Collocation methods for Volterra functional integral ...

Applied Mathematics and Computation 296 (2017) 198–214

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Applied Mathematics and Computation journal homepage: www.elsevier.com/locate/amc

Collocation methods for Volterra functional integral equations with non-vanishing delaysR Wanyuan Ming a,b, Chengming Huang a,∗ a b

School of Mathematics and Statistics, Huazhong University of Science and Technology, Wuhan 430074, PR China School of Mathematics and Information Sciences, Nanchang Hangkong University, Nanchang 330063, PR China

a r t i c l e

i n f o

Keywords: Volterra functional integral equations Non-vanishing delays Collocation methods Optimal order of superconvergence θ -invariant meshes

a b s t r a c t In this paper the existence, uniqueness, regularity properties, and in particular, the local representation of solutions for general Volterra functional integral equations with nonvanishing delays, are investigated. Based on the solution representation, we detailedly analyze the attainable (global and local) convergence order of (iterated) collocation solutions on θ -invariant meshes. It turns out that collocation at the m Gauss (-Legendre) points neither leads to the optimal global convergence order m + 1, nor yields the local convergence order 2m on the whole interval, which is in sharp contrast to the case of the classical Volterra delay integral equations. However, if the collocation is based on the m Radau II points, the local superconvergence order 2m − 1 will exhibit at all mesh points. Finally, some numerical experiments are performed to confirm our theoretical findings. © 2016 Elsevier Inc. All rights reserved.

1. Introduction Consider the following general Volterra functional integral equation (VFIE)

y(t ) = g(t ) + b(t )y(θ (t )) + (Vy )(t ) + (Vθ y )(t ), y(t ) = φ (t ),

t ∈ I := (t0 , T ],

t ∈ Iθ := [θ (t0 ), t0 ],

(1.1)

where the Volterra operators V and Vθ are given by

(Vy )(t ) :=

and

(Vθ y )(t ) :=

t

t0

K1 (t, s )y(s )ds, t ∈ I,

θ (t ) t0

K2 (t, s )y(s )ds, t ∈ I,

(1.2)

(1.3)

respectively. The kernel functions K1 (t, s) and K2 (t, s) in (1.2) and (1.3) are supposed to be continuous on their respective domains D := {(t, s): t0 ≤ s ≤ t ≤ T, t ∈ I} and Dθ := {(t, s): θ (t0 ) ≤ s ≤ θ (t), t ∈ I}. R ∗

Supported by the National Natural Science Foundation of China (Grant Nos. 11371157 and 11401294). Corresponding author. E-mail addresses: [email protected] (W. Ming), [email protected] (C. Huang).

http://dx.doi.org/10.1016/j.amc.2016.10.021 0 096-30 03/© 2016 Elsevier Inc. All rights reserved.

W. Ming, C. Huang / Applied Mathematics and Computation 296 (2017) 198–214

199

Obviously, if we set b(t) ≡ 0 in Eq. (1.1), then it reduces to the following Volterra integral equation (VIE)

y(t ) = g(t ) + (Vy )(t ) + (Vθ y )(t ),

t ∈ I.

(1.4)

The regularity of the exact solution, and the convergence properties of collocation solutions to (1.4) were well studied by Brunner (cf. [4], pp. 198–231). We will compare most of our results for Eq. (1.1) with the ones for Eq. (1.4) in the subsequent sections of this paper. Eq. (1.1) can be obtained from the differentiated form of first-kind Volterra integral equation

(Vy )(t ) + (Vθ y )(t ) = g(t ),

t ∈ I.

(1.5)

In addition, if K2 (t, s ) = −K1 (t, s ) on Dθ , VFIE (1.1) then reduces to

y(t ) = g(t ) + b(t )y(θ (t )) + (Wθ y )(t ),

with

(Wθ y )(t ) :=

t

θ (t )

K (t, s )y(s )ds,

t ∈ I,

(1.6)

t ∈ I.

This equation can be found in mathematical models in epidemiology, population growth and relevant phenomena in biology (cf. [16]). There have been a lot of papers concerning numerical methods for vanishing delay Volterra integral equations and differential equations (cf. [2,3,6–13,15,18–20,23,24]). More recently, Xie et al. [21] studied the attainable order of convergence of collocation solutions on uniform meshes for the vanishing delay VFIE

y(t ) = g(t ) + b(t )y(θ (t )) +

t 0

K1 (t, s )y(s )ds +

θ (t ) 0

K2 (t, s )y(s )ds,

t ∈ [0, T ],

(1.7)

where θ (0 ) = 0. They found that the iterated collocation solution does not possess local superconvergence at the mesh points. However, to the best of our knowledge, the numerical analysis for non-vanishing delay VFIE (1.1) with b(t ) ≡ 0 has not yet been studied. The main difficulty lies in the fact that the representation of the collocation error, which will play a key role in the analysis of superconvergence of collocation solutions (cf. [5]), is not yet understood. It is the aim of the present paper to derive such a solution representation, and use it to analyze the impact of the delay term b(t)y(θ (t)) on the convergence results for VFIE (1.1). It will be shown in this paper that our results are not only a generalization of the corresponding results for (1.4), but also indicate that the delay term b(t)y(θ (t)) will lead to (i) a lower regularity of the exact solution to (1.1) at the primary discontinuity points; (ii) a much more tedious deduction, and also a more complicated expression of the local representation of the exact solution to (1.1); (iii) a reduction in the convergence order of collocation solutions to (1.1). In addition, we will also show that collocation solutions may exhibit local superconvergence properties at certain nonmesh points. This paper is organized as follows. In Section 2, we discuss the existence, uniqueness, regularity, and local representation of the exact solution to Eq. (1.1). Some (iterated) collocation approximations to (1.1) are constructed in Section 3. Section 4 focuses on the attainable (global and local) convergence order of collocation solutions. In Section 5, we present several numerical experiments to illustrate our theoretical results. Finally, some concluding remarks are given in Section 6. 2. Existence, uniqueness, regularity, and representation of the exact solution We will assume that the delay function θ satisfies (D1) θ (t ) := t − τ (t ), τ ∈ Cd (I) for some d ≥ 0; (D2) θ is strictly increasing on I; (D3) τ (t) ≥ τ 0 > 0 for t ∈ I. It is well known that if the delay τ (t) satisfies condition (D3), then it will induce the so-called primary discontinuity points {ξ μ }, and these points can be generated by the recursion

θ (ξμ+1 ) = ξμ ,

μ = 0, 1, . . . ;

ξ0 := t0 .

(2.1)

Moreover, condition (D2) ensures that {ξ μ } is strictly increasing. For simplicity and without loss of generality, we assume that T = ξM (or, ξM < T < ξM+1 for some positive integer M). Since solutions of non-vanishing delay problems generally suffer from a loss of regularity at {ξ μ }, we divide the whole interval I into several ‘macro-intervals’ I (μ ) := (ξμ , ξμ+1 ] (0 ≤ μ ≤ M − 1 ) by {ξ μ }, and denote

Zμ :=

ξμ : t0 = ξ0 < ξ1 < · · · < ξM = T , θ (ξμ+1 ) = ξμ .

as the set of these primary discontinuity points. In order to study the contribution of the delay term b(t)y(θ (t)) in (1.1) on some well-posedness of the exact solution, and also on the convergence results of collocation solutions, we will assume that b(t ) ≡ 0 in Eq. (1.1) hereafter in this paper.

200


2.1. Existence and uniqueness of the exact solution Theorem 2.1. Suppose that the given data describing VFIE (1.1) are continuous on their respective domains, and θ is subject to conditions (D1)–(D3). Then there exists a unique (bounded) solution y(t) solving Eq. (1.1) on I for any initial function φ (t) ∈ C(Iθ ). In addition, this solution is continuous on each macro-interval I (μ ) = (ξμ , ξμ+1 ] (μ ≥ 0 ), but it has a finite (jump) discontinuity at each point {ξ μ } (μ ≥ 0) in general. Furthermore, (i) y(t) is continuous at t = t0 if and only if

g(t0 ) + b(t0 )φ (θ (t0 )) −

t0

θ (t0 )

K2 (t0 , s )φ (s )ds = φ (t0 ).

(2.2)

(ii) y(t) is continuous at t = ξμ (μ ≥ 1 ) if b(ξμ ) = 0 or y(t) is continuous at t = ξμ−1 . Moreover, if condition (2.2) holds, then y(t) is continuous at all points {ξμ } (μ = 0, 1, . . . , M ). Proof. For t ∈ I(μ) (μ ≥ 0), we can recast Eq. (1.1) as

y(t ) = gμ (t ) +

t

ξμ

t ∈ I (μ ) ,

K1 (t, s )y(s )ds,

(2.3)

where

gμ (t ) := g(t ) + b(t )y(θ (t )) +

ξμ t0

K1 (t, s )y(s )ds +

θ (t ) t0

K2 (t, s )y(s )ds.

(2.4)

Then, Eq. (2.3) can be viewed as a classical (non-delay) second-kind Volterra integral equation. Under the assumptions on the given data, we readily obtain g0 (t) ∈ C(I(0) ). According to the relevant theory for the classical second-kind VIEs (cf. [4], pp. 56), there exists a unique continuous solution to (1.1) on I(0) . Similarly, the unique continuous solution on each subsequent macro-interval I(μ) (μ ≥ 1) can be obtained successively. Now, we consider the continuity property of y(t) at t = t0 . It follows from (2.3) and (2.4) with μ = 0 that

lim+ y(t ) = g(t0 ) + b(t0 )φ (θ (t0 )) −

t →t0

t0

θ (t0 )

K2 (t0 , s )φ (s )ds.

(2.5)

Note that

lim y(t ) = y(t0 ) = φ (t0 ),

(2.6)

t →t0−

thus, y(t) is continuous at t = t0 if and only if (2.2) holds. For μ ≥ 1, recalling (2.1) and the regularity assumptions on the given data, we have

y(ξμ± ) = g(ξμ ) + b(ξμ )y(ξμ±−1 ) +

ξμ t0

K1 (ξμ , s )y(s )ds +

θ ( ξμ ) t0

K2 (ξμ , s )y(s )ds.

(2.7)

Hence, y(t) is continuous at t = ξμ if and only if

b(ξμ )y(ξμ+−1 ) = b(ξμ )y(ξμ−−1 ).

(2.8)

That is, b(ξμ ) = 0 or y(t) is continuous at t = ξμ−1 . Moreover, if y(t) is continuous at t = t0 , it follows from (2.7) and (2.8) that y(t) is also continuous at t = t1 , which allows us to derive that y(t) is continuous at all the subsequent points {ξ μ } (μ ≥ 2). This completes the proof of the theorem. Remark 2.1. It is worth remarking that for VIE (1.4), the solution y(t) is continuous at t = ξμ for all μ ≥ 1, which contrasts sharply to the case of VFIE (1.1). Remark 2.2. For vanishing delay VFIE (1.7), to guarantee the exitance and uniqueness of the exact solution, the function b(t) in (1.7) needs to satisfy ||b||∞ < 1 (cf. [21,22]). But for non-vanishing delay VFIE (1.1), this condition on b(t) is unnecessary. 2.2. Regularity of the exact solution Theorem 2.2. Suppose that the given data in Eq. (1.1) all have continuous derivatives up to order m ≥ 1 on their respective domains, and θ is subject to conditions (D1)–(D3) with d ≥ m. Then: (i) The (unique) solution y(t) lies in C m (ξμ , ξμ+1 ] (0 ≤ μ ≤ M − 1 ) and is bounded at each prime discontinuous point {ξ μ } (0 ≤ μ ≤ M). (ii) If y(k) (t) is continuous at t = t0 for some positive integer k ≤ m, then y(k) (t) is also continuous at t = ξμ (1 ≤ μ ≤ M ).


201

Proof. From Eq. (2.3), the solution y(t) on I(μ) can be represented as

y(t ) = gμ (t ) +

t

t ∈ I (μ ) ,

R1 (t, s )gμ (s )ds,

ξμ

(2.9)

where R1 denotes the resolvent kernel associated with K1 (t, s). According to the regularity assumptions on the given data, y(t ) ∈ C m (ξμ , ξμ+1 ] for all 0 ≤ μ ≤ M − 1. Furthermore, recalling (2.5) and (2.6), y(t) is bounded at t = t0 . Similarly, we can deduce the boundedness property of y(t) at t = ξμ (μ ≥ 1 ) from (2.7). Now, we consider the continuity property of y (t) at t = ξμ (μ ≥ 1 ). Setting t = ξμ+ and t = ξμ− respectively in the differentiated form of (1.1), we have

y (ξμ± ) = g (ξμ ) + b (ξμ )y(ξμ±−1 ) + b(ξμ )θ (ξμ )y (ξμ±−1 ) + θ (ξμ )K2 (ξμ , ξμ−1 )y(ξμ±−1 ) + K1 (ξμ , ξμ )y(ξμ± ) + G(ξμ± ), where

G(t ) :=

t

t0

∂ K1 (t, s ) y(s )ds + ∂t

θ (t ) t0

(2.10)

∂ K2 (t, s ) y(s )ds, t ∈ I. ∂t

Therefore, if y (t) is continuous at t = t0 (which implies y is continuous at t = t0 ), then we can deduce that y(t) is also continuous at t = ξ1 . Thus, from (2.10) and the assumed regularity conditions on the given data, we may obtain

lim y (t ) = lim− y (t ),

t→ξ1+

(2.11)

t→ξ1

that is, y (t) is continuous at t = ξ1 . The continuity property of y (t) at t = ξμ (μ ≥ 2 ) then can be established along very similar lines. Finally, we take the kth (1 < k ≤ m) derivative on both sides of Eq. (1.1). If y(k) (t) is continuous at t = t0 , then by using rather similar way mentioned above, we can derive successively that y(k) (t) is also continuous at t = ξμ (μ = 1, 2, . . . , M ). The proof is then completed. Remark 2.3. From Theorem 2.2 we can see that the regularity of y(t) at t = ξμ (0 ≤ μ ≤ M ) will not be improved as μ increases, which is different from the case of Eq. (1.4) (cf. [4], pp. 204). 2.3. Local representation of the exact solution It is well known that the resolvent representation of solutions to non-vanishing delay problems will play a crucial role in the analysis of the optimal convergence results for collocation solutions. Now, we are in a position to give the following representation theorem for VFIE (1.1). Theorem 2.3. Suppose the assumptions of Theorem 2.1 hold, and define

g0 (t ) := g(t ) + b(t )φ (θ (t )) −

t0

θ (t )

K2 (t, s )φ (s )ds

f or

t ∈ I (0 ) .

(2.12)

Then, for t ∈ I (μ ) = (ξμ , ξμ+1 ] (μ ≥ 1 ), the solution y of (1.1) can be represented as

y(t ) = Bμ (t ) +

t

ξμ

R1 (t, s )Bμ (s )ds + Fμ (t ) + μ (t ),

(2.13)

with μ −1

Bμ (t ) :=g(t ) + Fμ (t ) :=

ξ1

k=1

g

θ k (t )

−1 μ b θ k (t ) , θ i−1 (t ) + g0 θ μ (t )

k

b

i=1

k=0

Rμ,v (t, s )Bv (s )ds, v=1 ξv θ μ (t ) μ −1 θ μ−v (t ) μ (t ) := Qμ,0 (t, s )g0 (s )ds + Qμ,v (t, s )Bv (s )ds. t0 v=1 ξv t0

Rμ,0 (t, s )g0 (s )ds +

μ −1 ξ v+1

Here, R1 is the resolvent kernel corresponding to K1 in (1.2), Rμ,v and Qμ,v are continuous functions which depend on K1 , K2 , R1 and θ , and θ 0 (t) := t. On the first macro-interval I (0 ) = (t0 , ξ1 ], this representation is given by

y(t ) = g0 (t ) +

t

t0

R1 (t, s )g0 (s )ds.

Proof. For μ = 0, recalling (2.3), (2.4) and (2.9), the expression (2.14) follows immediately.

(2.14)

202


For μ = 1, using Dirichlet’s formula and (2.4), we have

g1 (t ) = g(t ) + b(t )y(θ (t )) +

ξ1 t0

= g(t ) + b(t )g0 (θ (t )) +

K1 (t, s )y(s )ds +

θ (t ) t0

K2 (t, s )y(s )ds

ξ1

ξ1 K1 (t, s ) + K1 (t, v )R1 (v, s )dv g0 (s )ds t0

s

θ (t )

θ (t ) + b(t )R1 (t, s ) + K2 (t, s ) + K2 (t, v )R1 (v, s )dv g0 (s )ds t0

(2.15)

s

:= g(t ) + b(t )g0 (θ (t )) +

ξ1

Q1(,11) (t, s )g0 (s )ds +

t0

θ (t ) t0

Q1(,10) (t, s )g0 (s )ds,

with obvious meaning of the functions Q1(,10) and Q1(,11) . Recalling (2.9) and using trivial algebraic manipulation, the solution on I(1) can be expressed as

y(t ) = B1 (t ) +

ξ1

t (1 ) (1 ) R1 (t, s )B1 (s )ds + Q1,1 (t, s ) + R1 (t, v )Q1,1 (v, s )dv g0 (s )ds

t

ξ1

ξ1

t0

θ (t )

t (1 ) (1 ) + Q1,0 (t, s ) + R1 (t, v )Q1,0 (v, s )dv g0 (s )ds t0

:= B1 (t ) +

ξ1

t

R1 (t, s )B1 (s )ds +

ξ1

ξ1 t0

R1,0 (t, s )g0 (s )ds +

θ (t ) t0

(2.16) Q1,0 (t, s )g0 (s )ds,

Now, the representation for y(t) on I(μ) (μ ≥ 2) can be obtained by a rather tedious induction argument, and we omit the remaining details. Remark 2.4. Obviously, the representation theorem for the solution of VFIE (1.1) generalizes the corresponding result for VIE (1.4) (cf. [4], pp. 201), since b(t) ≡ 0 implies Bμ (t) ≡ g(t) for all μ ≥ 1 in (2.13). Remark 2.5. The structure of the above formula (2.13) clearly reveals the contribution of the delay term b(t)y(θ (t)) on the structure of the representation for y(t). This solution representation will play a key role in the analysis of the optimal global and local superconvergence order for collocation solutions, which will be shown in Section 4. 3. Collocation methods in piecewise polynomial spaces In this section, we will construct collocation solutions to (1.1) on θ -invariant meshes. For ease of exploration, we assume that the delay θ (t) is a linear function hereafter in this paper, i.e.,

θ (t ) = qt − τ ,

( 0 < q ≤ 1, τ ≥ 0, ( 1 − q )2 + τ 2 = 0 ).

(3.1)

Obviously, (3.1) contains two typical kinds of delays: the constant delay θ (t ) = t − τ and the proportional delay θ (t ) = qt. Now, we divide each macro-interval I (μ ) (0 ≤ μ ≤ M − 1 ) into several subintervals. Let (μ )

Ih

(μ )

:= tn

(μ )

: ξμ = t 0

(μ )

< t1

(μ )

< · · · < tN

be a local mesh on I(μ) and denote Ih := the uniform local mesh. Moreover, we introduce

H (μ) := ξμ+1 − ξμ ,

h

M−1 (μ ) as the set of mesh points. For linear delay function θ (t), we often use μ=0 Ih

μ = 0, 1, . . . , M − 1,

(μ )

:= tn+1 − tn , n = 0, 1, . . . , N − 1,

(μ )

:= max{hn

hn

(μ )

= ξμ+1 ,

(μ )

(μ )

n

}, μ = 0, 1, . . . , M − 1,

and

h := max{h(μ) }, μ

to denote the macro steps, the local steps, the maximum local steps, and the maximum step on [t0 , T] respectively. For uniform local meshes, we readily have (μ )

hn

=

ξμ+1 − ξμ N

,

h=

ξM − ξM−1 N

(μ )

Xh

(μ )

(μ )

:= tn,i := tn

(3.2)

:= (tn

(μ )

: 0 < c1 < · · · < cm ≤ 1 , 0 ≤ n ≤ N − 1 ,

+ ci hn

(μ )

T − θ (T ) . N

(μ )

Finally, on each subinterval σn

=

(μ )

, tn+1 ], we choose the collocation points as


203

(μ ) and introduce Xh := M−1 to denote the set of the collocation points. Note that for linear delay θ (t), the uniform local μ=0 Xh meshes will lead to the θ -invariant property of Xh , too. That is, (μ ) (μ−1 ) θ (tn,i ) = tn,i ,

μ = 1, . . . , M − 1.

(−1 ) (0 ) For consistency, we define tn,i := θ (tn,i ). We will approximate the solution of (1.1) by collocation in the space (−1 ) Sm (I ) := −1 h

v : v|σn(μ) ∈ πm−1 , 0 ≤ μ ≤ M − 1, 0 ≤ n ≤ N − 1 .

Here, πm−1 denotes the space of all polynomials with degree not exceeding m − 1. The collocation solution uh (t) to (1.1) will be chosen to solve

uh (t ) = g(t ) + b(t )uh (θ (t )) + (Vuh )(t ) + (Vθ uh )(t ), uh (t ) = φ (t ),

t ∈ Xh ,

(3.3)

t ∈ Iθ .

Then, the corresponding iterated collocation solution uith (t ) can be defined as

uith (t ) := g(t ) + b(t )uh (θ (t )) + (Vuh )(t ) + (Vθ uh )(t ), uith (t ) := φ (t ),

t ∈ I,

(3.4)

t ∈ Iθ .

From (3.3) and (3.4), we can readily obtain

uith (t ) = uh (t ),

t ∈ Xh .

(3.5)

uith (t )

uith (t )

We will return to in Section 4 to discuss whether can exhibit a higher (global or local) convergence order than uh (t) itself, under the assumption that the collocation points are chosen judiciously. (μ ) (μ ) (μ ) On the subinterval σn = (tn , tn+1 ], the collocation solution uh (t) is given by (μ )

uh (tn

(μ )

+ vhn ) =

m

(μ )

L j (v )Un, j ,

0 < v ≤ 1,

(3.6)

j=1

(μ )

(μ )

with Un, j := uh (tn, j ). Here, L j (v ) ( j = 1, 2, . . . , m ) denote the Lagrange basis functions with respect to the collocation parameters {ci }, i.e.,

L j (v ) =

m i= j

v − ci , c j − ci

j = 1, 2, . . . , m. (μ )

(μ )

Noting that the first equation of (3.3) holds when t = tn,i ∈ Xh , thus the ‘stage values’ {Un,i } assume the form (μ )

(μ )

(μ )

(μ−1 )

Un,i = g(tn,i ) + b(tn,i )uh (tn,i

(i = 1, 2, . . . , m ). Here, (μ ) (μ ) (Vuh )(tn,i ) = Fn(μ) (tn,i ) + hn(μ)

(μ )

with the lag term Fn (μ )

Fn

(t ) :=

ξμ t0

(μ ) (μ ) ) + (Vuh )(tn,i ) + (Vθ uh )(tn,i )

ci 0

(μ )

K1 (t, s )uh (s )ds +

(μ )

(μ )

tn

(μ )

+ vhn )d v,

ξμ−1 t0

K2 (t, s )uh (s )ds +

(μ )

t ∈ σn

K1 (t, s )uh (s )ds,

ξμ

(μ ) (μ ) (Vθ uh )(tn,i ) = n(μ−1) (tn,i ) + hn(μ−1)

n(μ−1) (t ) :=

(μ )

+ vhn )uh (tn

(3.8)

(t ) given by

Similarly,

where

(μ )

K1 (tn,i , tn

(3.7)

ci

0

(μ )

(μ−1 )

K2 (tn,i , tn

.

(μ−1 )

+ vhn

(3.9)

)uh (tn(μ−1) + vhn(μ−1) )dv,

(μ−1 )

tn

ξμ−1

(μ )

t ∈ σn

K2 (t, s )uh (s )ds,

(3.10)

.

(3.11)

(μ )

In order to obtain the values of Un,i , we substitute (3.6), (3.8), and (3.10) into (3.7), thus (μ )

(μ )

(μ )

(μ−1 )

Un,i = g(tn,i ) + b(tn,i )Un,i (μ−1 )

+ n

(μ )

+ Fn

m (μ−1 )

(μ ) (tn,i ) + hn

j=1

(μ )

(μ )

(tn,i ) + hn

m

j=1

ci 0

(μ )

(μ−1 )

K2 (tn,i , tn

ci 0

(μ )

(μ )

K1 (tn,i , tn (μ−1 )

+ vhn

(μ )

(μ )

+ vhn )L j (v )dv Un, j

)L j (v )dv Un,(μj −1)

(3.12)

204


(i = 1, 2, . . . , m ). For ease of notation, we introduce the m-vectors (μ )

Un

(μ )

An

(μ )

(μ )

(μ )

:= (Un,1 , . . . , Un,m )T ,

(μ )

:= Fn

gn

(μ ) T (tn,(μ1) ), . . . , Fn(μ) (tn,m ) ,

(μ )

T

(μ )

:= g(tn,1 ), . . . , g(tn,m ) ,

(μ−1 )

(μ−1 )

:= n

Gn

(μ ) T (tn,(μ1) ), . . . , n(μ−1) (tn,m ) .

In addition, we define matrices in L(Rm ) given by

(μ )

(μ )

(μ )

:= diag b(tn,1 ), . . . , b(tn,m ) , m×m

c (μ ) (μ ) (μ ) i K1 (tn,i , tn + vhn )L j (v )dv (μ ) 0 Cn := , (i, j = 1, . . . , m )

c (μ ) (μ−1 ) (μ−1 ) i + vhn )L j ( v )d v (μ ) 0 K2 (tn,i , tn Dn := . (i, j = 1, . . . , m ) Bn

(μ )

Therefore, the linear algebraic system (3.12) about Un,i can be written more concisely as

(μ ) (μ )

Im − hn Cn

(μ )

Un

(μ−1 ) (μ )

= hn

Dn

where Im denotes the identity matrix in

(0 ) (0 )

Im − hn Cn

(0 )

Un

(0 )

(μ )

+ Bn

(−1 )

= gn + An + G n

(μ−1 )

(μ )

+ gn

Un

(μ )

+ An

(μ−1 )

+ Gn

,

(3.13)

On the first macro-interval (ξ 0 , ξ 1 ], the above system (3.13) reduces to

L ( R m ).

(0 )

, n = 0, 1, . . . , N − 1,

(3.14)

where the components of Gn(−1 ) ∈ Rm are given by (−1 )

n

(0 )

(tn,i ) :=

(−1 ) tn,i

t0

(0 ) K2 (tn,i , s )φ (s )ds.

Next, we present the existence and uniqueness of the collocation solution uh (t). (−1 ) (I ) is the collocation solution defined by Theorem 3.1. Assume that the assumptions of Theorem 2.1 hold, and uh (t ) ∈ Sm −1 h (3.3), then there exists an h¯ > 0 (depending on θ ) so that for any θ -invariant mesh Ih with h ∈ (0, h¯ ), each of the linear algebraic (μ ) systems (3.13) possesses a unique solution Un ∈ Rm . Hence, the collocation equation (3.3) defines a unique collocation solution (μ ) to (1.1), and the local expression on subinterval σn is given by (3.6). (μ )

Proof. Since the kernel function K1 (t, s) is continuous on D, the elements of the matrices Cn (0 ≤ n ≤ N − 1; 0 ≤ μ ≤ (μ ) (μ ) (μ ) M − 1 ) are all bounded. Therefore, for sufficient small hn > 0, hn Cn < 1 holds for some matrix norm. According (μ ) (μ ) (μ ) to the Neumann Lemma (cf. [1], pp. 492), each matrix Cn := Im − hn Cn has a uniformly bounded inverse. Hence, there (μ ) ¯ exists an h > 0 so that for all meshes Ih with 0 < h = max(n,μ ) hn < h¯ , the linear algebraic systems (3.13) possess a unique solution. (μ )

Once the collocation solution uh (t) on σn (μ )

t = tn

(μ )

+ vhn

has been computed, the corresponding iterated collocation solution uith (t ) at

(0 < v ≤ 1; 0 ≤ n ≤ N − 1 ) can be obtained by

uith (t ) =g(t ) + b(t )

m

(μ−1 )

L j (v )Un, j

(μ )

+ hn

j=1

(μ )

+ Fn

m

(t ) + n(μ−1) (t ) + hn

(μ )

v 0

j=1 m (μ−1 ) j=1

Here, Fn

v 0

(μ )

K1 (t , tn

(μ−1 )

K2 (t , tn

(μ )

(μ )

+ shn )L j (s )ds Un, j (μ−1 )

+ shn

)L j (s )ds Un,(μj −1) .

(3.15)

(t ) and n(μ−1) (t ) are defined in (3.9) and (3.11), respectively.

4. Convergence results In this section, we will study the optimal global and local convergence orders of the (iterated) collocation solutions uh (t) and uith (t ). The collocation error eh (t) and the iterated collocation error eith (t ) are given by eh (t ) := y(t ) − uh (t ) and eith (t ) := y(t ) − uith (t ), respectively. 4.1. Global convergence orders of uh (t) and uith (t ) We are now ready to give the attainable global convergence results for uh (t) and uith (t ) in the following theorems. Theorem 4.1. Assume: (a) The given data g, b ∈ Cm (I), φ ∈ Cm (Iθ ), K1 ∈ Cm (D) and K2 ∈ Cm (Dθ );


205

(b) θ (t) is defined in (3.1); (−1 ) (c) uh (t ) ∈ Sm (I ) is the collocation solution to VFIE (1.1) on θ -invariant meshes, with uniform local meshes Ih(μ) satisfying −1 h h ∈ (0, h¯ ). Then

eh ∞ := max |y(t ) − uh (t )| ≤ Chm

(4.1)

t∈I

holds for any set of collocation points Xh with 0 < c1 < < cm ≤ 1, where the constant C depends on {ci } but not on h. Proof. Since y(t ) ∈ C m (ξμ , ξμ+1 ] (0 ≤ μ ≤ M − 1 ) by Theorem 2.2, we may resort to Peano’s Theorem (cf. [14], pp. 270–273) (μ )

to express y(t) on σn (μ )

y(tn

as

(μ )

+ vhn ) =

m

(μ )

(μ ) m (μ )

L j (v )Yn, j + hn

Rm,n (v ),

v ∈ (0, 1],

(4.2)

j=1

(μ )

(μ )

with Yn, j := y(tn, j ). The Peano reminder term and Peano kernel are defined by (μ )

Rm,n (v ) :=

1

(μ )

(μ )

Km (v, σ )y(m ) (tn

0

+ σ hn )d σ ,

(4.3)

and

1 −1 −1 ( v − σ )m − Lk (v )(ck − σ )m , + + ( m − 1 )!

Km (v, σ ) :=

m

σ ∈ (0, 1],

(4.4)

k=1

respectively. Here, (v − σ )+ := 0 for v < σ and (v − σ )+ := (v − σ ) p for v ≥ σ . p

(μ )

(μ )

p

(μ )

Setting εn,i := Yn,i − Un,i , the collocation error eh (t) then possesses the local representation (μ )

(μ )

eh (tn

+ vhn ) =

m

(μ )

(μ ) m (μ )

L j (v )εn, j + hn

Rm,n (v ),

v ∈ (0, 1].

(4.5)

j=1

Now, we are ready to derive estimate (4.1). From (1.1) and (3.3) we readily obtain

eh (t ) = b(t )eh (θ (t )) + (Veh )(t ) + (Vθ eh )(t ),

t ∈ Xh ,

(4.6)

(0 ) with eh (t ) = 0 when t ∈ [θ (t0 ), t0 ]. On the first macro-interval I (0 ) = (t0 , ξ1 ], substituting (4.5) into (4.6) and setting t = tn,i , we have (0 ) εn,i =

tn(0)

t0

(0 ) K1 tn,i , s eh (s )ds +

n−1 = hl(0) l=0

+ hn(0)

1 0

ci 0

(0 ) tn,i (0 )

tn

(0 ) K1 tn,i , s eh (s )ds

m m ( 0 ) (0 ) (0 ) K1 tn,i , tl + vhl(0) L j (v )εl,(0j) + hl(0) Rm,l (v ) d v j=1

m (0 ) (0 ) m ( 0 ) (0 ) K1 tn,i , tn + vhn(0) L j (v )εn, + hn(0) Rm,n (v ) d v j j=1

(0 ) (n = 0, 1, . . . , N − 1 ). For convenience, we introduce the matrices Cn(0) and Cn,l in L(Rm ) by

c (0 )

Cn

:=

(0 ) Cn,l :=

0

i

(0 ) (0 ) K1 tn,i , tn + vhn(0) L j (v )dv

(i, j = 1, 2, . . . , m ) 1 (0 ) (0 ) + vhl(0) L j (v )dv 0 K1 tn,i , tl (i, j = 1, 2, . . . , m )

In addition, we define the m-vectors

,

( l < n ).

(4.7)

206


(0 )

ρn :=

c i 0

T (0 ) (0 ) (0 ) K1 tn,i , tn + vhn(0) Rm,n ( v )d v

,

( i = 1, 2, . . . , m )

1 ( 0 ) ( 0 ) T (0 ) + vhl(0) Rl,n ( v )d v (0 ) 0 K1 tn,i , tl ρn,l := ( l < n ). ( i = 1, 2, . . . , m ) (0 ) (0 ) (0 ) T Thus, εn(0 ) := (εn, , εn,2 , . . . , εn,m ) is the solution of the linear algebraic system 1

Im − hn(0)Cn(0)

εn(0) =

n−1 n−1 m+1 (0) (0) m+1 (0) (0 ) (0 ) hl(0)Cn,l εl + hl(0) ρn,l + hn ρn l=0

(4.8)

l=0

(n = 0, 1, . . . , N − 1 ). According to the proof of Theorem 3.1, we have the uniform bound

(Im − hn(0)Cn(0) )−1 1 ≤ Q0 ,

(4.9)

for all h ∈ (0, h¯ ). Due to the regularity of the given data, we may assume (0 ) Cn,l 1 ≤ Q1 ( l < n ).

Furthermore, we set

(4.10)

M1 := max K1 (t, s ), M2 := max K2 (t, s ), (t,s )∈D

(4.11)

(t,s )∈Dθ

Mm := y(m ) ∞ , km := max

1

τ ∈[0,1] 0

Km (τ , σ )dσ , and m := max max |L j (τ )|, j

then

εn(0) 1 ≤ Q0 Q1

0≤τ ≤1

n−1 n−1 m+1 (0) m+1 hl(0) εl(0) 1 + Q0 mkm M1 Mm hl(0) + hn l=0

(4.12)

l=0

n−1 m ≤ Q0 Q1 hl(0) εl(0) 1 + Q0 mkm M1 Mm H (0) h(0) l=0 n−1 m := γ0 hl(0) εl(0) 1 + γ1 h(0) ,

n = 0, 1, . . . , N − 1.

(4.13)

l=0

Here, the meaning of the constants γ 0 and γ 1 is clear. Therefore, it follows from the generalized discrete Gronwall inequality (cf. [4], pp. 81) that

m m εn(0) 1 ≤ γ1 h(0) exp(γ0 H (0) ) := B0 h(0) ,

n = 0, 1, . . . , N − 1.

(4.14)

This yields, recalling (4.5),

m m m |eh (tn(0) + ν hn(0) )| ≤ m εn(0) 1 + km Mm h(0) ≤ ( m B0 + km Mm ) h(0) := D0 h(0)

(4.15)

(n = 0, 1, . . . , N − 1 ). Moreover, we have

m eh 0,∞ := sup |eh (t )| ≤ D0 h(0) .

(4.16)

t∈I (0)

Since the number (M) of macro-intervals is finite, a simple induction argument may lead to

m eh μ,∞ := sup |eh (t )| ≤ Dμ h(μ) t∈I (μ)

( μ = 1, . . . , M − 1 ),

(4.17)

and hence it follows that

eh ∞ := max |y(t ) − uh (t )| ≤ Chm . t∈I

Here, the constant C depends on {ci } but not on h.

We note that uh (t) satisfies Eq. (3.3) only on the set of the collocation points Xh . For any point t ∈ I, however, it will no longer be true in general. Thus, we define

δh (t ) := −uh (t ) + g(t ) + b(t )uh (θ (t )) + (Vuh )(t ) + (Vθ uh )(t ),

t ∈ I,

(4.18)

to denote the ‘residual’ (or, ‘defect’) associated with uh (t). Obviously, δ h (t) vanishes at the points t ∈ Xh . Eqs. (1.1) and (4.18) enable us to rewrite δ h (t) as

δh (t ) = eh (t ) − b(t )eh (θ (t )) − (Veh )(t ) − (Vθ eh )(t ),

t ∈ I.

(4.19)


207

Therefore, the following result can be obtained immediately from Theorem 4.1. Corollary 4.1. Under the assumptions of Theorem 4.1, the estimate

δh ∞ ≤ Chm ,

(4.20)

holds for all h ∈ (0, h¯ ). Here, the constant C does not depend on h. We know that for VIE (1.4), if the collocation parameters {ci } are chosen judiciously, the optimal local convergence order of uith (t ) can attain p∗ = m + 1. However, for VFIE (1.1), this superconvergence order only exhibits on the first macro-interval I(0) , which will be made precise in the following theorem. Theorem 4.2. Assume: (a) (b) (c) (d)

The given data g, b ∈ C m+1 (I ), φ ∈ C m+1 (Iθ ), K1 ∈ C m+1 (D ) and K2 ∈ C m+1 (Dθ ); θ (t) is defined in (3.1); (−1 ) uith (t ) ∈ Sm (I ) is the iterated collocation solution given by (3.4), with uniform local meshes Ih(μ) satisfying h ∈ (0, h¯ ); −1 h The collocation parameters {ci } are subject to the orthogonality condition

J0 :=

m 1

0

(s − ci )ds = 0.

(4.21)

i=1

Then (i) on the first macro-interval I (0 ) = (t0 , ξ1 ],

eith 0,∞ := sup |y(t ) − uith (t )| ≤ C0 hm+1 ,

(4.22)

t∈I (0)

(ii) on the subsequent macro-intervals I (μ ) = (ξμ , ξμ+1 ] (1 ≤ μ ≤ M − 1 ),

eith μ,∞ := sup |y(t ) − uith (t )| ≤ Cμ hm ,

only.

(4.23)

t∈I (μ)

Here, the constants Cμ (0 ≤ μ ≤ M − 1 ) depend on {ci } but not on h. Proof. Recalling (4.19), the collocation error eh (t) satisfies

eh (t ) = δh (t ) + b(t )eh (θ (t )) + (Veh )(t ) + (Vθ eh )(t ), eh (t ) = 0,

t ∈ I,

(4.24)

t ∈ Iθ .

On the first macro-interval I(0) , based on (2.14) in Theorem 2.3 (where the roles of y, g, and φ are now replaced by eh , δ h , and 0, respectively), eh (t) can be expressed as

eh (t ) = δh (t ) +

t

t0

R1 (t, s )δh (s )ds.

(4.25)

Furthermore, from (1.1), (3.4), (4.19) and (4.25), we have

eith (t ) = eh (t ) − δh (t ) =

t

t0

R1 (t, s )δh (s )ds,

t ∈ I (0 ) .

(4.26)

Setting t = tn(0 ) + vhn(0 ) ∈ I (0 ) in (4.26), we then obtain

eith (tn(0) + vhn(0) ) =

n−1 hl(0) l=0

1 0

R1 (t , tl(0) + shl(0) )δh (tl(0) + shl(0) )ds + hn(0)

v 0

R1 (t , tn(0) + shn(0) )δh (tn(0) + shn(0) )ds

(4.27)

:= I1 + I2 . Now, all the integrals in I1 can be replaced by a summation of the interpolatory m-point quadrature formulas based on (0 ) {ci } and the resulting error terms En,l . The expressions of the quadrature formulas are equal to zero, since δh (t ) = 0 when t ∈ Xh . By the orthogonality conditions (4.21) on {ci }, together with the assumed regularity of the given data, the orders of these quadrature formulas are all m + 1 (cf. [17], pp. 168). To be more precise,

|I1 | =

n−1 m (0 ) hl(0) bi R1 (t , tl(0) + ci hl(0) )δh (tl(0) + ci hl(0) ) + En,l (t ) l=0

i=1

n−1 n−1 (0 ) = hl(0) En,l (t ) ≤ Chm+1 hl(0) ≤ C (T − t0 )hm+1 , l=0

l=0

(4.28)

208


where bi =

1 0

Li (s )ds. For the second term I2 , recalling (4.20), we have

I2 ≤ hn(0) · R¯ 1 · δh ∞ ≤ C R¯ 1 hm+1 .

(4.29)

Here, we have set R¯ 1 := max(t,s )∈D |R1 (t, s )|. Collecting (4.27)–(4.29), the desired result (4.22) is obtained immediately. We are now ready to estimate eith (t ) on other macro-intervals I(μ) (μ ≥ 1) by some simple analysis. Let K1 , K2 ≡ 0 in (1.1), then we get

it eh (t ) = b(t ) · eh (θ (t )) ≤ b∞ · eh (θ (t )),

t ∈ I (μ ) .

(4.30)

By virtue of Theorem 4.1, the order of eh cannot exceed m apparently. Furthermore, since the given function b does not depend on h, the optimal global convergence order of uith (t ) on I(μ) (μ ≥ 1) cannot exceed m in general. The proof of estimate (4.23) is then completed. 4.2. Local superconvergence of uh (t) and uith (t ) on Ih Now, we present the optimal local (on the mesh points) convergence orders of uh (t) and uith (t ) in the following theorems. Theorem 4.3. Assume: (a) The given data g, b ∈ C m+κ (I ), φ ∈ C m+κ (Iθ ), K1 ∈ C m+κ (D ) and K2 ∈ C m+κ (Dθ ) for some integer κ with 1 ≤ κ ≤ m; (b) θ (t) is defined in (3.1); (−1 ) (c) uh (t ) ∈ Sm (I ) is the collocation solution to Eq. (1.1) on θ -invariant meshes, with uniform local meshes Ih(μ) satisfying −1 h ¯ h ∈ ( 0, h ); (d) The collocation parameters {ci } are subject to the orthogonality conditions

Jl :=

1

sl

0

m

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

(4.31)

i=1

with Jκ = 0. Then for any h ∈ (0, h¯ ), (i) If cm = 1 (implying κ < m), we have

max |y(t ) − uh (t )| ≤ Chm+κ ,

(4.32)

t∈Ih \{t0 }

where the constant C depends on {ci } but not on h. (ii) If cm < 1, we only have

max |y(t ) − uh (t )| = O hm .

(4.33)

t∈Ih \{t0 }

(μ )

Proof. By (2.13) in Theorem 2.3, eh (tn (μ ) (μ ) eh (tn ) = B˜μ (tn ) +

+

μ −1 ξ v+1 v=1 ξv

(μ )

tn

ξμ

) (μ ≥ 1 ) can be expressed as

(μ )

R1 (tn , s )B˜μ (s )ds + (μ )

Rμ,v (tn

, s )B˜v (s )ds +

ξ1 t0

μ −1 t (v ) n v=1 ξv

(μ )

Rμ,0 (tn , s )δh (s )ds + (μ )

Qμ,v (tn

tn(0)

t0

(μ )

Qμ,0 (tn , s )δh (s )ds (4.34)

, s )B˜v (s )ds,

where

B˜μ (t ) :=δh (t ) +

μ −1

−1 k μ δh θ k (t ) b θ i−1 (t ) + δh θ μ (t ) b θ k (t ) .

k=1

i=1

(4.35)

k=0

(μ )

(μ )

We note that when cm = 1, each mesh point tn is also a collocation point. Therefore, δh (tn 0, . . . , M − 1 and n = 0, . . . , N − 1. Moreover, recalling that Xh is θ -invariant, we have

θ k (tn(μ) ) = tn(μ−k) ∈ Xh ,

) = 0 holds for all μ =

k = 1 , . . . , μ,

and thus (μ ) B˜μ (tn ) = 0,

μ = 1, . . . , M − 1.

(4.36)

Again, all the integrals at the right-hand side of (4.34) can be converted into sums of (scaled) integrals over [0, 1] with factors hl(v ) (1 ≤ v ≤ μ ≤ M − 1; 0 ≤ l ≤ n ≤ N − 1 ), and the method used to estimate I1 in the proof of Theorem 4.2 can be


209

adopted here. We illustrate it by considering

I3 :=

= =

=

μ −1 ξ v+1

v=1 ξv μ −1 N−1 v=1 n=0 μ −1 N−1 v=1 n=0 μ −1 N−1

(μ )

Rμ,v (tn , s )δh (s )ds

hn(v ) (v )

1 0

hn

(μ )

Rμ,v (tn , tn(v ) + shn(v ) )δh (tn(v ) + shn(v ) )ds

m

(μ )

(v )

(v )

(v )

bi Rμ,v (tn , tn + ci hn )δh (tn

(μ ) + ci hn ) + En,v (t ) (v )

(4.37)

i=1

(μ )

hn(v ) En,v (t ).

v=1 n=0

By the orthogonality conditions (4.31) on {ci }, all these quadrature formulas have degree of precision m + κ − 1. Therefore, according to the assumed conditions on the given data, there exists a constant Q¯ > 0 such that

(μ ) En,v ≤ Q¯ hm+κ ,

1 ≤ v ≤ μ ≤ M − 1; 0 ≤ n ≤ N − 1,

(4.38)

where Q¯ does not depend on h. Furthermore, μ μ −1 N−1 −1 (v ) I3 ≤ Q¯ hm+κ hn = Q¯ hm+κ ξv+1 − ξv ≤ T Q¯ hm+κ v=1 n=0

(4.39)

v=1

holds for all μ = 1, . . . , M − 1. Similarly, on the first macro-interval I (0 ) = (t0 , ξ1 ], the same order estimate can be obtained by (2.14). Collecting all these estimates, we readily establish the desired results (4.32). (μ ) (μ ) In the case of cm < 1, since the mesh points {tn } are no longer collocation points, δ h (t) does not vanish at t = tn ( μ ) generally, and thus B˜μ (tn ) ≡ 0 (μ = 1, . . . , M − 1 ). Therefore, by (4.20) and (4.34), the order of eh (t) cannot exceed m apparently. Corollary 4.2. Suppose κ = m (the collocation points {ci } are the Gauss–Legendre points) in Theorem 4.3, then we have

max |y(t ) − uh (t )| ≤ Chm .

(4.40)

t∈Ih \{t0 }

Corollary 4.3. Suppose κ = m − 1 and cm = 1 (the collocation points {ci } are the Radau II points) in Theorem 4.3, then we have

max |y(t ) − uh (t )| ≤ Ch2m−1 .

(4.41)

t∈Ih \{t0 }

It is well known that for VIE (1.4), if the collocation points are subject to (4.31), then uith (t ) can attain the optimal local superconvergence order p∗ = m + κ at all points t ∈ Ih . For VFIE (1.1), however, this local superconvergence property can only occur on a small part of Ih generally. The precise result is described in the following theorem. (−1 ) Theorem 4.4. Suppose the assumptions in Theorem 4.3 hold, uith (t ) ∈ Sm (I ) is the iterated collocation solution defined in −1 h Eq. (3.4), then

(i) on the first macro-interval I (0 ) = (t0 , ξ1 ],

max |y(t ) − uith (t )| = O hm+κ .

(4.42)

(0 )

t∈Ih

(ii) on the subsequent macro-interval I (μ ) = (ξμ , ξμ+1 ] (1 ≤ μ ≤ M − 1 ), (a) if cm = 1,

max |y(t ) − uith (t )| = O hm+κ ,

(4.43)

(μ )

t∈Ih

(b) if cm < 1,

max |y(t ) − uith (t )| = O hm , (μ )

only.

(4.44)

t∈Ih

Proof. The details of the proof is omitted since it is closely resemble those for Theorems 4.2 and 4.3. We note in passing that when cm = 1, each mesh point is also a collocation point, thus uh (t ) = uith (t ) for any t ∈ Ih (recalling (3.5)), and so estimate (4.43) is the same as (4.32).

210


4.3. Local superconvergence of uh (t) and uith (t ) on Xh As shown in Theorem 4.4, when cm = 1, the local superconvergence of uith on Ih can only occur on the first macro-interval of I. We now discuss whether the local convergence order p∗ = m can be improved at certain non-mesh points if cm = 1. Theorem 4.5. Under the assumptions of Theorem 4.3, the following estimates

max |y(t ) − uh (t )| = O hm+1 ,

(4.45)

t∈Xh

and

max |y(t ) − uith (t )| = O hm+1 ,

(4.46)

t∈Xh

hold for all h ∈ (0, h¯ ). The exponent m + 1 cannot be replaced by m + 2 in general. (μ )

Proof. Analogous to (4.34), we now set t = tn,i in the local representation of eh (t), then

(μ )

eh (tn,i ) =

(μ )

tn

ξμ

+

μ −1 ξ v+1 v=1 ξv

+

(μ )

R1 (tn,i , s )B˜μ (s )ds +

tn,i

tn

(μ )

Rμ,0 (tn,i , s )δh (s )ds +

t0

μ −1 t (v ) n

(μ )

Rμ,v (tn,i , s )B˜v (s )ds +

(μ )

(μ )

ξ1

(μ )

R1 (tn,i , s )B˜μ (s )ds +

v=1 ξv

(0 )

tn,i (0 )

tn

tn(0)

t0

(μ )

Qμ,0 (tn,i , s )δh (s )ds

(μ )

Qμ,v (tn,i , s )B˜v (s )ds

(μ )

Qμ,0 (tn,i , s )δh (s )ds +

μ −1 t (v ) n,i v=1

tn(v )

(μ )

(4.47)

Qμ,v (tn,i , s )B˜v (s )ds

:= I4 + I5 . For the first part I4 in (4.47), all integrals can again be converted into integrals over [0, 1]. Therefore, following similar lines in the analysis of (4.37) and using the orthogonality conditions (4.31) on {ci }, we may obtain that I4 = O(hm+κ ). Besides, the integrals in I5 may be converted into integrals over [0, ci ]. For these non-scaled integrals, we consider

μ−1 t (v) n,i (μ ) Qμ,v (tn,i , s )δh (s )ds (v ) v=1 tn ci μ −1 (μ ) (v ) (v ) (v ) (v ) (v ) ≤ hn Qμ,v (tn,i , tn + shn )δh (tn + shn )ds 0

v=1

≤Q Chm

μ −1

hn(v ) = Q Chm

v=1

μ −1

ξv − ξv−1 N

v=1

≤

T Th Q Chm = Q Chm ≤ Q Chm+1 , N T − θ (T )

where Q := maxμ,v max(t,s )∈D |Qμ,v (t, s )|. Then, the estimates for other terms in (4.47) can be obtained similarly, and thus assertion (4.45) holds. Assertion (4.46) is due to the fact that on the set of collocation points Xh , uith (t ) ≡ uh (t ) (recalling (3.5)), which finishes the proof. 5. Numerical experiments In this section, we take two typical examples, a proportional delay VFIE and a constant delay VFIE, to confirm our theoretical findings derived in Section 4. In our numerical experiments, we solve (1.1) in the piecewise polynomial spaces S1(−1 ) (Ih ) (m = 2 ) and S2(−1 ) (Ih ) (m = 3 ). For the collocation parameters {ci }, we choose the Radau II points and the Gauss points. That is,

c1 = 1 /3 , for m = 2, while

c2 = 1 ,

c1 = ( 4 −

√ 6 )/10,

c1 = ( 5 −

√ 15 )/10,

and

for m = 3.

and

c1 = ( 3 −

c2 = ( 4 +

√ 3 )/6,

√ 6 )/10,

c2 = 1 /2 ,

c2 = ( 3 +

c3 = 1 ,

c3 = ( 5 +

√ 15 )/10

√ 3 )/6


211

Fig. 1. The global convergence orders of uh (t) and uith (t ) by 2-point Radau II collocation.

Fig. 2. The local (on Ih ) convergence orders of uh (t) and uith (t ) by 3-point Gauss collocation.

Example 5.1. Consider the proportional delay VFIE

⎧ ⎨ y(t ) = et + e1−t − e − 1 + y(qt ) + t e−t y(s )ds − qt y(s )ds, 1 1 ⎩ y(t ) = et , t ∈ [q, 1],

t ∈ I := (1, 10],

(5.1)

with the exact analytical solution y(t ) = et . Fig. 1 is obtained by 2-point Radau II collocation with different delay parameters q = 0.1, q = 0.5, q = 0.9 and q = 0.95. From the left picture one may observe that the numerical convergence orders of the collocation solution uh (t) are all p∗ = m = 2 for widely varying values q ∈ (0, 1). However, in the case of q = 0.1, the convergence order of the iterated collocation solution uith (t ) is 3 (see the right picture). This is because when q = 0.1, the computational interval (1, 10] is just the first (and unique) macro-interval of I. According to our theoretical result (4.22), this order should be p∗ = m + 1 = 3. In Fig. 2 we show the numerical local convergence results for uh (t) and uith (t ) by 3-point Gauss collocation. In this case, the convergence orders of uh (t) are all p∗ = m = 3 for different delay parameter values q ∈ (0, 1) (see the left picture). But for uith (t ), the corresponding results change again. We observe from the right picture that when q = 0.1 (which implies the whole interval I is the first macro-interval itself), the local convergence order is p∗ = m + 3 = 6, and thus conforming the result given in Theorem 4.4.

212


Fig. 3. The local (on Ih and Xh ) convergence orders of uh (t) by 3-point Radau II collocation.

Fig. 4. The global (on I) and local (on Xh ) convergence orders by 2-point Radau II collocation.

In order to illustrate the local error estimates of uh (t) when the collocation parameter cm = 1, we choose the 3-point Radau II collocation in Fig. 3. It is easy to see that the local (on Ih ) convergence orders of uh (t) are all p∗ = m + 2 = 5 for different values q ∈ (0, 1) (see the left picture). We note in passing that when cm = 1, we have uith (t ) = uh (t ) for all t ∈ Ih (recalling (3.5)), and so the picture for the convergence order of uith (t ) coincide exactly with the one for uh (t). To avoid repetition, we do not list the picture of the corresponding results for uith (t ) here. The numerical local (on Xh ) convergence results for uh (t) are shown at the right picture of Fig. 3. Obviously, these convergence orders are all p∗ = m + 1 = 4 for different values q ∈ (0, 1). Example 5.2. Consider the problem

⎧ ⎨ y(t ) = 1 + sint · y(t − τ ) − t y(s )ds + 2 t−τ y(s )ds, 0 0 ⎩ y(t ) = e−t , t ∈ [−τ , 0],

t ∈ I := (0, 5],

(5.2)

with τ = 0.5. We note that the exact solution of (5.2) is discontinuous at t = kτ , k = 0, 1, . . . , 9. The numerical results are displayed in Figs. 4 and 5.


213

Fig. 5. The local (on Ih ) convergence orders by 2-point Gauss (left) and 2-point Radau II (right) collocations. Table 1 Comparisons. VIE (1.4)

VFIE (1.1)

C μ−1

C −1

max{|y(t ) − uh (t )| : t ∈ I} max{|y(t ) − uith (t )| : t ∈ I}

O(h m ) O(hm+1 )

O(hm ) O(hm+1 ) only on I(0)

max{|y(t ) − uh (t )| : t ∈ Ih }

O(hm ) when cm < 1 O(hm+κ ) when cm = 1

max{|y(t ) − uith (t )| : t ∈ Ih }

O ( h m+κ )

Regularity at t = ξμ (μ ≥ 1 )

O(hm ) on I(μ) (μ ≥ 1) O(hm ) when cm < 1 O(hm+κ ) when cm = 1 O(hm+κ ) on Ih(0) (μ )

O(hm ) on Ih max{|y(t ) − uh (t )| : t ∈ Xh }

O(hm+1 )

(μ ≥ 1 ) when cm < 1 (μ ≥ 1 ) when cm = 1

(μ )

O(hm+κ ) on Ih

Fig. 4 is obtained by 2-point Radau II collocation. We observe that the global convergence order of uith (t ) is indeed m +

1 = 3 on I(0) , and is only m = 2 on I(μ) (μ ≥ 1) (see the left picture). In addition, we can see that the numerical convergence order of uh (t) is m + 1 = 3 on Xh (see the right picture). Fig. 5 reflects the local (on Ih ) convergence order of uith (t ) by 2-point Gauss collocation (left) and 2-point Radau II collo-

cation (right). It is easy to see that for the Gauss collocation, the local convergence order of uith (t ) is m + 2 = 4 on I(0) , while

it is only m = 2 on I(μ) (μ ≥ 1) (see the left picture). But for the Radau II collocation, this order is m + 1 = 3 both on I(0) and on I(μ) (μ ≥ 1) (see the right picture), which coincides with the results given in Theorem 4.4.

6. Conclusion In this work, we analyze the existence, uniqueness, regularity properties and the local representation of the exact solution of VFIE (1.1). In addition, the optimal global and local convergence order of the collocation solutions are also investigated. Compared with VIE (1.4), the delay term b(t)y(θ (t)) in VFIE (1.1) not only results in a lower regularity of the exact solution, but also leads to a reduction in the (global and local) convergence orders of the collocation solutions. For convenience of comparison, we summarize the main results derived in the preceding sections for VFIE (1.1) and the corresponding results for VIE (1.4) in the following Table 1. Remark 6.1. The letter ‘m’ in Table 1 denotes the number of the collocation points in each subinterval, and ‘κ ’ is the precision of the m collocation points defined in (4.31). The symbol ‘’ means that there is no related result for VIE (1.4) yet, but we can easily obtain the same order estimate as the one for VFIE (1.1) from the proof of Theorem 4.5. Obviously, all the results obtained in this paper encompass the corresponding ones for Eq. (1.6), and the differentiated form of (1.5). However, the related results for VFIE (1.1) with nonlinear delay remain to be investigated.

214


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