
p X ðV qk uh ÞðtÞ;
ð16Þ
k¼1
then the iterated error eith ¼ u uith is given by
eith ðtÞ ¼ eh ðtÞ dh ðtÞ ¼
p X
ðV qk uÞðtÞ ðV qk uh ÞðtÞ ;
k¼1
where
dh ðtÞ :¼ uh ðtÞ þ f ðtÞ þ
p X ðV qk uh ÞðtÞ: k¼1
In particular, for the collocation points, we have dh ðtÞ ¼ 0. Hence, we obtain eith ðtÞ ¼ eh ðtÞ for t 2 X h . In order to study superconvergence, we present some notations and preliminary lemmas firstly. For given abscissas a 6 n1 < < nm 6 b, define the quadrature formula
Q m ðgÞ :¼
m X
xj gðnj Þ;
j¼1
Rb to approximate the integral Q ðgÞ ¼ a xðtÞgðtÞdt, where the weight function xðtÞ is assumed to satisfy xðtÞ 2 L1 ½a; b. Let the corresponding error be denoted by Em ðgÞ ¼ Q ðgÞ Q m ðgÞ. We have following indentity for the quadrature formula error Em ðgÞ. Lemma 4.1. (c.f. [10]) Assume the quadrature formula Q m ðgÞ has algebraic degree of precision a P 1 and g 2 C d ½a; b with 1 6 d 6 a þ 1. Then Em ðgÞ possesses the integral representation
Em ðgÞ ¼
Z
b
K d ðsÞg ðdÞ ðsÞds;
a
where K d is given by
K d ðsÞ ¼
1 ðd 1Þ!
(Z
b a
xðzÞðz sÞd1 þ dz
m X
)
: xk ðnk sÞd1 þ
k¼1
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K. Zhang et al. / Applied Mathematics and Computation 218 (2012) 10848–10860
Consider the following integral equation
yðtÞ ¼ f ðtÞ þ
Z
t
Kðt; sÞyðsÞds;
t 2 I :¼ ½0; T;
ð17Þ
0
where f and K are bounded and integrable on I and D :¼ fðt; sÞ : 0 6 s 6 t 6 Tg respectively. Define iterated kernels by K 1 ðt; sÞ :¼ Kðt; sÞ and
Z
K n ðt; sÞ :¼
t
K 1 ðt; v ÞK n1 ðv ; sÞdv ;
n P 2:
ð18Þ
s
Lemma 4.2. (c.f. [3]) Set p 1 X X K n ðt; sÞ ¼ lim K n ðt; sÞ ¼: Rðt; sÞ; p!1
n¼1
ðt; sÞ 2 D:
n¼1
Then the so-called resolvent kernel Rðt; sÞ converges absolutely and uniformly in D, and satisfies
Z
Z
t
Kðt; sÞyðsÞds ¼
0
t
Rðt; sÞf ðsÞds:
0
Now, we are ready to show the superconvergence result about the iterated collocation solution. Theorem 4.3. Assume that the given functions in (2) satisfy f 2 C mþ1 ðIÞ; Gi 2 C mþ1 ðDÞ; i ¼ 1; 2 and K k 2 C mþ1 ðDqk Þ; k ¼ 1; 2; . . . ; p. If the collocation parameters fci g are chosen so that the orthogonality condition
J 0 :¼
Z
m 1Y
0
ðs ci Þds ¼ 0
i¼1
holds, then the iterated collocation solution defined by (16) exhibits the superconvergence on the entire interval I:
ku uith k1 6 Ch
mþ1
;
where the constant C is independent on h. Proof. Using the definitions of eh ðtÞ and dh ðtÞ, we know that
eh ðtÞ ¼ dh ðtÞ þ
p Z X
K k ðt; sÞðG1 ðuðt sÞÞ G1 ðuh ðt sÞÞÞG2 ðuðsÞÞds þ
0
k¼1
¼ dh ðtÞ þ
¼ dh ðtÞ þ
Z
t
qk t
K k ðt; sÞ/ðt; sÞG2 ðuðsÞÞeh ðt sÞds þ 0
(Z
p X k¼1
¼ dh ðtÞ þ
Z
1
0
uðt; sÞ ¼
Z
1
0
e ðt; sÞ ¼ K
K k ðt; sÞG1 ðuh ðt sÞÞðG2 ðuðsÞÞ G2 ðuðsÞÞÞds
qk t
K k ðt; sÞG1 ðuh ðt sÞÞuðt; sÞeh ðsÞds
0
K k ðt; t sÞ/ðt; t sÞG2 ðuðt sÞÞeh ðsÞds þ
tqk t
Z 0
e ðt; sÞeh ðsÞds ¼ dh ðtÞ þ K
Z
t
Rðt; sÞdh ðsÞds;
0
G01 ðuh ðt sÞ þ hðuðt sÞ uh ðt sÞÞÞdh; G02 ðuh ðsÞ þ hðuðsÞ uh ðsÞÞÞdh;
p X
Z
t
0
/ðt; sÞ ¼
qk t
0
p Z X k¼1
where
Z
qk t
xIk K Ik ðt; sÞ þ xIIk K IIk ðt; sÞ ;
k¼1
K Ik ðt; sÞ ¼ K k ðt; t sÞ/ðt; t sÞG2 ðuðt sÞÞ; K IIk ðt; sÞ ¼ K k ðt; sÞG1 ðuh ðt sÞÞuðt; sÞ;
0; s 2 ð0; t qk tÞ; 1; s 2 ð0; qk tÞ; xIk ¼ xIIk ¼ 1; s 2 ðt qk t; tÞ; 0; s 2 ðqk t; tÞ:
qk t
) K k ðt; sÞG1 ðuh ðt sÞÞuðt; sÞeh ðsÞds
10857
K. Zhang et al. / Applied Mathematics and Computation 218 (2012) 10848–10860 m
Using Theorem 4.2, it is easy to see that dh ðtÞ ¼ Oðh Þ. For t ¼ t n þ v h, we get
Z
t
Rðt; sÞdh ðsÞds ¼
0
n1 Z X h l¼0
1
Rðt; t l þ shÞdh ðtl þ shÞds þ h
Z v
Rðt; tn þ shÞdh ðt n þ shÞds:
0
0
Suppose the integrals in above equality are approximated by the interpolatory m-point quadrature formula with abscissas fci g, then we obtain
Z
1
Rðt; tl þ shÞdh ðt l þ shÞds ¼
0
m X bj Rðt; tl þ cj hÞdh ðt l þ cj hÞ þ EðlÞ n ðv Þ;
v 2 ½0; 1;
j¼1
R1 where bj ¼ 0 Lj ðsÞds and EðlÞ n ðv Þ denote the quadrature errors induced by these quadrature approximations. The orthogonalmþ1 ity condition J 0 ¼ 0 and Lemma 4.1 imply that the induced quadrature errors EnðlÞ ðv Þ ¼ Oðh Þ, which leads to
Z
t
Rðt; sÞdh ðsÞds ¼ Oðh
mþ1
Þ:
0
Together with the relation eith ðtÞ ¼ eh ðtÞ dh ðtÞ, we get the conclusion.
h
5. Numerical examples In this section, we apply the collocation methods described in Section 3 to several NCVIEMPDs examples. Results of the numerical simulations verify our convergence and superconvergence analysis in Section 4. Example 5.1. We consider the Bernstein and Doetsch equation (c.f. [2]), which is a degenerate case of NCVIEMPDs (2) with p ¼ 1; I ¼ ½0; 1 and
G1 ðtÞ ¼ t;
G2 ðtÞ ¼ t;
K 1 ðt; sÞ ¼
1 ; t
f ðtÞ ¼ 0:
pffiffiffiffi pffiffiffiffi
ð1Þ We use the piecewise quadratic space S2 ðIh Þ with collocation points C 1 ¼ ð1=4; 1=2; 3=4Þ and C 2 ¼ 51015 ; 1=2; 5þ1015 1 substituting the exact solution and the results are shown in respectively. Here we use fine mesh solutions with h ¼ 5000 Fig. 1. From the figure, we know that the convergence order of eith (which is equal to eh at the collocation point) is four as C 1 and C 2 both satisfy the orthogonality condition in Theorem 4.3, but is three at the grid points. Example 5.2. We consider the nonlinear case of Liu and Li’s model (c.f. [14]), which is a degenerate case of NCVIEMPDs (2) with p ¼ 2; I ¼ ½0; 1 and
G1 ðtÞ ¼ 1; q1 ¼
1 ; 2
1 K 1 ðt; sÞ ¼ es sin s; 2
G2 ðtÞ ¼ sin t; q2 ¼
1 ; 4
1 K 2 ðt; sÞ ¼ e3s cos s; 2
1 t ½e ðsin t cos tÞ þ e1 cos 1: 2
f ðtÞ ¼
pffiffiffiffi pffiffiffiffi
ð1Þ We use the piecewise quadratic space S2 ðIh Þ with collocation points C 1 ¼ ð1=4; 1=2; 3=4Þ and C 2 ¼ 51015 ; 1=2; 5þ1015 1 substituting the exact solution and the results are shown in respectively. Here we use fine mesh solutions with h ¼ 5000 −6
−6
10
10 log(||u−uh||∞,col)/log(N)
log(||u−u ||
log(||u−u ||
log(||u−uh||∞,grid)/log(N)
h ∞,grid
−8
10
−10
−10
10
10
−12
−12
10
10
−14
−14
10
10
−16
10
slope=−3 slope=−4
−8
slope=−3 slope=−4
10
)/log(N)
h ∞,col
)/log(N)
−16
1
10
2
10
3
10
ð1Þ
Fig. 1. Example 5.1. The errors for S2
10
1
10
2
10
ðIh Þ, left is by choice C 1 and right is by C 2 .
3
10
10858
K. Zhang et al. / Applied Mathematics and Computation 218 (2012) 10848–10860
−6
−6
10
10 log(||u−uh||∞,col)/log(N)
log(||u−uh||∞,col)/log(N)
log(||u−uh||∞,grid)/log(N)
−7
10
log(||u−uh||∞,grid)/log(N)
−7
10
slope=−3 slope=−4
slope=−3 slope=−4
−8
−8
10
−9
10
10
−9
10
−10
10 −10
10
−11
10 −11
10
−12
10 −12
10
−13
10 −13
10
−14
10
−14
10
−15
10
−15
−16
10
1
2
10
10
3
10
1
10
ð1Þ
Fig. 2. Example 5.2. The errors for S2
2
10
3
10
10
ðIh Þ, left is by choice C 1 and right is by C 2 .
Fig. 2. From the figure, we know that the convergence order of eith (which is equal to eh at the collocation point) is four as C 1 and C 2 both satisfy the orthogonality condition in Theorem 4.3, but is three at the grid points. Example 5.3. Consider the NCVIEMPDs (2) with I ¼ ½0; 1; p ¼ 2 and
1 1 G2 ðtÞ ¼ t2 ; K 1 ðt; sÞ ¼ ; K 2 ðt; sÞ ¼ ; f ðtÞ 4 2 i 1 h i pffiffiffiffiffiffiffiffiffiffiffi 1 h 3 3 5 5 ð1 þ t q1 tÞ2 ð1 þ tÞ2 ¼ 1 þ t þ ð1 þ q1 tÞð1 þ t q1 tÞ2 ð1 þ tÞ2 þ 6 15 i 2 h i 3 3 5 5 1h 2 2 ð1 þ t q2 tÞ2 ð1 þ tÞ2 : þ ð1 þ q2 tÞð1 þ t q2 tÞ ð1 þ tÞ þ 3 15
G1 ðtÞ ¼ t;
pffiffiffiffiffiffiffiffiffiffiffi Here, we let G2 admit polynomial growth, and the exact solution is uðtÞ ¼ 1 þ t . Firstly, we consider a special case with q1 ¼ q2 ¼ 0:9 (i.e. is single proportional delay), and use the piecewise quadratic pffiffiffiffi pffiffiffiffi
ð1Þ space S2 ðIh Þ with the collocation parameters C 3 ¼ ð1=5; 2=5; 3=5Þ and C 2 ¼ 51015 ; 1=2; 5þ1015 respectively. The results are presented in Fig. 3. We can observe that the convergence order of left picture and right picture is three and four at the collocation points respectively. It’s because that C 2 satisfies the orthogonality condition in Theorem 4.3 but C 3 does not, which is consistent with the theoretic result.
−4
−5
10
10 log(||u−u ||
log(||u−u ||
)/log(N)
−6
10
)/log(N)
h ∞,col
h ∞,col
log(||u−uh||∞,grid)/log(N)
log(||u−uh||∞,grid)/log(N)
slope=−3
slope=−3 slope=−4
−8
10
−10
10 −10
10
−12
10
−14
10
−15
1
10
2
10
3
10
ð1Þ
Fig. 3. Example 5.3(a). The errors for S2
10
1
10
2
10
ðIh Þ, left is by choice C 3 and right is by C 2 .
3
10
10859
K. Zhang et al. / Applied Mathematics and Computation 218 (2012) 10848–10860 ð1Þ
Next, we consider NCVIEMPDs (2) with q1 – q2 . In our numerical implementation, we use space S2 ðIh Þ with the pffiffiffiffi pffiffiffiffi
collocation points C 1 ¼ ð1=4; 1=2; 3=4Þ, delay parameters ðq1 ; q2 Þ ¼ ð0:99; 0:8Þ and collocation points C 2 ¼ 51015 ; 1=2; 5þ1015 , delay parameters ðq1 ; q2 Þ ¼ ð0:5; 0:75Þ respectively. The results are shown in Fig. 4. It is easy to see that the convergence order of eith (which is equal to eh at the collocation point) is four as C 1 and C 2 both satisfy the orthogonality condition in Theorem 4.3, but is three at the grid points.
Example 5.4. Consider the NCVIEMPDs (2) with p ¼ 3, that is with three proportional delays. Let
G1 ðtÞ ¼ t;
G2 ðtÞ ¼ sinðtÞ;
f ðtÞ ¼ e þ cosðeq1 t Þ cos 1 þ et cosðeq2 t Þ cos 1 þ e2t cosðeq3 t Þ cos 1 ; t
K 1 ðt; sÞ ¼ etþ2s ;
K 2 ðt; sÞ ¼ e2ðtsÞ ;
K 3 ðt; sÞ ¼ e3tþ2s ; ð1Þ
then the exact solution is uðtÞ ¼ et . In this experiment, we use space S2 ðIh Þ with the collocation parameters pffiffiffiffi pffiffiffiffi
C 2 ¼ 51015 ; 1=2; 5þ1015 , delay parameters ðq1 ; q2 ; q3 Þ ¼ ð0:5; 0:75; 0:9Þ and ðq1 ; q2 ; q3 Þ ¼ ð0:99; 0:9; 0:8Þ. The results are presented in Fig. 5, which clearly exhibit eith (which is equal to eh at the collocation point) can achieve a higher order convergence, order of four.
−5
−4
10
10 log(||u−u ||
)/log(N)
log(||u−u ||
h ∞,col
log(||u−u ||
h ∞,grid
)/log(N)
h ∞,col
)/log(N)
log(||u−u ||
h ∞,grid
slope=−3 slope=−4
)/log(N)
slope=−3 slope=−4
−6
10
−8
10 −10
10
−10
10
−12
10
−15
10
−14
1
10
2
3
10
10
10 ð1Þ
Fig. 4. Example 5.3(b). The errors for S2 ðq1 ; q2 Þ ¼ ð0:5; 0:75Þ.
1
10
2
10
ðIh Þ, left is by choice C 1 ¼ ð1=4; 1=2; 3=4Þ; ðq1 ; q2 Þ ¼ ð0:99; 0:8Þ and right is by C 2 ¼
−5
pffiffiffiffi pffiffiffiffi
5 15 ; 1=2; 5þ1015 ; 10
−5
10
10 log(||u−u ||
)/log(N)
log(||u−u ||
h ∞,col
log(||u−u ||
h ∞,grid
)/log(N)
h ∞,col
)/log(N)
log(||u−u ||
h ∞,grid
slope=−3 slope=−4
)/log(N)
slope=−3 slope=−4
−10
−10
10
10
−15
10
3
10
−15
1
10
2
3
10
10 ð1Þ
Fig. 5. Example 5.4. The errors for S2
10
1
10
2
10
ðIh Þ, left is by choice ðq1 ; q2 ; q3 Þ ¼ ð0:5; 0:75; 0:9Þ and right is by ðq1 ; q2 ; q3 Þ ¼ ð0:99; 0:9; 0:8Þ.
3
10
10860
K. Zhang et al. / Applied Mathematics and Computation 218 (2012) 10848–10860
6. Conclusion remarks and future work In this paper, we present collocation methods for nonlinear convolution Volterra integral equations with multiple proportional delays (NCVIEMPDs). We have shown the existence, uniqueness and regularity properties of analytic solution for (2), and then show that the convergence and superconvergence order of the collocation solution is m and m þ 1 respectively. The numerical results validate our theoretical analysis. The collocation methods presented in this paper can be extended to obtain analogous results for nonlinear convolution Volterra integral equations with singular kernels. Acknowledgments The authors thank the referees for their comments and suggestions that help improve the manuscript. The authors would also like to thank high performance computing center of Jilin University of China. The first author is partially supported by Forefront of Science and Interdisciplinary Innovation Projects of Jilin University and NNSF (No. 11071102 of China). References [1] I. Ali, H. Brunner, T. Tang, Spectral methods for pantograph-type differential and integral equations with multiple delays, Front. Math. China 4 (2009) 49–61. [2] L. Berg, L.V. Wolfersdorf, On a class of generalized autoconvolution equations of the third kind, Z. Anal. Anwend. 24 (2005) 217–250. [3] H. Brunner, Collocation Methods for Volterra Integral and Related Functional Differential Equations, Cambridge University Press, Cambridge, 2004. [4] H. Brunner, Collocation methods for pantograph-type Volterra functional equations with multiple delays, Comput. Methods Appl. Math. 8 (2008) 207– 222. [5] H. Brunner, Q.Y. Hu, Superconvergence of iterated collocation solutions for Volterra integral equations with variable delays, SIAM J. Numer. Anal. 43 (2005) 1934–1949. [6] G. Capobianco, D. Conte, I.D. Prete, E. Russo, Fast Runge–Kutta methods for nonlinear convolution systems of Volterra integral equations, BIT 47 (2007) 259–275. [7] Y.P. Chen, T. Tang, Convergence analysis of the Jacobi spectral-collocation methods for Volterra integral equations with a weakly singular kernel, Math. Comp. 79 (2010) 147–167. [8] J.A. Cuminato, On the uniform convergence of a collocation method for a class of singular integral equations, BIT 27 (1987) 190–202. [9] P.P.B. Eggermont, C. Lubich, Fast numerical solution of singular integral equations, J. Integral Equat. Appl. 6 (1994) 335–351. [10] D. Germund, B. Ake, Numerical Methods in Scientific Computing, Society for Industrial & Applied Mathematics, United States, 2008. [11] Q.G. Guan, Y.K. Zou, R. Zhang, Analysis of collocation solutions for nonstandard Volterra integral equations. IMA Numer. Anal., http://dx.doi.org/ 10.1093/imanum/drr038. [12] F.C. Hoppensteadt, Z. Jackiewicz, K.B. Zubik, Numerical solution of Volterra integral and integro-differential equations with rapidly vanishing convolution kernels, BIT 47 (2007) 325–350. [13] J. Janno, Nonlinear equations with operators satisfying generalized Lipschitz conditions in scales, J. Anal. Appl. 18 (1999) 287–295. [14] M.Z. Liu, D. Li, Properties of analytic solution and numerical solution and multi-pantograph equation, Appl. Math. Comput. 155 (2004) 853–871. [15] J.T. Ma, H. Brunner, A posteriori error estimates of discontinuous Galerkin methods for non-standard Volterra integro-differential equations, IMA Numer. Anal. 26 (2006) 78–95. [16] J.T. Ma, K.L. Xiang, Y.J. Jiang, An integral equation method with high-order collocation implementations for pricing American put options intern, J. Econ. Fin. 2 (2010) 102–112. [17] A. Iserles, On the generalized pantograph functional differential equation, Eur. J. Appl. Math. 4 (1993) 1–38. [18] L.V. Wolfersdorf, Autoconvolution equations of the third kind with Abel integral, J. Integral Equat. Appl. 23 (2011) 113–136.
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Collocation methods for nonlinear convolution Volterra integral equations with multiple proportional delays Kai Zhang ⇑, Jie Li, Haiming Song Department of Mathematics, Jilin University, Changchun, Jilin 130023, PR China
a r t i c l e
i n f o
a b s t r a c t In this paper, we apply the collocation methods to a class of nonlinear convolution Volterra integral equations with multiple proportional delays (NCVIEMPDs). We shall present the existence, uniqueness and regularity properties of analytic solution for this type equation, and then analyze the convergence and superconvergence properties of the collocation solution. The numerical results verify our theoretical analysis. Crown Copyright Ó 2012 Published by Elsevier Inc. All rights reserved.
Keywords: Collocation methods Nonlinear convolution Volterra integral equation Multiple proportional delays
1. Introduction Modeling many problems of physics, economics, stochastics, and other disciplines leads to nonlinear convolution Volterra integral equations with multiple proportional delays, where the rate of change of the process is not only determined by its present state but also by a certain past state. These are usually difficult to solve analytically and in many cases the solutions must be approximated. Numerical methods based on finite difference methods, Runge–Kutta methods, discontinuous Galerkin methods and spectral methods etc., have also been developed for various nonlinear Volterra integral equations and we refer to [1,3,5,7–9,11,17], and references therein for details about the rich literature. In [15], Ma and Brunner derived a posteriori error estimates for nonlinear Volterra integro-differential equations (NVIDEs) possessing the nonstandard memory term, and studied the discontinuous Galerkin (DG) method for NVIDEs with fully discretized memory terms. The assumption of nonstandard memory term in [15] is that the derivatives of nonstandard memory term are bounded. Ma also used high order collocation methods for Black Scholes equation in economics (c.f. [16]) under the frame of previous work. Guan et al. relaxed the assumption on nonstandard memory term to Lipschitz continuity (c.f. [11]), but the Lipschitz function is constrained to small enough in some integral sense, which is little strict to extend. In [6,12], some Volterra integral equations with nonlinear convolution are also studied, and Brunner also gives some convergence and superconvergence results for Volterra functional equations with multiple proportional delays (c.f. [4]). To the best of our knowledge, there are few works about convergence of collocation methods for nonlinear convolution Volterra integral equations with multiple proportional delays, which means the memory is extended from nonlinear term to nonlinear convolution term. Motivated by the work in [15,11], in this paper, we shall study the collocation method for nonlinear convolution Volterra integral equations with multiple delay (or: lag) functions hk ¼ hk ðtÞ; k ¼ 1; 2; . . . ; p of the form
uðtÞ ¼ f ðtÞ þ
p X
V hk u ðtÞ;
t 2 I :¼ ½0; T;
k¼1
where p is some positive integer. The Volterra integral operators V hk ðk ¼ 1; 2; . . . ; pÞ : CðIÞ ! CðIÞ are defined by
V hk u ðtÞ :¼
Z
hk ðtÞ
K k ðt; sÞG1 ðuðt sÞÞG2 ðuðsÞÞds;
0
⇑ Corresponding author. E-mail addresses: [email protected] (K. Zhang), [email protected] (J. Li), [email protected] (H. Song). 0096-3003/$ - see front matter Crown Copyright Ó 2012 Published by Elsevier Inc. All rights reserved. http://dx.doi.org/10.1016/j.amc.2012.04.045
ð1Þ
K. Zhang et al. / Applied Mathematics and Computation 218 (2012) 10848–10860
10849
where f ; G1 ; G2 and K k are given smooth functions. The delay functions hk ðtÞ; k ¼ 1; 2; . . . ; p are assumed to have the following properties: (1) hk ð0Þ ¼ 0, and hk is strictly increasing on I; k t on I for some q k 2 ð0; 1Þ; (2) hk ðtÞ 6 q (3) hk 2 C mk ðIÞ for some integer mk P 0. An important special case is the linear vanishing delay or proportional delay, i.e. hk ðtÞ ¼ qk t ¼ t ð1 qk Þt :¼ t sk ðtÞ with 0 < qk < 1, which are known as the pantograph delay functions (c.f. [17]). We shall concern on the corresponding NCVIEMPDs given by
uðtÞ ¼ f ðtÞ þ
p X
V qk u ðtÞ :¼ f ðtÞ þ LðG1 ðuÞ; G2 ðuÞÞðtÞ;
t 2 I;
ð2Þ
k¼1
where L is an operator which is given by
LðG1 ðuÞ; G2 ðuÞÞðtÞ ¼
p Z X k¼1
qk t
K k ðt; sÞG1 ðuðt sÞÞG2 ðuðsÞÞds :¼ Lu;
t2I
0
Rq t and V qk u ðtÞ :¼ 0 k K k ðt; sÞG1 ðuðt sÞÞG2 ðuðsÞÞds, k ¼ 1; 2; . . . ; p. Here, the nonstandard memory terms G1 ðuðt sÞÞG2 ðuðsÞÞ in each V qk u can be different, we assume they are uniform just for simplification in the rest of this paper. To the best of our knowledge, there exists few work on collocation method for NCVIEMPDs of form (2). In order to gain some insight approaches for nonlinear convolution Volterra integral equations, we present a study of piecewise polynomial collocation solutions for (2). There are several challenges for these NCVIEMPDs: the nonlinear convolution memory term is more tough than nonstandard memory term, and admits polynomial growth; the convergence and superconvergence proofs of integral equation are more difficult than corresponding integro-differential equations; the situations for the multiple proportional delays V qk u in (2) are more complex than single proportional delay. The rest of this paper is organized as follows: In Section 2, the existence, uniqueness and regularity of the analytic solution to (2) is presented. Section 3 is devoted to construct the collocation schemes. The conditions for the uniqueness of numerical schemes, convergence and superconvergence results of collocation scheme are shown in Section 4, and in Section 5, we give some numerical experiments to verify our theoretic results. The last section is devoted to some conclusion remarks and future work. 2. Existence, uniqueness and regularity of the analytic solution In order to show the existence, uniqueness and regularity of the analytic solution, we first introduce weighted norm space and some notations. Suppose that the space CðIÞ is endowed with exponentially weighted norm
kzkr ¼ maxjert zðtÞj; t2I
r P 0; 8z 2 CðIÞ
and kv k1 :¼ max jv ðtÞj. Then there exists positive number t2I
jðrÞ such that
jðrÞkzk1 6 kzkr 6 kzk1 ; z 2 CðIÞ: We assume the operators L and Gi in (2) satisfy following assumptions throughout this paper. Assumption A: There exist positive parameters L; M; N, l and kðrÞ such that, for i ¼ 1; 2 following inequalities hold
kGi ðu1 Þ Gi ðu2 Þkr 6 Mku1 u2 kr ; kGi ðu1 Þ Gi ðu2 Þk1 6 Mku1 u2 k1 ; kGi ðuÞk1 6 Lkukl1 ; kLðv 1 ; v 2 Þkr 6 Nkv 1 kr kv 2 kr ; kLðv 1 ; v 2 Þkr 6 kðrÞ minfkv 1 k1 kv 2 kr ; kv 1 kr kv 2 k1 g with
lim kðrÞ ¼ 0:
r!1
Define the linear operator T : L1 ðIÞ ! L1 ðIÞ by
T ðuÞðtÞ :¼ f ðtÞ þ LðG1 ðuÞ; G2 ðuÞÞðtÞ;
t 2 I;
then the Eq. (2) can be rewritten as
u ¼ T ðuÞ:
ð3Þ
By using the weighted norm technique (c.f. [13,18]), we can have following theorem about the existence and uniqueness of (2) or (3). Theorem 2.1. Assume that the given functions and operators in (2) satisfy (i) f 2 CðIÞ and K k 2 CðDqk Þ with Dqk ¼ fðt; sÞ : 0 6 s 6 qk tg; k ¼ 1; 2; . . . ; p ; (ii) L and Gi ði ¼ 1; 2Þ under the Assumption A.
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K. Zhang et al. / Applied Mathematics and Computation 218 (2012) 10848–10860
Then the NCVIEMPDs (2) has a unique solution u 2 CðIÞ.
Proof. Let Bq;r ðf Þ be a ball given by
Bq;r ðf Þ :¼ fu 2 CðIÞ : ku f kr 6 qg;
ðq > 0;
r P 0; f 2 CðIÞÞ
and we shall prove the conclusion by Banach fixed point theorem in Bq;r ðf Þ. We first show that the operator T : Bq;r ðf Þ ! Bq;r ðf Þ is a contraction mapping with q is small enough and enough. For any v ; v 1 and v 2 2 Bq;r ðf Þ,
ð4Þ
r is large
kT v f kr ¼ kLðG1 ðv Þ G1 ðf Þ þ G1 ðf Þ; G2 ðv Þ G2 ðf Þ þ G2 ðf ÞÞkr 6 kLðG1 ðv Þ G1 ðf Þ; G2 ðv Þ G2 ðf ÞÞkr þ kLðG1 ðv Þ G1 ðf Þ; G2 ðf ÞÞkr þ kLðG1 ðf Þ; G2 ðv Þ G2 ðf ÞÞkr þ kLðG1 ðf Þ; G2 ðf ÞÞkr 6 NM2 kv f k2r þ kðrÞMkG2 ðf Þk1 kv f kr þ kðrÞMkG1 ðf Þk1 kv f kr þ kðrÞkG1 ðf Þk1 kG2 ðf Þk1 6 NM2 q2 þ kðrÞMðkG2 ðf Þk1 þ kG1 ðf Þk1 Þq þ kðrÞkG1 ðf Þk1 kG2 ðf Þk1 and
kT v 1 T v 2 kr ¼ kLðG1 ðv 1 Þ; G2 ðv 1 ÞÞ LðG1 ðv 2 Þ; G2 ðv 2 ÞÞkr 6 kLðG1 ðv 1 Þ G1 ðv 2 Þ; G2 ðv 1 Þ G2 ðf Þ þ G2 ðf ÞÞkr þ kLðG1 ðv 2 Þ G1 ðf Þ þ G1 ðf Þ; G2 ðv 1 Þ G2 ðv 2 ÞÞkr 6 Mkv 1 v 2 kr NMkv 1 f kr þ kðrÞkG2 ðf Þk1 þ Mkv 1 v 2 kr NMkv 2 f kr þ kðrÞkG1 ðf Þk1 h i 6 2NM2 q þ kðrÞM kG2 ðf Þk1 þ kG1 ðf Þk1 kv 1 v 2 kr : From the continuity of Gi and f, we know that kG1 ðf Þk1 and kG2 ðf Þk1 are bounded. Since limr!1 kðrÞ ¼ 0, there exist sufficiently large numbers r1 and small q1 such that when r P r1 and q 6 q1 ; kT v f kr 6 q; kT v 1 T v 2 kr 6 12 kv 1 v 2 kr hold, which imply T is a contraction mapping in Bq;r ðf Þ. Therefore, Eq. (2) has a unique solution in Bq;r ðf Þ. Next, we show the uniqueness of the solution of (2) in the whole space CðIÞ. Suppose that u1 ; u2 2 CðIÞ are two solutions of (2), then
kui f kr ¼ kLðG1 ðui Þ; G2 ðui ÞÞkr 6 kðrÞkG1 ðui Þk1 kG2 ðui Þk1 ;
i ¼ 1; 2:
Since kG1 ðu1 Þk1 and kG2 ðu2 Þk1 are bounded, and limr!1 kðrÞ ¼ 0, we know that
lim kui f kr ¼ 0;
i ¼ 1; 2:
r!1
Thus for any q 6 q1 , there exists a sufficiently large number r2 such that when r P r2 , we have that kui f kr 6 q for i ¼ 1; 2, which implies u1 ; u2 2 Bq;r ðf Þ. Together with T is a contraction mapping in Bq;r ðf Þ, we obtain the uniqueness. h In order to analyze the regularity of the solution of the Eq. (2), we introduce some preliminary lemmas as follows. Lemma 2.1. [3]Consider the multiple delays functional equation
uðtÞ ¼ f ðtÞ þ
p Z X k¼1
qk t
K k ðt; sÞuðt sÞds;
t 2 I:
ð5Þ
0
If f 2 CðIÞ and K k 2 CðDqk Þ, then (5) has a unique solution u 2 CðIÞ. Lemma 2.2. For g 2 C 1 ðRÞ, the following identity holds
gðtÞ gðsÞ ¼ ðt sÞ
Z
1
g 0 ðs þ ðt sÞhÞdh;
8t; s 2 R:
0
Now, we are in the stage of showing the regularity result about the solution of (2). Theorem 2.2. Assume that the given functions in (2) satisfy f 2 C m ðIÞ; Gi 2 C m ðRÞ and K k 2 C m ðDqk Þ; k ¼ 1; 2; . . . ; p, for some integer v P 1, then the NCVIEMPDs (2) has a unique solution u 2 C m ðIÞ. Proof. By Theorem 2.1, we have already known that u 2 CðIÞ. For f 2 C 1 ðIÞ; Gi 2 C 1 ðRÞ and K k 2 C 1 Dqk , differentiate both sides of the Eq. (2) formally leading to
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K. Zhang et al. / Applied Mathematics and Computation 218 (2012) 10848–10860 p X qk K k ðt; qk tÞG1 ðuðt qk tÞÞG2 ðuðqk tÞÞ
u0 ðtÞ ¼ f 0 ðtÞ þ
k¼1
þ
p Z X
qk t
0
k¼1
@K k ðt; sÞG1 ðuðt sÞÞG2 ðuðsÞÞ þ K k ðt; sÞu0 ðt sÞG01 ðuðt sÞÞG2 ðuðsÞÞ ds: @t
Since the existence of a differentiable solution to the Eq. (2) is still unknown, consider the equation
e ðtÞ ¼ f 0 ðtÞ þ u
p X qk K k ðt; qk tÞG1 ðuðt qk tÞÞG2 ðuðqk tÞÞ k¼1
þ
p Z X k¼1
qk t 0
@K k e ðt sÞG01 ðuðt sÞÞG2 ðuðsÞÞ ds: ðt; sÞG1 ðuðt sÞÞG2 ðuðsÞÞ þ K k ðt; sÞ u @t
ð6Þ
e 2 CðIÞ. Lemma 2.1 implies that the above equation has a unique solution u e ðtÞ equals to u0 ðtÞ, that is to show limh!0 u e ðtÞ uðtþhÞuðtÞ Next, we shall prove u ¼ 0. For the simplification, we introduce h notations
x½g; v ; s ¼ sup gðtÞ t2I;0
;
v ðt þ sÞ v ðtÞ s
f ðt þ hÞ f ðtÞ A1 ¼ f 0 ðtÞ ; h " # Z p X 1 qk ðtþhÞ A2 ¼ qk K k ðt; qk tÞG1 ðuðt qk tÞÞG2 ðuðqk tÞÞ K k ðt þ h; sÞG1 ðuðt þ h sÞÞG2 ðuðsÞÞds ; h qk t k¼1 Z p Z qk t X @K k 1 qk t ðK k ðt þ h; sÞ K k ðt; sÞÞG1 ðuðt þ h sÞÞG2 ðuðsÞÞds ; A3 ¼ ðt; sÞG1 ðuðt sÞÞG2 ðuðsÞÞds h 0 @t 0 k¼1 Z Z p q t X k 1 qk t e ðt sÞG01 ðuðt sÞÞG2 ðuðsÞÞds K k ðt; sÞ u K k ðt; sÞG2 ðuðsÞÞðG1 ðuðt þ h sÞÞ G1 ðuðt sÞÞÞds : A4 ¼ h 0 0 k¼1 Then for any t; t þ h 2 ½0; d with sufficiently small d > 0 and h – 0, using (2) and (6), we obtain
uðt þ hÞ uðtÞ u e ðtÞ 6 jA1 j þ jA2 j þ jA3 j þ jA4 j: h
ð7Þ
Noting that
G1 ðuðt þ h sÞÞ G1 ðuðt sÞÞ ¼ ðuðt þ h sÞ uðt sÞÞ
Z
1
0
G01 ðuðt sÞ þ hðuðt þ h sÞ uðt sÞÞÞdh
and K k ; Gi ; G01 ; u are continuous, we can choose a positive number d such that
jA4 j 6
1 x½ ue ; u; h 2
and then taking sup-norm on both sides of (7), we have
1 x½ ue ; u; h 6 jA1 j þ jA2 j þ jA3 j: 2 e ðtÞ ¼ u0 ðtÞ for any t 2 ½0; d. Since f 2 C 1 ðIÞ and u 2 CðIÞ, we know that limh!0 jAi j ¼ 0ði ¼ 1; 2; 3Þ, which imply u e ðtÞ ¼ u0 ðtÞ holds on ½0; ld for some positive integer l, then for t; t þ h 2 ½ld; ðl þ 1Þd, we have Assume u
e ðtÞ u
uðt þ hÞ uðtÞ e4; ¼ A1 þ A 2 þ A 3 þ A h
where
e4 ¼ A
"Z p X k¼1
þ
0
tld
e ðt sÞG01 ðuðt sÞÞG2 ðuðsÞÞds K k ðt; sÞ u
p Z qk t X k¼1
tld
1 h
K k ðt; sÞu0 ðt sÞG01 ðuðt sÞÞG2 ðuðsÞÞds
Z
#
tld
K k ðt; sÞG2 ðuðsÞÞðG1 ðuðt þ h sÞÞ G1 ðuðt sÞÞÞds
0
1 h
Z
qk t
K k ðt; sÞG2 ðuðsÞÞðG1 ðuðt þ h sÞÞ G1 ðuðt sÞÞÞds :
tld
e ðtÞ ¼ u0 ðtÞ on ½ld; ðl þ 1Þd, therefore (2) has a solution u 2 C 1 ðIÞ by Using the same arguments on interval ½0; d, it follows that u mathematical induction. For integer v P 2, by the similar arguments, we can obtain the conclusion. h
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K. Zhang et al. / Applied Mathematics and Computation 218 (2012) 10848–10860
3. Collocation methods Let Ih ¼ ftn ¼ nh; n ¼ 0; . . . ; Ng with tN ¼ Nh ¼ T be a given uniform mesh on I and set hn :¼ ðt n ; t nþ1 ; n ¼ 0; . . . ; N 1. We shall be concerned with the collocation solution uh lying in the piecewise polynomial space ð1Þ
Sm1 ðIh Þ :¼ fv : v jhn 2 P m1 ; 0 6 n 6 N 1g;
ð8Þ
where P m1 ðm P 1Þ denotes the set of polynomials of degree not exceeding m 1. The dimension of the space equals Nm, and it is natural to choose the set of collocation points to be
ð1Þ Sm1 ðIh Þ
X h :¼ ftn;i ¼ tn þ ci h : 0 < c1 < < cm 6 1; n ¼ 0; . . . ; N 1g; since its cardinality is also Nm. Here, fci gm i¼1 is a given set of collocation parameters in ð0; 1. Since
qk tn;i ¼ qk ðt n þ ci hÞ ¼ qk ðnh þ ci hÞ ¼ qk ðn þ ci Þh; let
ðkÞ ðkÞ ðkÞ ðkÞ ðkÞ qk tn;i :¼ qn;i þ cn;i h ¼ qn;i h þ cn;i h ¼ tqðkÞ þ cn;i h 2 tqðkÞ ; tqðkÞ þ1 ; n;i
ðkÞ
ðkÞ
n;i
n;i
ðkÞ
with qn;i ¼ bqk ðn þ ci Þc, cn;i ¼ qk ðn þ ci Þ qn;i 2 ð0; 1. Here for any x 2 R; bxc is the greatest integer not exceeding x. ð1Þ We are looking for uh 2 Sm1 ðIh Þ satisfying the collocation equation
uh ðtÞ ¼ f ðtÞ þ
p X
V qk uh ðtÞ;
t 2 Xh :
ð9Þ
k¼1
Setting U n;j ¼ uh ðt n þ cj hÞðj ¼ 1; . . . ; mÞ, we can express unh (the restriction of uh on interval hn ) by interpolation
uh jhn ¼ unh ðtÞ ¼ uh ðtn þ shÞ ¼
m X Lj ðsÞU n;j ;
0 < s 6 1;
ð10Þ
j¼1
with Lagrange interpolation polynomials
Lj ðsÞ ¼
m Y s ck ; c ck k¼1;k–j j
0 < s 6 1; j ¼ 1; . . . ; m:
ð11Þ
Therefore, the global collocation solution uh on I is given by
uh ðtÞ ¼
N1 X
vn ðtÞunh ðtÞ;
n¼0
where vn ðtÞ is the characteristic function on hn . For each collocation point t n;i 2 X h , the collocation Eq. (9) becomes
uh ðt n;i Þ ¼ f ðt n;i Þ þ
p X
V qk uh ðtn;i Þ;
ð12Þ
k¼1
where
V qk uh ðt n;i Þ ¼ P1 þ P2 þ P 3 : ðkÞ
Z
qn;i 1
¼h
X
ðkÞ qn;i 1
K k t n;i ; t l þ sh G1
X Z
0
m X
! Lp ðci sÞU nl;p G2
p¼1
Z
1
K k ðtn;i ; tl þ shÞG1
ci
l¼0
þh
0
l¼0
þh
ci
cðkÞ n;i
! ! m X Lq ðsÞU l;q ds q¼1
! ! ! m m X X Lp ð1 þ ci sÞU nl1;p G2 Lq ðsÞU l;q ds p¼1
K k ðt n;i ; t qðkÞ n;i
q¼1
! ! m X þ shÞG1 uh tnqðkÞ þ ðci sÞh G2 Lq ðsÞU qðkÞ ;q ds : n;i
n;i
q¼1 ðkÞ
There are two cases for P 3 in ðV qk uh Þðtn;i Þ corresponding to tnqðkÞ þ ðci sÞh, s 2 ð0; cn;i : n;i
ðkÞ
(i) if ci 6 cn;i , then
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K. Zhang et al. / Applied Mathematics and Computation 218 (2012) 10848–10860
ðV qk uh Þðt n;i Þ ¼ P 1 þ P2 þ h þh
Z ci
(ii) if ci > c
ðkÞ n;i ,
cðkÞ n;i
Z
ci
! ! ! m m X X K k t n;i ; tqðkÞ þ sh G1 Lp ðci sÞU nqðkÞ ;p G2 Lq ðsÞU qðkÞ ;q ds n;i
0
n;i
p¼1
q¼1
n;i
! ! ! m m X X K k t n;i ; t qðkÞ þ sh G1 Lp ð1 þ ci sÞU nqðkÞ 1;p G2 Lq ðsÞU qðkÞ ;q ds ; n;i
n;i
p¼1
q¼1
n;i
then
ðV qk uh Þðt n;i Þ ¼ P 1 þ P2 þ h
Z
ðkÞ cn;i
! ! ! m m X X K k t n;i ; t qðkÞ þ sh G1 Lp ðci sÞU nqðkÞ ;p G2 Lq ðsÞU qðkÞ ;q ds : n;i
0
n;i
p¼1
q¼1
n;i
4. Theoretic results for the collocation solution on uniform mesh I h In this section, we shall present the existence and uniqueness of the collocation solution. Using projection operators, we propose a theorem about convergence of collocation method (12). The last part of this section is devoted to some results on superconvergence of collocation method for NCVIEMPDs. 4.1. The existence and uniqueness of the collocation solution For the simplification, we introduce notations
U n ¼ ðU n;1 ; . . . ; U n;m ÞT ;
F n ¼ ðf ðt n þ c1 hÞ; . . . ; f ðt n þ cm hÞÞT ;
n ¼ 0; 1; . . . ; N 1:
Then the existence and uniqueness of (12) is given by following theorem. Theorem 4.1. Assume that the given functions and operators in (2) satisfy (i) f 2 CðIÞ; Gi 2 CðRÞ and K k 2 CðDqk Þ, k ¼ 1; 2; . . . ; p; (ii) Gi under Assumption A. the Eq. (12) defines a unique collocation solution > 0, for any uniform mesh I with h < h, Then there exists a constant h h ð1Þ uh 2 Sm1 for all qk 2 ð0; 1Þ.
Proof. We use mathematical induction method with respect to index n to prove this result. For n ¼ 0, let u ¼ ðu1 ; . . . ; um ÞT . Define the operator K : Rm ! Rm as follows
ðKuÞi ¼ h
p Z X
cðkÞ 0;i
K k ðci h; shÞG1
0
k¼1
! ! m m X X Lp ðci sÞup G2 Lq ðsÞuq ds; p¼1
q¼1
then the Eq. (12) can be rewritten as
ðI KÞU 0 ¼ F 0 ;
ð13Þ
where I denotes the identity operator. Setting Br ð0Þ ¼ fu 2 R : kuk1 6 rg with kuk1 ¼ maxi jui j, then for /; u 2 Br ð0Þ, we have m
(Z ðkÞ ! " ! !# p c0;i m m m X X X X ðK/Þ ðKuÞ 6 h K k ðci h; shÞG1 Lp ðci sÞ/p G2 Lq ðsÞ/q G2 Lq ðsÞuq ds i i 0 p¼1 q¼1 q¼1 k¼1 Z ðkÞ " ! !# ! ) c0;i m m m X X X þ K k ðci h; shÞ G1 Lp ðci sÞ/p G1 Lp ðci sÞup G2 Lq ðsÞuq ds 0 p¼1 p¼1 q¼1 6 hk/ uk1 nR
p n X
o KMðmLÞ1þl Lrl þ KMðmLÞ1þl Lrl ;
k¼1
o where K ¼ maxt2I jK k ðt; sÞjds ; L ¼ maxj kLj k1 and L is the constant defined in Assumption A. For u 2 Br ð0Þ, we have t 0
! ! p Z cðkÞ p m m X X X X 0;i jðKuÞi j 6 h K k ðci h; shÞG1 Lp ðci sÞup G2 Lq ðsÞuq ds 6 h KðmLÞ2l L2 r 2l : 0 p¼1 q¼1 k¼1 k¼1
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K. Zhang et al. / Applied Mathematics and Computation 218 (2012) 10848–10860
~ such that for any h 2 ð0; hÞ, ~ we have Then there exists a positive number h
1 k/ uk1 2
kK/ Kuk1 6
and kKuk1 6 r:
Therefore, we know that K : Br ð0Þ ! Br ð0Þ is a contraction map. Hence the operator I K has a bounded inverse, which implies (12) has a unique bounded solution U 0 . Assume (12) has a unique bounded solution on n 6 l 1 for some positive integer l, we want to prove that it is also true for n ¼ l. Noting that when n – 0 and s 2 ð0; qk tn;i Þ, the points tn;i s and s can not belong to ðtn ; tnþ1 simultaneously. For n ¼ l, define the operator K on Rm as follows
ðKU l Þi ¼ f ðtl;i Þ þ
p Z X k¼1
qk t l;i
K k ðt l;i ; sÞG1 ðuh ðt l;i sÞÞG2 ðuh ðsÞÞds:
0
For /; u 2 Rm , we get
" ! !# ! m m m X X X jðK/Þi ðKuÞi j 6 h Lp ðci sÞ/p G1 Lp ðci sÞup Lq ðsÞU 0;q ds G2 K k ðtl;i ; shÞ G1 0 p¼1 p¼1 q¼1 k¼1 Z ðkÞ ! " ! !# ) cn;i m m m X X X þ K k tl;i ; tqðkÞ þ sh G1 Lp ðci sÞU 0;p G2 Lq ðsÞ/q G2 Lq ðsÞuq ds 0 l;i p¼1 q¼1 q¼1 (Z p X
ci
6 hk/ uk1
p n o X KMðmLÞ1þl LkU 0 kl1 þ KMðmLÞ1þl LkU 0 kl1 : k¼1
such that for any h 2 ð0; hÞ ð0; hÞ, ~ we have Then there exists a positive number h
kK/ Kuk1 6
1 k/ uk1 : 2
Hence K is a contraction map on Rm for n ¼ l, which implies there exists a unique solution U l in Rm as n ¼ l. This completes the proof. h 4.2. The convergence results for collocation solution ð1Þ
We now analyze the convergence order of the collocation solution uh 2 Sm1 ðIh Þ for the NCVIEMPDs (2) by using projection operators. ð1Þ
Theorem 4.2. Let uh 2 Sm1 ðIh Þ is the collocation solution defined in (12). Assume that the given functions and operators in (2) satisfy (i) f 2 CðIÞ; Gi 2 CðRÞ and K k 2 CðDqk Þ, k ¼ 1; 2; . . . ; p; (ii) Gi under Assumption A. Then for all sufficiently small h > 0, we have
ku uh k1 6 C kðI P h Þf k1 þ kðI P h ÞLuk1 ;
ð14Þ
where P h is the Lagrange interpolation operator corresponding to the collocation parameters fci g, and the constant C is independent on h. Furthermore, if the exact solution u 2 W m;1 ðIÞ, we obtain m
ku uh k1 6 Ch kukm;1 ; where kv km;1
j :¼ max sup d v ðtÞ . dt j 06j6m
t2I
Proof. The operator formulations for NCVIEMPDs (2) and its collocation Eq. (12) are given by
u ¼ f þ Lu; uh ¼ P h f þ P h Luh :
Based on the solvability of the NCVIEMPDs and its collocation equation, we obtain
ð15Þ
K. Zhang et al. / Applied Mathematics and Computation 218 (2012) 10848–10860
(
10855
u ¼ ðI LÞ1 f ; uh ¼ ðI P h LÞ1 P h f :
The error between u and uh can be expressed by
eh :¼ u uh ¼ ðI LÞ1 f ðI P h LÞ1 P h f ¼ ðI LÞ1 P h f ðI P h LÞ1 P h f þ ðI LÞ1 ðf P h f Þ ¼ ðI P h LÞ1 ðK P h LÞðI LÞ1 P h f þ ðI LÞ1 ðf P h f Þ ¼ ðI P h LÞ1 ðL P h LÞðI LÞ1 ðP h f f Þ þ ðI P h LÞ1 ðK P h LÞðI LÞ1 f þ ðI LÞ1 ðI P h Þf ¼ ðI P h LÞ1 ðL P h LÞðI LÞ1 ðP h I Þf þ ðI P h LÞ1 ðI P h ÞLu þ ðI LÞ1 ðI P h Þf ; which implies
ku uh k1 6 C kðI P h Þf k1 þ kðI P h ÞLuk1 : If u 2 W m;1 , from the error estimates of the interpolation operator P h , we know that m
m
kðI P h Þf k1 6 Ch kf km;1 6 Ch kukm;1 ; m
m
kðI P h ÞLuk1 6 Ch kLukm;1 6 Ch kukm;1 ; which leads to m
ku uh k1 6 Ch kukm;1 : h 4.3. The superconvergence results for collocation solution In the rest of this section, we discuss the superconvergence of collocation method for NCVIEMPDs. Define the iterated collocation solution uith associated with uh by
uith ðtÞ ¼ f ðtÞ þ
p X ðV qk uh ÞðtÞ;
ð16Þ
k¼1
then the iterated error eith ¼ u uith is given by
eith ðtÞ ¼ eh ðtÞ dh ðtÞ ¼
p X
ðV qk uÞðtÞ ðV qk uh ÞðtÞ ;
k¼1
where
dh ðtÞ :¼ uh ðtÞ þ f ðtÞ þ
p X ðV qk uh ÞðtÞ: k¼1
In particular, for the collocation points, we have dh ðtÞ ¼ 0. Hence, we obtain eith ðtÞ ¼ eh ðtÞ for t 2 X h . In order to study superconvergence, we present some notations and preliminary lemmas firstly. For given abscissas a 6 n1 < < nm 6 b, define the quadrature formula
Q m ðgÞ :¼
m X
xj gðnj Þ;
j¼1
Rb to approximate the integral Q ðgÞ ¼ a xðtÞgðtÞdt, where the weight function xðtÞ is assumed to satisfy xðtÞ 2 L1 ½a; b. Let the corresponding error be denoted by Em ðgÞ ¼ Q ðgÞ Q m ðgÞ. We have following indentity for the quadrature formula error Em ðgÞ. Lemma 4.1. (c.f. [10]) Assume the quadrature formula Q m ðgÞ has algebraic degree of precision a P 1 and g 2 C d ½a; b with 1 6 d 6 a þ 1. Then Em ðgÞ possesses the integral representation
Em ðgÞ ¼
Z
b
K d ðsÞg ðdÞ ðsÞds;
a
where K d is given by
K d ðsÞ ¼
1 ðd 1Þ!
(Z
b a
xðzÞðz sÞd1 þ dz
m X
)
: xk ðnk sÞd1 þ
k¼1
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K. Zhang et al. / Applied Mathematics and Computation 218 (2012) 10848–10860
Consider the following integral equation
yðtÞ ¼ f ðtÞ þ
Z
t
Kðt; sÞyðsÞds;
t 2 I :¼ ½0; T;
ð17Þ
0
where f and K are bounded and integrable on I and D :¼ fðt; sÞ : 0 6 s 6 t 6 Tg respectively. Define iterated kernels by K 1 ðt; sÞ :¼ Kðt; sÞ and
Z
K n ðt; sÞ :¼
t
K 1 ðt; v ÞK n1 ðv ; sÞdv ;
n P 2:
ð18Þ
s
Lemma 4.2. (c.f. [3]) Set p 1 X X K n ðt; sÞ ¼ lim K n ðt; sÞ ¼: Rðt; sÞ; p!1
n¼1
ðt; sÞ 2 D:
n¼1
Then the so-called resolvent kernel Rðt; sÞ converges absolutely and uniformly in D, and satisfies
Z
Z
t
Kðt; sÞyðsÞds ¼
0
t
Rðt; sÞf ðsÞds:
0
Now, we are ready to show the superconvergence result about the iterated collocation solution. Theorem 4.3. Assume that the given functions in (2) satisfy f 2 C mþ1 ðIÞ; Gi 2 C mþ1 ðDÞ; i ¼ 1; 2 and K k 2 C mþ1 ðDqk Þ; k ¼ 1; 2; . . . ; p. If the collocation parameters fci g are chosen so that the orthogonality condition
J 0 :¼
Z
m 1Y
0
ðs ci Þds ¼ 0
i¼1
holds, then the iterated collocation solution defined by (16) exhibits the superconvergence on the entire interval I:
ku uith k1 6 Ch
mþ1
;
where the constant C is independent on h. Proof. Using the definitions of eh ðtÞ and dh ðtÞ, we know that
eh ðtÞ ¼ dh ðtÞ þ
p Z X
K k ðt; sÞðG1 ðuðt sÞÞ G1 ðuh ðt sÞÞÞG2 ðuðsÞÞds þ
0
k¼1
¼ dh ðtÞ þ
¼ dh ðtÞ þ
Z
t
qk t
K k ðt; sÞ/ðt; sÞG2 ðuðsÞÞeh ðt sÞds þ 0
(Z
p X k¼1
¼ dh ðtÞ þ
Z
1
0
uðt; sÞ ¼
Z
1
0
e ðt; sÞ ¼ K
K k ðt; sÞG1 ðuh ðt sÞÞðG2 ðuðsÞÞ G2 ðuðsÞÞÞds
qk t
K k ðt; sÞG1 ðuh ðt sÞÞuðt; sÞeh ðsÞds
0
K k ðt; t sÞ/ðt; t sÞG2 ðuðt sÞÞeh ðsÞds þ
tqk t
Z 0
e ðt; sÞeh ðsÞds ¼ dh ðtÞ þ K
Z
t
Rðt; sÞdh ðsÞds;
0
G01 ðuh ðt sÞ þ hðuðt sÞ uh ðt sÞÞÞdh; G02 ðuh ðsÞ þ hðuðsÞ uh ðsÞÞÞdh;
p X
Z
t
0
/ðt; sÞ ¼
qk t
0
p Z X k¼1
where
Z
qk t
xIk K Ik ðt; sÞ þ xIIk K IIk ðt; sÞ ;
k¼1
K Ik ðt; sÞ ¼ K k ðt; t sÞ/ðt; t sÞG2 ðuðt sÞÞ; K IIk ðt; sÞ ¼ K k ðt; sÞG1 ðuh ðt sÞÞuðt; sÞ;
0; s 2 ð0; t qk tÞ; 1; s 2 ð0; qk tÞ; xIk ¼ xIIk ¼ 1; s 2 ðt qk t; tÞ; 0; s 2 ðqk t; tÞ:
qk t
) K k ðt; sÞG1 ðuh ðt sÞÞuðt; sÞeh ðsÞds
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K. Zhang et al. / Applied Mathematics and Computation 218 (2012) 10848–10860 m
Using Theorem 4.2, it is easy to see that dh ðtÞ ¼ Oðh Þ. For t ¼ t n þ v h, we get
Z
t
Rðt; sÞdh ðsÞds ¼
0
n1 Z X h l¼0
1
Rðt; t l þ shÞdh ðtl þ shÞds þ h
Z v
Rðt; tn þ shÞdh ðt n þ shÞds:
0
0
Suppose the integrals in above equality are approximated by the interpolatory m-point quadrature formula with abscissas fci g, then we obtain
Z
1
Rðt; tl þ shÞdh ðt l þ shÞds ¼
0
m X bj Rðt; tl þ cj hÞdh ðt l þ cj hÞ þ EðlÞ n ðv Þ;
v 2 ½0; 1;
j¼1
R1 where bj ¼ 0 Lj ðsÞds and EðlÞ n ðv Þ denote the quadrature errors induced by these quadrature approximations. The orthogonalmþ1 ity condition J 0 ¼ 0 and Lemma 4.1 imply that the induced quadrature errors EnðlÞ ðv Þ ¼ Oðh Þ, which leads to
Z
t
Rðt; sÞdh ðsÞds ¼ Oðh
mþ1
Þ:
0
Together with the relation eith ðtÞ ¼ eh ðtÞ dh ðtÞ, we get the conclusion.
h
5. Numerical examples In this section, we apply the collocation methods described in Section 3 to several NCVIEMPDs examples. Results of the numerical simulations verify our convergence and superconvergence analysis in Section 4. Example 5.1. We consider the Bernstein and Doetsch equation (c.f. [2]), which is a degenerate case of NCVIEMPDs (2) with p ¼ 1; I ¼ ½0; 1 and
G1 ðtÞ ¼ t;
G2 ðtÞ ¼ t;
K 1 ðt; sÞ ¼
1 ; t
f ðtÞ ¼ 0:
pffiffiffiffi pffiffiffiffi
ð1Þ We use the piecewise quadratic space S2 ðIh Þ with collocation points C 1 ¼ ð1=4; 1=2; 3=4Þ and C 2 ¼ 51015 ; 1=2; 5þ1015 1 substituting the exact solution and the results are shown in respectively. Here we use fine mesh solutions with h ¼ 5000 Fig. 1. From the figure, we know that the convergence order of eith (which is equal to eh at the collocation point) is four as C 1 and C 2 both satisfy the orthogonality condition in Theorem 4.3, but is three at the grid points. Example 5.2. We consider the nonlinear case of Liu and Li’s model (c.f. [14]), which is a degenerate case of NCVIEMPDs (2) with p ¼ 2; I ¼ ½0; 1 and
G1 ðtÞ ¼ 1; q1 ¼
1 ; 2
1 K 1 ðt; sÞ ¼ es sin s; 2
G2 ðtÞ ¼ sin t; q2 ¼
1 ; 4
1 K 2 ðt; sÞ ¼ e3s cos s; 2
1 t ½e ðsin t cos tÞ þ e1 cos 1: 2
f ðtÞ ¼
pffiffiffiffi pffiffiffiffi
ð1Þ We use the piecewise quadratic space S2 ðIh Þ with collocation points C 1 ¼ ð1=4; 1=2; 3=4Þ and C 2 ¼ 51015 ; 1=2; 5þ1015 1 substituting the exact solution and the results are shown in respectively. Here we use fine mesh solutions with h ¼ 5000 −6
−6
10
10 log(||u−uh||∞,col)/log(N)
log(||u−u ||
log(||u−u ||
log(||u−uh||∞,grid)/log(N)
h ∞,grid
−8
10
−10
−10
10
10
−12
−12
10
10
−14
−14
10
10
−16
10
slope=−3 slope=−4
−8
slope=−3 slope=−4
10
)/log(N)
h ∞,col
)/log(N)
−16
1
10
2
10
3
10
ð1Þ
Fig. 1. Example 5.1. The errors for S2
10
1
10
2
10
ðIh Þ, left is by choice C 1 and right is by C 2 .
3
10
10858
K. Zhang et al. / Applied Mathematics and Computation 218 (2012) 10848–10860
−6
−6
10
10 log(||u−uh||∞,col)/log(N)
log(||u−uh||∞,col)/log(N)
log(||u−uh||∞,grid)/log(N)
−7
10
log(||u−uh||∞,grid)/log(N)
−7
10
slope=−3 slope=−4
slope=−3 slope=−4
−8
−8
10
−9
10
10
−9
10
−10
10 −10
10
−11
10 −11
10
−12
10 −12
10
−13
10 −13
10
−14
10
−14
10
−15
10
−15
−16
10
1
2
10
10
3
10
1
10
ð1Þ
Fig. 2. Example 5.2. The errors for S2
2
10
3
10
10
ðIh Þ, left is by choice C 1 and right is by C 2 .
Fig. 2. From the figure, we know that the convergence order of eith (which is equal to eh at the collocation point) is four as C 1 and C 2 both satisfy the orthogonality condition in Theorem 4.3, but is three at the grid points. Example 5.3. Consider the NCVIEMPDs (2) with I ¼ ½0; 1; p ¼ 2 and
1 1 G2 ðtÞ ¼ t2 ; K 1 ðt; sÞ ¼ ; K 2 ðt; sÞ ¼ ; f ðtÞ 4 2 i 1 h i pffiffiffiffiffiffiffiffiffiffiffi 1 h 3 3 5 5 ð1 þ t q1 tÞ2 ð1 þ tÞ2 ¼ 1 þ t þ ð1 þ q1 tÞð1 þ t q1 tÞ2 ð1 þ tÞ2 þ 6 15 i 2 h i 3 3 5 5 1h 2 2 ð1 þ t q2 tÞ2 ð1 þ tÞ2 : þ ð1 þ q2 tÞð1 þ t q2 tÞ ð1 þ tÞ þ 3 15
G1 ðtÞ ¼ t;
pffiffiffiffiffiffiffiffiffiffiffi Here, we let G2 admit polynomial growth, and the exact solution is uðtÞ ¼ 1 þ t . Firstly, we consider a special case with q1 ¼ q2 ¼ 0:9 (i.e. is single proportional delay), and use the piecewise quadratic pffiffiffiffi pffiffiffiffi
ð1Þ space S2 ðIh Þ with the collocation parameters C 3 ¼ ð1=5; 2=5; 3=5Þ and C 2 ¼ 51015 ; 1=2; 5þ1015 respectively. The results are presented in Fig. 3. We can observe that the convergence order of left picture and right picture is three and four at the collocation points respectively. It’s because that C 2 satisfies the orthogonality condition in Theorem 4.3 but C 3 does not, which is consistent with the theoretic result.
−4
−5
10
10 log(||u−u ||
log(||u−u ||
)/log(N)
−6
10
)/log(N)
h ∞,col
h ∞,col
log(||u−uh||∞,grid)/log(N)
log(||u−uh||∞,grid)/log(N)
slope=−3
slope=−3 slope=−4
−8
10
−10
10 −10
10
−12
10
−14
10
−15
1
10
2
10
3
10
ð1Þ
Fig. 3. Example 5.3(a). The errors for S2
10
1
10
2
10
ðIh Þ, left is by choice C 3 and right is by C 2 .
3
10
10859
K. Zhang et al. / Applied Mathematics and Computation 218 (2012) 10848–10860 ð1Þ
Next, we consider NCVIEMPDs (2) with q1 – q2 . In our numerical implementation, we use space S2 ðIh Þ with the pffiffiffiffi pffiffiffiffi
collocation points C 1 ¼ ð1=4; 1=2; 3=4Þ, delay parameters ðq1 ; q2 Þ ¼ ð0:99; 0:8Þ and collocation points C 2 ¼ 51015 ; 1=2; 5þ1015 , delay parameters ðq1 ; q2 Þ ¼ ð0:5; 0:75Þ respectively. The results are shown in Fig. 4. It is easy to see that the convergence order of eith (which is equal to eh at the collocation point) is four as C 1 and C 2 both satisfy the orthogonality condition in Theorem 4.3, but is three at the grid points.
Example 5.4. Consider the NCVIEMPDs (2) with p ¼ 3, that is with three proportional delays. Let
G1 ðtÞ ¼ t;
G2 ðtÞ ¼ sinðtÞ;
f ðtÞ ¼ e þ cosðeq1 t Þ cos 1 þ et cosðeq2 t Þ cos 1 þ e2t cosðeq3 t Þ cos 1 ; t
K 1 ðt; sÞ ¼ etþ2s ;
K 2 ðt; sÞ ¼ e2ðtsÞ ;
K 3 ðt; sÞ ¼ e3tþ2s ; ð1Þ
then the exact solution is uðtÞ ¼ et . In this experiment, we use space S2 ðIh Þ with the collocation parameters pffiffiffiffi pffiffiffiffi
C 2 ¼ 51015 ; 1=2; 5þ1015 , delay parameters ðq1 ; q2 ; q3 Þ ¼ ð0:5; 0:75; 0:9Þ and ðq1 ; q2 ; q3 Þ ¼ ð0:99; 0:9; 0:8Þ. The results are presented in Fig. 5, which clearly exhibit eith (which is equal to eh at the collocation point) can achieve a higher order convergence, order of four.
−5
−4
10
10 log(||u−u ||
)/log(N)
log(||u−u ||
h ∞,col
log(||u−u ||
h ∞,grid
)/log(N)
h ∞,col
)/log(N)
log(||u−u ||
h ∞,grid
slope=−3 slope=−4
)/log(N)
slope=−3 slope=−4
−6
10
−8
10 −10
10
−10
10
−12
10
−15
10
−14
1
10
2
3
10
10
10 ð1Þ
Fig. 4. Example 5.3(b). The errors for S2 ðq1 ; q2 Þ ¼ ð0:5; 0:75Þ.
1
10
2
10
ðIh Þ, left is by choice C 1 ¼ ð1=4; 1=2; 3=4Þ; ðq1 ; q2 Þ ¼ ð0:99; 0:8Þ and right is by C 2 ¼
−5
pffiffiffiffi pffiffiffiffi
5 15 ; 1=2; 5þ1015 ; 10
−5
10
10 log(||u−u ||
)/log(N)
log(||u−u ||
h ∞,col
log(||u−u ||
h ∞,grid
)/log(N)
h ∞,col
)/log(N)
log(||u−u ||
h ∞,grid
slope=−3 slope=−4
)/log(N)
slope=−3 slope=−4
−10
−10
10
10
−15
10
3
10
−15
1
10
2
3
10
10 ð1Þ
Fig. 5. Example 5.4. The errors for S2
10
1
10
2
10
ðIh Þ, left is by choice ðq1 ; q2 ; q3 Þ ¼ ð0:5; 0:75; 0:9Þ and right is by ðq1 ; q2 ; q3 Þ ¼ ð0:99; 0:9; 0:8Þ.
3
10
10860
K. Zhang et al. / Applied Mathematics and Computation 218 (2012) 10848–10860
6. Conclusion remarks and future work In this paper, we present collocation methods for nonlinear convolution Volterra integral equations with multiple proportional delays (NCVIEMPDs). We have shown the existence, uniqueness and regularity properties of analytic solution for (2), and then show that the convergence and superconvergence order of the collocation solution is m and m þ 1 respectively. The numerical results validate our theoretical analysis. The collocation methods presented in this paper can be extended to obtain analogous results for nonlinear convolution Volterra integral equations with singular kernels. Acknowledgments The authors thank the referees for their comments and suggestions that help improve the manuscript. The authors would also like to thank high performance computing center of Jilin University of China. The first author is partially supported by Forefront of Science and Interdisciplinary Innovation Projects of Jilin University and NNSF (No. 11071102 of China). References [1] I. Ali, H. Brunner, T. Tang, Spectral methods for pantograph-type differential and integral equations with multiple delays, Front. Math. China 4 (2009) 49–61. [2] L. Berg, L.V. Wolfersdorf, On a class of generalized autoconvolution equations of the third kind, Z. Anal. Anwend. 24 (2005) 217–250. [3] H. Brunner, Collocation Methods for Volterra Integral and Related Functional Differential Equations, Cambridge University Press, Cambridge, 2004. [4] H. Brunner, Collocation methods for pantograph-type Volterra functional equations with multiple delays, Comput. Methods Appl. Math. 8 (2008) 207– 222. [5] H. Brunner, Q.Y. Hu, Superconvergence of iterated collocation solutions for Volterra integral equations with variable delays, SIAM J. Numer. Anal. 43 (2005) 1934–1949. [6] G. Capobianco, D. Conte, I.D. Prete, E. Russo, Fast Runge–Kutta methods for nonlinear convolution systems of Volterra integral equations, BIT 47 (2007) 259–275. [7] Y.P. Chen, T. Tang, Convergence analysis of the Jacobi spectral-collocation methods for Volterra integral equations with a weakly singular kernel, Math. Comp. 79 (2010) 147–167. [8] J.A. Cuminato, On the uniform convergence of a collocation method for a class of singular integral equations, BIT 27 (1987) 190–202. [9] P.P.B. Eggermont, C. Lubich, Fast numerical solution of singular integral equations, J. Integral Equat. Appl. 6 (1994) 335–351. [10] D. Germund, B. Ake, Numerical Methods in Scientific Computing, Society for Industrial & Applied Mathematics, United States, 2008. [11] Q.G. Guan, Y.K. Zou, R. Zhang, Analysis of collocation solutions for nonstandard Volterra integral equations. IMA Numer. Anal., http://dx.doi.org/ 10.1093/imanum/drr038. [12] F.C. Hoppensteadt, Z. Jackiewicz, K.B. Zubik, Numerical solution of Volterra integral and integro-differential equations with rapidly vanishing convolution kernels, BIT 47 (2007) 325–350. [13] J. Janno, Nonlinear equations with operators satisfying generalized Lipschitz conditions in scales, J. Anal. Appl. 18 (1999) 287–295. [14] M.Z. Liu, D. Li, Properties of analytic solution and numerical solution and multi-pantograph equation, Appl. Math. Comput. 155 (2004) 853–871. [15] J.T. Ma, H. Brunner, A posteriori error estimates of discontinuous Galerkin methods for non-standard Volterra integro-differential equations, IMA Numer. Anal. 26 (2006) 78–95. [16] J.T. Ma, K.L. Xiang, Y.J. Jiang, An integral equation method with high-order collocation implementations for pricing American put options intern, J. Econ. Fin. 2 (2010) 102–112. [17] A. Iserles, On the generalized pantograph functional differential equation, Eur. J. Appl. Math. 4 (1993) 1–38. [18] L.V. Wolfersdorf, Autoconvolution equations of the third kind with Abel integral, J. Integral Equat. Appl. 23 (2011) 113–136.