Monte Carlo Methods Appl. 19 (2013), 281 – 330
DOI 10.1515 / mcma-2013-0013
© de Gruyter 2013
Worst case error for integro-differential equations
by a lattice-Nyström method
Davoud Rostamy, Mohammad Jabbari and Mahshid Gadirian
Abstract. In this paper, we make an offer of the lattice approximate method for solving
a class of multi-dimensional integro-differential equations with the initial conditions. Also,
we analyze the worst case error measured in weighted Korobov spaces for these equations.
Finally, numerical examples complete this work.
Keywords. QMC-Nyström, lattice quadrature, worst case error, multi-dimensional
integral equation, multi-dimensional integro-differential equations.
2010 Mathematics Subject Classification. 65N60, 65N15.
1
Introduction
The paper aims to extend the integral equation to the following system of Fredholm
integro-differential equations given by
Z
Z
!
p.u/ D g.x/ C 1
1 .x; y/u.y/d y C 2
2 .x; y/ ru.y/d y;
(1.1)
D
D
with the initial conditions
8
ˆ
< u.x/ D b0 in x 2 @D1 ;
or
ˆ
!
:
ru.x/ D b2 in x 2 @D2 ;
(1.2)
where @D1 ; @D2 @D and
!.x/ ru.x/
p.u/ D ˛0 .x/u.x/ C ˛
1
(1.3)
! are given
for x 2 D D Œ0; 1d , 1 ; 2 2 R and d 2 N. Also, 1 , !
2 , g, ˛0 and ˛
1
functions from a weighted Korobov space. This problem describes several interesting physical and financial phenomena (see [7–9, 13, 14, 22]).
On the other hand, the existence and uniqueness of the above problem were
proved by many authors (see [3, 11, 12, 17, 18, 24]). The paper proposes a convergence analysis of discretization schemes for these equations. The convergence
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D. Rostamy, M. Jabbari and M. Gadirian
analysis is cast into the general setting of information based complexity (IBC). Numerical solution by collocation discretization is proposed, with collocation points
chosen as lattice points of a QMC integration rule.
1.1
Preliminary
(See [4, 5, 10].) The weighted Korobov spaces are characterized by a smoothness parameter ˛ > 1 and weights 1 1 2 > 0 where j moderates
the behavior of the function with respect to the j th variable; a small j means
d .D/ denote
that the function depends weakly on the j th variable. Let H D H;
˛
a weighted Korobov space, where D .j /j 1 is a sequence of positive weights
and ˛ > 1 is a smoothness parameter. For any
Z
X
2 i h:x
u.x/ D
u.h/e
O
with u.h/
O
D
u.x/e 2 ih:x d x;
D
h2Zd
the norm of u in H is given by
kukH D
X
1=2
ju.h/j
O
ra .; h/
;
2
h2Zd
therefore, if we consider
X
@u
D
u.h/2
O
ihj e 2 i h:x
@xj
d
for j D 1; : : : ; d;
(1.4)
h2Z
then we have
kru.x/kH;1 D
X
1=2
ku.h/2
O
ihk21 r˛ .; h/
;
h2Zd
where
r˛ .; h/ D
d
Y
r˛ .j ; hj /;
(1.5)
j D1
and
´
1
r˛ .j ; hj / D
j 1 jhj j˛
if hj D 0;
otherwise.
Thus, we have r˛ .; h/ 1 for all h 2 Zd . If we use the Cauchy–Schwarz inequality, it shows that for all u 2 H the following inequality is made:
X
1=2 X
1=2
X
1
2
kuksup ju.h/j
O
ju.h/j
O
r˛ .; h/
:
r˛ .; h/
d
d
d
h2Z
h2Z
h2Z
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Worst case error for integro-differential equations
283
Therefore we have
d
Y
kuksup kukH
1 C 2.˛/j
1=2
;
(1.6)
j D1
where
.x/ WD
1
X
j
x
j D1
denotes the Riemann zeta function. Furthermore the inequality in (1.6) becomes
equality when u is a multiple of the function
X e 2 ih:x
:
r˛ .; h/
d
h2Z
The rest of this paper is organized as follows. The paper investigates two cases
of (1.1) in Sections 2 and 3. In these sections, we will obtain that the worst case
˛
error achieves the optimal rate of convergence O.n 2 Cı /, ı > 0, in weighted Korobov spaces, for a sufficiently large n. We assume that t1 ; : : : ; tn 2 D form a set
of rank-1 lattice points. On the other hand, tractability in the absolute sense means
that the minimal value of n is needed in the Quasi Monte Carlo Nyström (QMCNyström) method to reduce the worst case error to " 2 .0; 1/ and it is a bounded
polynomial in d and " 1 . Also, the tractability and strong tractability of the QMCNyström method in the absolute or normalized sense are investigated. Of course,
we know that strong tractability means that the bound is independent of d . Also,
we will show that strong QMC-Nyström tractability in the absolute sense holds iff
1
X
j < 1;
(1.7)
j D1
and QMC-Nyström tractability in the absolute sense holds iff
Pd
j D1 j
lim sup
< 1:
d !1 log.d C 1/
(1.8)
Moreover, strong tractability in the normalized sense is defined in term of the
normalized error with respect to the initial error. The conditions (1.7) and (1.8) are
also sufficient conditions for strong QMC-Nyström tractability in the normalized
sense [23,25]. Finally, some numerical results for two cases show that the proposed
method has merited. Also, some important propositions and results are proved in
appendices.
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284
2
D. Rostamy, M. Jabbari and M. Gadirian
First case study of (1.1)
In this section, we will consider 1 .x; y/ D .x; y/a1 .y/, !
2 .x; y/ D .x; y/!
a2 .y/,
!
!
D 1 D 2 , ˛0 .x/ D 1 and ˛1 .x/ D 0 in (1.1). Therefore we have the following equation:
Z
Z
!
u.x/ D g.x/ C .x; y/a1 .y/u.y/d y C
.x; y/a2 .y/ ru.y/d y ; (2.1)
D
D
with the initial conditions (1.2), where the kernel is assumed to be of the form
.x; y/ WD k.x y/, with k.x/ having period one in each component of x. Further,
we assume that g, k belong to a weighted Korobov space H.
We approximate u in (2.1) by using the Nyström method based on QMC rules,
that is, equal-weight integration rules. We assume that t1 ; : : : ; tn 2 D and the approximation of u is given by
n
X
un .x/ WD g.x/ C
k.x ti /a1 .ti /un .ti / C k.x ti /!
a2 .ti / run .ti / ; (2.2)
n
i D1
by differentiating from (2.2), we have
n
rx un .x/ WD rx g.x/ C
X
rx k.x
n
ti / a1 .ti /un .ti / C !
a2 :.ti /run .ti / ; (2.3)
i D1
where the function values un .t1 /; : : : ; un .tn / are obtained by solving the following
linear system from (2.2) and (2.3):
8
n
X
ˆ
ˆ
ˆ
k.tj ti /a1 .ti /un .ti /
u
.t
/
WD
g.t
/
C
n
j
j
ˆ
ˆ
n
ˆ
ˆ
i
D1
ˆ
ˆ
<
C k.tj ti /!
a2 .ti / run .ti / ;
(2.4)
n
ˆ
X
ˆ
ˆ
ˆ
rx k.tj ti / a1 .ti /un .ti /
rx un .tj / WD rx g.tj / C
ˆ
ˆ
ˆ
n
ˆ
i
D1
ˆ
:
C!
a2 .ti / run .ti / ;
for j D 1; 2; : : : ; n.
We recall that the system of (2.4) is as a reduced linear system by required
conditions given in equations (1.2).
Therefore, in this section we investigate the worst case error of the QMCNyström method which essentially the worst possible error u un , measured in
sup norm. Also, we assume that
!
k1 .x y/ WD k.x y/a1 .y/; k2 .x y/ WD k.x y/!
a2 .y/:
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285
Hence we make a good set of points t1 ; : : : ; tn which leads to a as small worst
case error as possible. There are alternative methods for tending to this goal, so
we select a rank-1 lattice rule as a QMC-rule with points given by ti D ¹ inz º with
i D 1; 2; : : : ; n (see [6,16]). In particular, we need a class of lattice-Nyström methods such that z is introduced by the generating vector which is an integer vector
having no factor in common with n, and the braces around a vector indicate that
each component of the vector is to be replaced by its fractional part.
2.1
Lower and upper bounds on the worst case error for (2.1)
Let C 1;v .D/ be a set of continuous functions u W D ! R which are one time
continuously differentiable in D and such that for all t 2 Dn@D, the following
estimate holds:
8
ˆ
if v < 0;
<1
kruk c.u/ 1 C jlog .t/j if v D 0;
ˆ
: v
.t /
if v > 0;
where 1 < v < 1, c.u/ is a positive constant and .t/ D min0<t<1 ¹t; 1 tº
is the distance from t 2 .0; 1/ to the boundary of the interval .0; 1/. We equip
the space of bounded linear operators from C 1;v .D/ to C 1;v .D/ with the usual
operator norm
kKk D sup kKuksup ;
kuksup 1
for a given kernel, we are interested in the integral operator
K W C 1;v .D/ ! C 1;v .D/
given by
Z
.x; y/ a1 .y/u.y/ C !
a2 .y/:ru.y/ d y
D
Z
Z
D
1 .x; y/u.y/d y C
3 .x; y/d y;
Ku D
D
D
where 3 .x; y/ D !
2 .x; y/ ru.y/ and !
2 .x; y/ D 2 .x; y/!
a2 .y/ with
Z
Z
kKk D max
j1 .x; y/jd y C max
j3 .x; y/jd y;
x2D
x2D
D
D
and the corresponding discrete operator
Kn W C 1;v .D/ ! Pn
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D. Rostamy, M. Jabbari and M. Gadirian
given by
n
Kn u D
1X
k.x
n
ti / a1 .ti /u.ti / C !
a2 .ti / ru.ti /
i D1
n
1X
D
k1 .x
n
n
ti /u.ti / C
i D1
with
1 X!
k2 .x
n
n
kKn k D max
x2D
ti / ru.ti /;
i D1
1X
jk1 .x
n
n
ti /j C max
x2D
i D1
1X
jk3 .x
n
ti /j;
i D1
where t1 ; : : : ; tn 2 D. Thus
Z
Z
jk3 .y/jd y kk1 ksup C kk3 ksup
jk1 .y/jd y C
kKk D
D
D
and
n
kKn k D max
x2D
1X
jk1 .x
n
n
ti /j C max
x2D
i D1
1X
jk3 .x
n
ti /j kk1 ksup C kk3 ksup ;
i D1
where the inequalities become equalities when k1 and k3 are constant functions.
Here, we assume that the operator K is a compact and bounded operator (see
Appendix 1).
!
If g, k1 , k2 2 H are given, then we study the solution of (2.1) to the re-solvent
formula
!
S.g; k1 ; k2 / WD u;
which we express as u D g C Ku or as .I
K/u D g, where
I W C 1;v .D/ ! C 1;v .D/
denotes the identity operator Iu D u also, we assume that the operator .I K/ 1
exists. Thus, by using the Fredholm alternative, we have k.I K/ 1 k < 1 and
u D .I K/ 1 g.
Therefore we have
!
X
2 i h:x O
.Ku/.x/ D
u.h/e
O
k1 .h/ C kO2 .h/:2 ih
h2Zd
implying
!
u.h/
O
D g.h/
O
C kO1 .h/ C kO2 .h/ 2 i h u.h/;
O
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287
hence we conclude the following results:
g.h/
O
;
!
O
O
k1 .h/ C k2 .h/ 2 i h
u.h/
O
D
1
(2.5)
and
kukH
Xˇ
ˇ
ˇ
D
ˇ
h2Zd
k.I
ˇ2
1=2
ˇ
g.h/
O
ˇ r˛ .; h/
!
ˇ
kO1 .h/ C kO2 .h/ 2 ih
1
1
K/
(2.6)
kC kgkH ;
!
where the inequality becomes equality when g, k1 and k2 are constant functions.
We use QMC-Nyström method by the algorithm
!
An .g; k1 ; k2 / WD un
or un D g C Kn un :
Suppose that
n WD k.I
Then the operator .I
1
Kn /
k.I
Kn /
1
K/
kk.K
Kn /Kn k < 1:
exists and
1
1 C k.I
k
K/ 1 kkKn k
:
1 n
Then un is well defined and we have
un D .I
1
Kn /
g:
Therefore we observe that n < 1 is essentially as a related condition on the value
of n and the equality of the points t1 ; : : : ; tn .
We assume that ˇ > 0 and > 1 are fixed. Therefore we recall that
!
S.g; k1 ; k2 / D .I
and
K/
!
An D An .g; k1 ; k2 / D .I
1
g
Kn /
1
g:
Hence we define the worst case error of a QMC-Nyström method by
en;d .An / WD
sup
!!
k1 ;k2 ;k4 ;g2Æ
!
kS.g; k1 ; k2 /
!
An .g; k1 ; k2 /ksup ;
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D. Rostamy, M. Jabbari and M. Gadirian
where
® !
!
Æ D k1 ; k2 ; g W kgkH 1; kk1 kH ˇ; kk2 kH;1 ˇ;
krki kH;1 ˇ for i D 1; 2; k.I
and
!
k4 .x
!
y/ D r k2 .x
K/
1
¯
k ;
y/:ru.tj /:
Also, we write the following inequality:
!
kS.g; k1 ; k2 /
!
An .g; k1 ; k2 /ksup en;d .An / kgkH :
Note that the constants ˇ and in Æ are mutually independent. We define the
initial error associated with the zero algorithm A0 0 as
e0;d .A0 / WD
sup
!!
k1 ;k2 ;k4 ;g2Æ
!
kS.g; k1 ; k2 /ksup :
For " 2 .0; 1/, we are interested in finding the smallest value of n for which
en;d .An / ";
corresponding to tractability in the absolute sense, or
en;d .An / "e0;d .A0 /;
corresponding to tractability in the normalized sense. For " 2 .0; 1/ and d 1, we
define the following set (see [13, 14]):
nabs ."; d / WD min¹n W 9QMC-Nyström method An with en;d .An / "º:
The integral equation in this section is said to be QMC-Nyström tractable in the
absolute sense iff there exist nonnegative constants C , p and q independent of "
and d such that (see [4])
nabs ."; d / C "
p
d q;
for all " 2 .0; 1/; d 1;
and the problem is said to be strongly QMC-Nyström tractable in the absolute
sense iff the above condition holds with q D 0.
Furthermore, tractability and strong tractability in the normalized sense are defined in a similar way, with nabs ."; d / replaced by
nnor ."; d / WD min¹n W 9QMC-Nyström method An with en;d .An / "e0;d º:
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Worst case error for integro-differential equations
On the other hand, we obtain the initial error, from (1.6) and (2.6),
!
kS.g; k1 ; k2 /ksup k.I
K/
1
kkgkH
d
Y
1 C 2.˛/j
1=2
j D1
d
Y
1 C 2.˛/j
1=2
;
j D1
that is an upper bound on the initial error e0;d and does not depend on ˇ.
Now, we obtain a lower bound on initial error. We assume that
L 1
1 .x; y/ D c WD min ˇ;
;
where
Z
LD
D
!
k2 rgd y
!
!
!
and k2 is a constant vector such that kk2 kH;1 ˇ and kr k2 kH;1 ˇ. We define g such that
1
g.h/
O
1
D
;
!
Gr
˛ .; h/
O
O
.k1 .h/ C k2 .h/:2 ih/
where
G WD
d
Y
1 C 2.˛/j
1=2
j D1
and kgkH 1.
On the other hand, if we put ai .x/ D 1, bi .y/ D c, and ki .x y/ D ai .x/bi .y/,
i D 1; 2, in the equation
Z
Z
!
u.x/ D g.x/ C k1 .x y/u.y/d y C
k2 .x y/ ru.y/d y ;
(2.7)
D
D
then we have
u.x/ D g.x/ C c1 C c2 ;
(2.8)
R !
where c1 D D cu.y/d y and c2 D D k2 :ru.y/d y. By differentiating from (2.8),
we have
ru.x/ D rg.x/:
(2.9)
R
Substituting (2.8) and (2.9) in (2.7), we write
Z
Z
!
1
c1 D
c
g.y/d y C
k2 rgd y C c2 .c
1 c
D
D
1/ :
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D. Rostamy, M. Jabbari and M. Gadirian
Therefore c2 is a free parameter and c1 is obtained in terms of c2 ; then we have
1 C L
k.I K/ 1 k D kuk 1 C
c C L C c2 .c 1/ C c2 D
;
1 c
1 c
L 1
such that c .
By the above assumptions, (1.6) and (2.5), we know
d
Y
1 X e 2 i h:x 1=2
!
D
kS.g; k1 ; k2 /ksup D 1 C 2.˛/j
;
G
r˛ .; h/ sup
d
j D1
h2Z
!
and we know that for proving the above inequality, S.g; k1 ; k2 / is a factor of
X e 2 ih:x
:
r˛ .; h/
d
h2Z
Then we have
d
Y
1=2
!
kuksup D kS.g; k1 ; k2 /k D kukH
1 C 2.˛/j
:
j D1
On the other hand, we have kukH D 1, because
X
1=2
2
kukH D
r˛ .; h/
ju.h/j
O
h2Zd
D
X
h2Zd
D
1=2
1
r˛ .; h/
jGr˛ .; h/j2
1=2
1 X
1
G2
r˛ .; h/
d
h2Z
X
1
1
1
C
D
G r˛ .; 0/
r˛ .; h/
1=2
h¤0
d
XY
1
1
D
C
j jhj
Qd
G
j D1 1
h¤0 j D1
d
d
Y
X
1 Y
D
1C
j 2
jhj
G
j D1
j D1
˛
1=2
˛
1=2
D 1:
h¤0
In the above, we assume that g D k1 D c and we show another lower bound for
k.I K/ 1 k D 1 cc . Thus, we have the following proposition.
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Worst case error for integro-differential equations
!
!
Proposition 2.1. If k1 ; k2 ; k3 ; k4 2 Æ, then
!
d
d
Y
Y
1=2
1=2
c
max
;
e0;d 1 C 2.˛/j
1 C 2.˛/j
:
1 c
j D1
j D1
Hence we have a lower bound and upper bound on the initial error with the same
dependence on d ; in other words we know exactly how the initial error increases
with d .
Proposition 2.2. If c WD min.ˇ; Nyström method (2.4) satisfies
L 1
/,
then the worst case error for the QMC-
d
c
2.˛/1 1 Y
;
en;d .An / max
.1 C 2.˛/w˛ j /
1 c
n˛
n
!1=2
1
;
j D1
where w˛ 1 is a constant independent of n and d .
R !
L 1 !
Proof. We consider k1 D c WD min.ˇ; /, k2 and L D D k2 rgd y such
that
!
!
kk2 kH;1 < ˇ and kr k2 k ˇ;
hence we write
u
" Z
un D
c
g.y/d y
1 c
D
!#
n
1X
g.ti / :
n
i D1
Therefore we have the inequality
!
sup kS.g; k1 ; k2 /
en;d .An / !
An .g; k1 ; k2 /ksup
kgkH 1
1 c
D
1
sup
kgkH 1
ˇ!
n
ˇ
1X
ˇ
g.ti /ˇ
ˇ
n
ˇZ
ˇ
ˇ
c ˇ g.y/d y
ˇ D
i D1
c wor-int
e
.t1 ; : : : ; tn /:
c n;d
Hence we conclude
wor-int
en;d
.t1 ; : : : ; tn /
wor-int
en;1
1 2
n 1
0; ; ; : : : ;
n n
n
D
2.˛/1
n˛
1=2
;
(2.10)
wor-int
where en;d
.t1 ; : : : ; tn / denotes the worst case integration error in H using quadrature points t1 ; : : : ; tn . If we consider Sharygin’s lower bound [4], then this rate
of convergence of O.n ˛=2 / is optimal for the integration problem in weighted
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D. Rostamy, M. Jabbari and M. Gadirian
Korobov spaces. In fact, it was proved in [4, 5] that a generating vector z for a
rank-1 lattice rule constructs component-by-component to achieve the rate of convergence O.n ˛=2Cı /; ı > 0.
Also, some authors have proved (see [20, 21]) that
wor-int
en;d
.t1 ; : : : ; tn / d
1 Y
.1 C 2.˛/w˛ j /
n
!1=2
1
;
(2.11)
j D1
where w˛ WD min.1; 1=.21 jmin j// 1, with
the minimum of the function
1 < min <
1C2
˛
denoting
1
X
cos.2hx/
.x/ D
h˛
hD1
(also, see [15, 16]).
Moreover, it was proved in [25,26] that the integration problem in weighted Korobov spaces is strongly QMC tractable iff (1.7) holds, and QMC tractable iff (1.8)
holds.
In the following we obtain an upper bound for the worse case error. We recall
that
.I Kn /un D g
and
.I
Kn /u D .I
K/u C .K
Kn /u D g C .K
Kn /u:
Therefore we obtain
u
un D .I
Kn /
1
.K
Kn /u:
Thus, we have the following inequality:
!
kS.g; k1 ; k2 /
!
An .g; k1 ; k2 /ksup D ku
un ksup
k.I
Kn /
1
kk.K
Kn /uksup :
Therefore we have
k.I
Kn /
1
k
1 C k.I
K/ 1 kkKn k
;
1 n
where
n WD k.I
K/
1
kk.K
Kn /Kn k < 1:
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Worst case error for integro-differential equations
Therefore we conclude
kKn k kk1 ksup C kk3 ksup .kk1 kH C kk3 kH /
d
Y
1 C 2.˛/j
1=2
: (2.13)
j D1
Hence we write
ku
un ksup 1 C k.I
1 k.kk
K/
1 kH
k.I
1
k.K
C kk3 kH /
K/
Qd
1 kk.K
j D1
1 C 2.˛/j
1=2
Kn /Kn k
Kn /uksup :
The term k.K Kn /Kn k controls whether or not n < 1 while k.K Kn /uksup
determines the rate of convergence. It remains to obtain bounds on these two terms.
Let t1 ; : : : ; tn be rank-1 lattice points generated by z, that is, ti D ¹ inz º where
¹xº D x Œx. We have
Z
Z
!
k2 .x y/ ru.y/d y
k1 .x y/u.y/d y C
..K Kn /u/.x/ D
D
D
n
1X
k1 .x
n
n
1 X!
k2 .x
n
ti /u.ti /
i D1
X
D
ti / ru.ti /
i D1
X
UO 1x .h/
h2Zd ¹0º
hz0 .mod n/
UO 2x .h/;
h2Zd ¹0º
hz0 .mod n/
where
U1x .y/ D k1 .x
y/u.y/;
!
U2x .y/ D k2 .x y/ ru.y/;
Z
O
U1x .h/ D
k1 .x y/u.y/e 2 ih:y d y
D
Z X
D
kO1 .l/e 2 i l:.x y/ u.y/e
D
D
X
l2Zd
D
X
2 i h:y
dy
l2Zd
kO1 .l/e 2 i l:x
Z
u.y/e
2 i.hCl/:y
dy
D
kO1 .l/e 2 i l:x u.h
O C l/;
l2Zd
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D. Rostamy, M. Jabbari and M. Gadirian
and
UO 2x .h/ D
!
k2 .x
Z
y/ ru.y/e
D
Z X !
D
kO2 .l/e 2 i l:.x
D
2 i h:y
y/
dy
:ru.y/e
2 i h:y
dy
l2Zd
Z
X!
2 il:x
O
D
k2 .l/e
:
ru.y/e
2 i.hCl/:y
dy
D
l2Zd
X!
kO2 .l/e 2 il:x u.h
O C l/2 i:.h C l/:
D
l2Zd
Hence we have
X
Kn /u D
.K
X
kO1 .l/u.h
O C l/e 2 i l:x
h2Zd ¹0º l2Zd
hz0 .mod n/
X
X!
kO2 .l/u.h
O C l/e 2 il:x 2 i:.h C l/:
h2Zd ¹0º l2Zd
hz0 .mod n/
Proposition 2.3. Suppose there exists an integer vector z for Sn;d .z/ defined by
Sn;d .z/ D
X
l2Zd
X
h2Zd
¹0º
hz0 .mod n/
1
r˛ .; h C l/r˛ .; h/
1=2
(2.14)
such that
Sn;d .z/ <
1
:
4ˇ 2
Then the worst case error for the lattice-Nyström method satisfies
en;d .An / .1 C 2ˇ/
1
Qd
j D1 1 C 2.˛/j
4ˇ 2 Sn;d .z/
1=2
ˇ. C 1/Sn;d .z/:
Proof. See Appendix 2.
In [4, 8, 9, 15, 16, 19], we observe the following CBC algorithm for constructing
a generating vector z.
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Worst case error for integro-differential equations
Algorithm 2.4. If n is a prime number, then:
Step 1: Set z1 D 1.
Step 2: For t D 2; 3; : : : ; d , with z1 ; : : : ; z t 1 already chosen and fixed, find a
z t 2 ¹1; 2; : : : ; n 1º to minimize Sn;t .z1 ; : : : ; z t 1 ; z t /.
In this case, the components of the generating vector z can be restricted to the
set ¹1; 2; : : : ; n 1º. It causes to the optimal rate of convergence O.n ˛=2Cı /
for ı > 0.
Proposition 2.5. If we assume that
1
X
1=.˛ 2ı/
j
< 1;
j D1
and n is a prime number such that
n .8ˇ 2 /2 26˛
d
Y
1 C 2.1 C 2
3˛ 1=2
/
.˛/j
2
;
(2.15)
j D1
then the generating vector z constructed by Algorithm 2.4 completes the optimal
rate of convergence, with
en;d.An /
CQ d;ı n ˛=2Cı
en;d .An / Cd;ı n ˛=2Cı
e0;d
for all ı 2 .0; min.2
n D n.ı; d /.
3˛ ; .˛
1/=2//, where Cd;ı and CQ d;ı are independent of
Proof. According to [9], we suppose that z 2 ¹1; 2; : : : ; n 1ºd is constructed
by Algorithm 2.4, such that n is a prime number. Then we have
Sn;d .z / 1
ın1=.2p/
d
Y
p 1=p
1 C 2.1 C ı p /1=2 .˛p/j
;
j D1
for all p 2 .1=˛; 1 and ı 2 .0; 2 3˛ . We now obtain a sufficient condition on n
to ensure that
1
Sn;d .z/ <
:
4ˇ 2
It is enough to choose n such that the upper bound in the above inequality with
1
p D 1 and ı D 2 3˛ is not greater than 8ˇ
2 . In other words, if we write (2.15),
then
1
Sn;d .z/ :
4ˇ 2
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D. Rostamy, M. Jabbari and M. Gadirian
If we consider
Sn;d .z/ d
Y
1
ın1=.2p/
p 1=p
1 C 2.1 C ı p /1=2 .˛p/j
j D1
1
;
8ˇ 2
then we have
d
Y
ın1=.2p/ 8ˇ 2
p 1=p
1 C 2.1 C ı p /1=2 .˛p/j
:
j D1
Now, we put p D 1 and ı D 2
n .8ˇ 2 /2 26˛
3˛ ;
then
d
Y
1 C 2.1 C 2
3˛ 1=2
/
.˛/j
2
;
j D1
and we conclude from the above propositions that
en;d .An / 2.1 C 2ˇ/ˇ. C 1/
ın1=.2p/
d
Y
p 1=p
1 C 2.1 C ı p /1=2 .˛p/j
.1 C 2.˛/j /1=2 ;
j D1
for all p 2 .1=˛; 1 and ı 2 .0; 2
d
Y
3˛ .
On the other hand, we know from [4]
.1 C xj / .d C 1/
xj
j D1 log.d C1/
Pd
;
(2.16)
j D1
for all xj > 0, we see that the requirement (2.15) on n does not grow with d
if (1.7) holds, and it grows only polynomially with d when (1.8) holds. The
conditions (1.7) and/or (1.8) are also sufficient to ensure that en;d .An / does not
grow faster than polynomially with d . If we assume that p D 1=.˛ 2ı/ and
ı min.2 3˛ ; .˛ 1/=2/, then we have
en;d .An / D O.n
1=.2p/
˛=2Cı
/
.˛ 2ı/=2
because in the above formula n
Dn
D n ˛=2Cı . However, we
will need to assume stronger conditions on the weights if we have the optimal rate
of convergence at the same time.
On the other hand, we analyze tractability in the normalized sense. For given
" 2 .0; 1/, we find the smallest n for which en;d .An / ". We observe that it is
sufficient to insist that
1
Sn;d .z/ ; (2.17)
1=2
Q
" 1 .1 C 2ˇ/ˇ. C 1/ jdD1 1 C 2.˛/j
C 4ˇ 2
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Worst case error for integro-differential equations
the right-hand side of which is less than 1=.4ˇ 2 /. Using Algorithm 2.4, we generate a vector z satisfying (2.17) such that
"
d
1 Y
p 2
n pr
min
1 C 2.1 C ı p /1=2 .˛p/j
2p
p2.1=˛;1 and ı2.0;2 3˛  ı
j D1
"
1
.1 C 2ˇ/ˇ. C 1/
d
Y
!2p #!
.1 C 2.˛/j /1=2 C 4ˇ 2
;
(2.18)
j D1
where pr.x/ denotes the smallest prime number greater than or equal to x. Hence
we conclude that nnor.";d // is less than or equal to the right-hand side of (2.18).
Hence, for tractability in the normalized sense we obtain
"
d
1 Y
p 2
nor
n ."; d / pr
min
1 C 2.1 C ı p /1=2 .˛p/j
2p
3˛
ı
2.1=˛;1 and ı2.0;2

j D1
#!
1
2 2p
" .1 C 2ˇ/ˇ. C 1/ C 4ˇ
:
2.2
Numerical experiments for the first case study
In this section, we present some numerical results for the proposed scheme (2.4) by
using the CBC algorithm. We carry out (2.4) by using an AMD Opteron computer
with 15 Gigabytes RAM memory with 2.2 GHz CPU for these experiments.
In (2.4), we assume that
Qd
2
d
Y
e xi
e xi
k.x/ D i D1
; p.x/ D
p ;
p.x/
2 2
i D1
D 1, @D1 D @D; D D Œ0; 1d and
u.x/ D
d
Y
.xi e
xi2
1/:
i D1
Hence we obtain g.x/ by the following cases. Therefore we will compare exact solution with approximation solution. The evolution of the absolute error of
this method, ku un k1 , for d D 10; 20 and n D 107; 523; 1009 are given in Tables 1–4 based on CPU times. In this case, we will consider two different examples
in (2.1).
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Example 2.6. We have
8
a .y/ D 1; y 2 D;
ˆ
ˆ
< 1
b0 D 0;
ˆ
ˆ
!
:!
a 2 .y/ D 0 ; y 2 D:
Example 2.7. We have
8
a1 .y/ D 1; y 2 D;
ˆ
ˆ
< !
!
b2 D 0;
ˆ
ˆ
!
:!
a 2 .y/ D 1 ; y 2 D:
n
CPU time(s)
107
523
1009
10.515
139.036
479.259
ku
un k
0.251e-9
0.103e-8
0.631e-7
Table 1. d D 10 for Example 2.6.
n
CPU time(s)
107
523
1009
23.675
305.823
1123.862
ku
un k
0.121e-8
0.3213e-8
0.321e-7
Table 2. d D 20 for Example 2.6.
n
CPU time(s)
107
523
1009
12.764
175.523
387.246
ku
un k
0.745e-8
0.194e-8
0.971e-7
Table 3. d D 10 for Example 2.7.
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Worst case error for integro-differential equations
n
CPU time(s)
107
523
1009
52.876
426.985
1432.765
ku
un k
0.642e-7
0.544-8
0.398e-7
Table 4. d D 20 for Example 2.7.
3
Second case study of (1.1) or integro-differential equation with
convection
! as a vector in (1.1); therefore we study
In this case, we consider ˛0 D 0, and ˛
1
the equation
Z
p.u/ D g.x/ C 1
D
Z
1 .x; y/u.y/d y C 2
D
!
2 .x; y/ ru.y/d y;
(3.1)
with the initial conditions (1.2) and (1.3). Therefore we have
Z
Z
!
!
a1 ru.x/ D g.x/ C 1
1 .x; y/u.y/d y C 2
2 .x; y/ ru.y/d y: (3.2)
D
D
We approximate u using the Nyström method based on quasi-Mont-Carlo rules.
Let t1 ; : : : ; tn be selected points in D that we use to approximate u:
n
n
1 X
2 X !
!
2 .x; ti / run .ti /
a1 run .x/ D g.x/ C
1 .x; ti /un .ti / C
n
n
i D1
i D1
by integrating the above equation and we assume that there is a suitable vector !
a1
such that
Z
u.x/ D !
a :ru.x/d xI
1
therefore we have
Z
un .x/ D
g.x/d x C
n Z
1 X
1 .x; ti /un .ti /d x
n
i D1
C
2
n
n Z
X
!
2 .x; ti / run .ti /d x:
i D1
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D. Rostamy, M. Jabbari and M. Gadirian
Here, we obtain un .t1 /; : : : ; un .tn /, run .t1 /; : : : ; run .tn / by solving the following linear system:
8
n
ˆ
1 X
ˆ
!
ˆ
1 .tj ; ti /un .ti /
a
ru
.t
/
D
g.t
/
C
ˆ
1
n j
j
ˆ
ˆ
n
ˆ
ˆ
i
D1
ˆ
ˆ
ˆ
n
ˆ
ˆ
2 X !
ˆ
ˆ
ˆ
C
2 .tj ; ti / run .ti /;
ˆ
ˆ
n
<
i D1
!
(3.3)
Z
n Z
ˆ
1 X
ˆ
ˆ
ˆ
un .tj / D
1 .x; ti /un .ti /dx
g.x/d x
C
ˆ
ˆ
n
ˆ
tj
ˆ
i
D1
tj
ˆ
ˆ
ˆ
!
ˆ
Z
n
ˆ
ˆ
2 X
ˆ
ˆ
C
2 .x; ti / run .ti /d x ;
ˆ
ˆ
:
n
i D1
tj
where j D 1; : : : ; n, and therefore we have 2n equations and 2n unknowns. We
e W C 1;v .D/ ! C 1;v .D/ by
define the integral operator K
Z
Z
!
e D 1
2 ru.y/d y;
(3.4)
Ku
1 .x; y/u.y/d y C 2 .x; y/
D
D
and we consider
Z
Z
e D 1 max
kKk
x2D
D
jk1 jd y C 2 max
x2D
D
jk3 jd y;
(3.5)
!
such that k1 .x y/ D 1 .x; y/, k2 .x y/ D !
2 .x; y/ and k3 D !
2 ru.y/. The
1;v
e n W C .D/ ! C 1;v .D/ is given by
corresponding discrete operator K
n
X
e n u D 1
k1 .x
K
n
ti /un .ti / C
i D1
with
e n k D max
kK
x2D
n
2 X !
k2 .x
n
ti / run .ti /;
(3.6)
i D1
n
1 X
jk1 .x
n
i D1
ti /j C max
x2D
n
2 X
jk3 .x
n
ti /j;
(3.7)
i D1
where t1 ; : : : ; tn 2 D. Thus the above kernels in (3.7) are convolution kernels with
periodic form. Therefore we define
Z
Z
e
kKk D 1
jk1 .y/jd y C 2
jk3 .y/jdy 1 kk1 ksup C 2 kk3 ksup ; (3.8)
D
D
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Worst case error for integro-differential equations
and we write
n
e n k D 1 max
kK
x2D
1X
jk1 .x
n
n
ti /j C 2 max
x2D
i D1
1X
jk3 .x
n
ti /j:
i D1
Therefore we conclude that
fn k 1 kk1 ksup C 2 kk3 ksup :
kK
(3.9)
3.1
Lower and upper bounds on the worst case error for (3.1)
!
We assume that g; k1 ; k2 2 H are given functions and we study the solutions of
!
S.g; k1 ; k2 / D u
from (3.2) as
!
e
a1 ru D g C Ku;
e D!
where Ku
a1 ru
g. Hence we have
e 1 .!
a ru
uDK
(3.10)
g/;
1
(3.11)
since .x; y/ D k.x y/.
On the other hand, we write
Z
Z
!
e
Ku.x/
D 1
k1 .x y/u.y/d y C 2
k2 .x y/ ru.y/d y
D
D
X
Z X
2 ih:.x y/
2 i h:y
O
D 1
k1 .h/e
u.h/e
O
dy
D
h2Zd
h2Zd
Z X !
C 2
kO2 .h/e 2 i h:.x
D
D 1
X
X
h2Zd
2 ih:x
kO1 .h/u.h/e
O
C 2
h2Zd
y/
u.h/2
O
ih:e
2 i h:y
dy
h2Zd
X !
kO2 .h/u.h/:2
O
ihe 2 i h:x :
h2Zd
Therefore we have
e
Ku.x/
D
X
!
2 i h:x
u.h/e
O
1 kO1 .h/ C 2 kO2 .h/ 2 i h :
h2Zd
Hence, if we put
v.h/ D !
a1 ru.h/;
then we have
!
v.h/
O
D g.h/
O
C u.h/
O
1 kO1 .h/ C 2 kO2 .h/ 2 ih ;
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D. Rostamy, M. Jabbari and M. Gadirian
or we write
u.h/
O
D
v.h/
O
g.h/
O
:
!
1 kO1 .h/ C 2 kO2 .h/ 2 i h
(3.13)
Thus we conclude that
kukH
ˇ2
!1=2
ˇ
g.h/
O
ˇ
D
ˇ r˛ .; h/
!
ˇ
h2Z 1 kO1 .h/ C 2 kO2 .h/ 2 ih
e 1 k k!
kK
a ru gk :
ˇ
X ˇˇ
ˇ
ˇ
d
v.h/
O
1
(3.14)
H
!
We use QMC-Nyström method by the algorithm An .g; k1 ; k2 / WD un or
e n 1 .!
un D K
a1 ru
g/:
(3.15)
e K
e n k < 1;
K/
(3.16)
Also, if we assume that
e n WD kK
e
4
1
en
kk.K
e n 1 exists and we have
then the operator K
e
kK
1
k
e 1 kkK
e nk
1 C kK
:
en
1 
(3.17)
Now, we consider ˇ > 0, > 1,
!
e
u D S.g; k1 ; k2 / D K
and
1
.!
a1 ru
!
e n 1 .!
un D An .g; k1 ; k2 / D K
a1 ru
g/
g/:
Also, we assume that the worst case error of the QMC-Nyström method is introduced by
en;d .An / WD
sup
!

k1 ;k2 ;g2Æ
!
kS.g; k1 ; k2 /
!
An .g; k1 ; k2 /ksup ;
(3.18)
where
®
!
e D g; k1 ; k2 W k!
Æ
a1 ru
!
e
gkH 1; kk1 kH ; kk2 kH;1 ˇ; kK
1
¯
k :
!
Thus, if g 2 H is a linear function, then for all k1 ; k2 we have
!
kS.g; k1 ; k2 /
!
An .g; k1 ; k2 /ksup en;d .An /kgkH :
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(3.19)
303
Worst case error for integro-differential equations
Moreover, we define the initial error associated with the zero algorithm A0 0 as
follows:
e0;d WD
sup kS.g; k; k1 /ksup :
(3.20)
!

k1 ;k2 ;g2Æ
Therefore, the initial error follows from (1.6) and (3.15) is
e
kS.g; k1 ; k2 /ksup kK
1
kk!
a1 ru
gkH
d
Y
1 C 2.˛/j
1=2
;
j D1
thus we write
!
8g; k1 ; k2 ; 2 Æ;
kS.g; k1 ; k2 /ksup d
Y
1 C 2.˛/j
1=2
:
(3.21)
j D1
Now, we obtain a lower bound on the initial error. Let
y/ D k1 ;
k1 .x
!
k2 .x
!
y/ D ˛
1
!k
where k1 is a constant so that kk1 kH D k1 ˇ and we assume k˛
1 H;1 ˇ and
!
1
e k . Now g.x/ is defined such that ka1 ru gkH 1 and
kK
u.h/
O
D
!
1
a1 ru.h/ g.h/
O
D
;
!
Gr
.;
h/
˛
O
O
1 k1 .h/ C 2 k2 .h/ 2 i h
(3.22)
!
thus for this choice of g, k1 and k2 , we have
d
Y
1 X e 2 i h:x 1=2
!
D
1 C 2.˛/j
:
kS.g; k1 ; k2 /ksup D G
r˛ .; h/ sup
d
(3.23)
j D1
h2Z
Hence we conclude the following proposition for the initial error based on the
before section.
Proposition 3.1. Let k1 .x
kk1 kH ˇ;
!
y/ D k1 and k2 .x
! where
y/ D ˛
1
!k D k!
k˛
k 2 kH ˇ
1 H
e
and kK
1
k :
Then
d
Y
1 C 2.˛/j
j D1
1=2
e0;d d
Y
1 C 2.˛/j
1=2
:
j D1
Proof. See Appendix 3.
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D. Rostamy, M. Jabbari and M. Gadirian
Proposition 3.2. Suppose there exists an integer vector z for which Sn;d .z/ is defined in (2.14) and we assume that
Sn;d .z/ <
1
:
.1 C 2 /2 ˇ 2
Then the worst case error for the lattice-Nyström method satisfies
Q
.1 C 2 /ˇ 1 C ˇ.1 C 2 / jdD1 1 C 2.˛/j
en;d .An / Sn;d .z/;
1 ˇ 2 .1 C 2 /2 Sn;d .1 C 2 ˇ/
˛
2 Cı
and we conclude en;d .An / D O.n
/.
Proof. We obtain a sufficient condition on n to ensure that
Sn;d .z/ <
1
:
C 2 / 2
ˇ 2 .1
It is enough to choose n such that the upper bound in Section 2 with p D 1 and
ı D 2 3˛ is not greater than 2ˇ 2 .1 C /2 . In other words, if
1
2
n 2.1 C 2 /2 ˇ 2 26˛
2
d
Y
3˛ 1=2
1 C 2.1 C 2
/
.˛/j
2
;
(3.24)
j D1
then
Sn;d .z/ 1
2ˇ 2 .
1
C 2 /2
because
Sn;d .z/ d
Y
1
ın1=.2p/
p 1=p
1 C 2.1 C ı p /1=2 .˛p/j
j D1
1
:
2ˇ 2 .1 C 2 /2
Therefore we have the following inequality:
ın
1=.2p/
2
2
2ˇ .1 C 2 /
d
Y
p 1=p
1 C 2.1 C ı p /1=2 .˛p/j
:
j D1
Moreover, by Propositions 2.5 and 3.2, we conclude
en;d .An / d
2.1 C ˇ.1 C 2 // Y
.1 C 2.˛/j /.1 C 2 /ˇ
ın1=.2p/
j D1
d
Y
p 1=p
1 C 2.1 C ı p /1=2 .˛p/j
:
j D1
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Worst case error for integro-differential equations
On the other hand, if we put
Sn;d .z/ <
1
;
2.1 C 2 /2 ˇ 2
then
ˇ 2 .1 C 2 /2
1
D :
2
2
2ˇ .1 C 2 /
2
Therefore we have the following inequality:
ˇ 2 .1 C 2 /2 Sn;d 1
ˇ 2 .1
1
2
C 2 /2 Sn;d .z/
Also, if we put p D 1=.˛
1
for all p 2 .˛
2ı/ with ı min.2
en;d .An / D O.n
because in the above formula we have n
3˛ ; .˛
˛=2Cı
1=.2p/
Dn
; 1 and ı 2 .0; 2
3˛
:
1/=2/, then we obtain
/
.˛ 2ı/=2
Dn
˛=2Cı .
Using the property
d
Y
xj
j D1 log.d C1/
Pd
.1 C xj / .d C 1/
for all xj > 0;
j D1
we see that the requirement (3.24) on n does not grow with d if
holds, and it grows only polynomially with d when
Pd
j D1 j
lim sup
<1
log.d C 1/
d !1
P1
j D1 j
<1
holds.
Proposition 3.3. Suppose n is a prime number satisfying (3.24). Then the generating vector z is constructed by Algorithm 2.4, so it achieves the optimal rate of
convergence, with
en;d.An /
en;d .An / Cd;ı n ˛=2Cı and
CQ d;ı n ˛=2Cı ;
e0;d
for all ı 2 .0; min.2 3˛ ; .˛ 1/=2//, where Cd;ı and CQ d;ı are independent of n
but depend on ı and d additionally if we write
1
X
1=.˛ 2ı/
j
< 1;
j D1
then the numbers Cd;ı , CQ d;ı and the requirement (3.24) on n are bounded independently of d .
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D. Rostamy, M. Jabbari and M. Gadirian
On the other hand, for tractability in the absolute sense, we find the smallest n
for en;d .An / ". From the before proposition, we see that it is sufficient to insist
that
1
Sn;d .z/ 1
" .1 C 2 /ˇ.1 C ˇ.1 C 2 //
(3.25)
1
;
Qd
2
2
j D1 .1 C 2.˛/j /ˇ .1 C 2 /
the right-hand side of which is less than 1=ˇ 2 .1 C 2 /2 . Using Proposition 2.5,
we observe that Algorithm 2.4 will generate a vector z satisfying (3.25) if we
demand that
"
d
1 Y
p 2
n pr
min
1 C 2.1 C ı p /1=2 .˛p/j
(3.26)
2p
3˛
p2.1=˛;1 and ı2.0;2
 ı
j D1
"
1
.1 C 2 /ˇ.1 C ˇ.1 C 2 //
d
Y
!2p #!
2
.1 C 2.˛/j / C ˇ .1 C 2 /
2
; (3.27)
j D1
where pr.x/ denotes the smallest prime number greater than or equal to x. Hence
we conclude that nabs ."; d / is less than or equal to the right-hand side of (3.27).
Similarly, for tractability in the normalized sense we obtain
"
d
1 Y
p 2
nor
1 C 2.1 C ı p /1=2 .˛p/j
n ."; d / pr
min
2
p2.1=˛;1 and ı2.0;2 3˛  ı p
j D1
#!
1
2
2 2p
" .1 C 2 /ˇ.1 C ˇ.1 C 2 // C ˇ .1 C 2 /
:
3.2
Numerical experiments for the second case study
In this section, we present some numerical results for the proposed scheme (3.3) by
using the CBC algorithm. We carry out (3.3) by using an AMD Opteron computer
with 15 Gigabytes RAM memory with 2:2 GHz CPU for these experiments.
In (3.3), we assume that
Qd
2
d
xi
Y
e xi
i D1 e
; p.x/ D
1 .x/ D 2 .x/ D
p ;
p.x/
2 2
i D1
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Worst case error for integro-differential equations
307
1 D 2 D 1, @D1 D @D; D D Œ0; 1d and
u.x/ D
d
Y
.xi e
xi2
1/:
i D1
Hence we obtain g.x/ by the following cases. Therefore we will compare exact solution with approximation solution. The evolution of the absolute error of
this method, ku un k1 , for d D 10; 20 and n D 107; 523; 1009 are given in Tables 5–8 based on CPU times. In this case, we will consider two different examples
in (3.3).
Example 3.4. We have
´
!
!
a 1 .y/ D 1 ; y 2 D;
b0 D 0;
Example 3.5. We have
8
!
<!
a 1 .y/ D 1 ; y 2 D;
!
: !
b2 D 0:
n
CPU time(s)
107
523
1009
238.515
743.036
1654.259
ku
un k
0.643e-9
0.543e-8
0.687e-6
Table 5. d D 10 for Example 3.4.
n
CPU time(s)
107
523
1009
197.695
875.985
1487.092
ku
un k
0.121e-8
0.459e-9
0.043e-7
Table 6. d D 20 for Example 3.4.
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D. Rostamy, M. Jabbari and M. Gadirian
n
CPU time(s)
107
523
1009
226.987
654.841
1304.765
ku
un k
0.942e-7
0.044e-8
0.654e-6
Table 7. d D 10 for Example 3.5.
n
CPU time(s)
107
523
1009
152.876
426.985
1752.765
ku
un k
0.642e-7
0.544-8
0.398e-7
Table 8. d D 20 for Example 3.5.
A
Appendix 1
Consider the following sequence of linear operators:
Kn W C 1;v .D/ ! Pn
given by
n
Kn u D
1X
k1 .x
n
n
ti /u.ti / C
i D1
1 X!
k2 .x
n
ti / ru.ti /;
i D1
where Pn is the space of polynomials of degree less than or equal to n in D.
Therefore the dimension of this space is finite. On the other hand, we assume that
the kernel functions satisfy the following conditions:
max
x2D
and
max
x2D
n
X
jk1 .x
ti /j D B1;n < 1
i D1
n
X
!
kk2 .x
ti /k D B2;n < 1:
i D1
Also, we consider the following linear operator:
K W C 1;v .D/ ! C 1;v .D/
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Worst case error for integro-differential equations
309
given by
Z
Ku D
Z
k1 .x
D
t/u.t/dt C
!
k2 .x
D
t/ ru.t/dt:
Theorem A.1. Each operator Kn W C 1;v .D/ ! Pn has the following properties:
(1) it is a bounded operator,
(2) dim Kn .Pn / < 1,
(3) the operator Kn is compact.
Proof. It is clear that we can prove that Kn is a sequence of bounded operators
(see [1, Chapter 12]) and according to the theorem of finite dimensional domain
or rang (see [2, Chapter 8]), the proof is completed.
Theorem A.2 (Sequence of compact linear operators). Let ¹Kn º be a sequence
of compact linear operators from the normed space C 1;v .D/ into the Banach
space Pn . If ¹Kn º is uniformly operator convergent, say, kKn Kk ! 0, then
the limit operator K is compact.
Proof. See [2].
Theorem A.3 (Continuity). Let X and Y be normed spaces. Then every compact
linear operator K W X ! Y is bounded, hence continuous.
Proof. See [2].
B Appendix 2. Proof of Proposition 2.3
It follows from the Cauchy–Schwarz inequality that
ˇ
X
X
ˇ
k.K Kn /uksup D sup ˇˇ
UO 1x .h/ C
x2D
h2Zd ¹0º
hz0 .mod n/
ˇ
ˇ
D sup ˇˇ
x2D
X
h2Zd ¹0º
hz0 .mod n/
X
ˇ
ˇ
UO 2x .h/ˇˇ
kO1 .l/u.h
O C l/e 2 i l:x
h2Zd ¹0º l2Zd
hz0 .mod n/
C
X
ˇ
X!
ˇ
2 i l:x
O
k2 .l/u.h
O C l/e
2 i.h C l/ˇˇ
h2Zd ¹0º l2Zd
hz0 .mod n/
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D. Rostamy, M. Jabbari and M. Gadirian
DI
‚
ƒ
X…„
O
jk1 .l/jju.h
O C l/j
X
h2Zd ¹0º l2Zd
hz0 .mod n/
DII
…„
ƒ
X ˇ!
ˇ
ˇkO2 .l/u.h
O C l/2 i.h C l/ˇ :
‚
X
C
h2Zd ¹0º l2Z d
hz0 .mod n/
We have
X
X
I D
jkO1 .l/jju.h
O C l/j
h2Zd ¹0º l2Zd
hz0 .mod n/
"
X
jkO1 .l/j
l2Zd
X
ju.h
O C l/j2 r˛ .; h C l/
h2Zd ¹0º
hz0 .mod n/
X
h2Zd ¹0º
hz0 .mod n/
by the Cauchy–Schwarz inequality
X
X
jkO1 .l/j2
l2Zd
X
l2Zd
X
1=2 #
;
1=2
ju.h
O C l/j r˛ .; h C l/
2
X
h2Zd ¹0º
hz0 .mod n/
jkO1 .l/jkukH
l2Zd
D kukH
1
r˛ .; h C l/
h2Zd ¹0º
hz0 .mod n/
D
1=2
X
l2Zd
X
1
r˛ .; h C l/
X
h2Zd ¹0º
hz0 .mod n/
jkO1 .l/j2 r˛ .; h/1=2
1=2
1
r˛ .; h C l/
1=2
l2Zd
X
l2Zd
X
h2Zd ¹0º
hz0 .mod n/
1
r˛ .; h C l/r˛ .; h/
1=2
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311
Worst case error for integro-differential equations
D kukH kk1 kH Sn;d .z/
kk1 kH k.I
K/
1
kkgkH Sn;d .z/
ˇSn;d .z/;
and
X
II D
Xˇ
ˇ
ˇkO2 .l/ u.h
O C l/2 i.h C l/ˇ
h2Zd ¹0º l2Zd
hz0 .mod n/
X !
kkO2 .l/k21
l2Zd
X
ku.h
O C l/2 i.h C
l/k21 r˛ .; h
1=2
C l/
h2Zd ¹0º
hz0 .mod n/
X
h2Zd ¹0º
hz0 .mod n/
1
r˛ .; h C l/
1=2 by the Cauchy–Schwarz inequality
1=2
X !
X
2
2
O
ku.h
O C l/2 i.h C l/k1 r˛ .; h C l/
kk2 .l/k1
h2Zd ¹0º
hz0 .mod n/
l2Zd
X
l2Zd
D
X
h2Zd ¹0º
hz0 .mod n/
1
r˛ .; h C l/
X !
1=2
X
kkO2 .l/k21
krukH;1
l2Zd
l2Zd
1=2
X
h2Zd ¹0º
hz0 .mod n/
1
r˛ .; h C l/
1=2
X !
1=2
2
O
D krukH;1
kk2 .l/k1 r˛ .; l/
l2Zd
X
l2Zd
X
h2Zd ¹0º
hz0 .mod n/
1
r˛ .; l/r˛ .; h C l/
1=2
!
D krukH;1 kk2 kH Sn;d .z/
ˇSn;d .z/:
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D. Rostamy, M. Jabbari and M. Gadirian
In the above equation, we assume that krukH;1 1. Thus, we obtain
k.K
Kn /uksup . C 1/Sn;d .z/:
Similarly, we write
.K
Kn /Kn D KKn Kn Kn
Z
n
1X
D
k1 .x y/
k1 .y
n
D
tj /u.tj /
j D1
n
1 X!
k2 .y
C
n
!
tj / ru.tj / dy
j D1
!
k2 .x
Z
C
1
n
y/
D
n
X
rk1 .y
tj /u.tj /
j D1
n
1X !
C
r k2 .y
n
!
tj /ru.tj / d y
j D1
n
1X
k1 .x
n
ti /
i D1
n
1X
k1 .ti
n
tj /u.tj /
j D1
n
1 X!
C
k2 .ti
n
!
tj /ru.tj /
j D1
n
1 X!
k2 .x
n
i D1
n
1X
ti /
rk1 .ti
n
tj /u.tj /
j D1
n
1X !
C
r k2 .ti
n
!
tj /ru.tj /
j D1
D
1
n
n
X
Z
k1 .x
y/k1 .y
tj /u.tj /d y
j D1
n
1X
k1 .x
n
!
ti /k1 .ti
tj /u.tj /
i D1
Z
n
1X
C
k1 .x
n
!
y/k2 .y
tj /ru.tj /d y
j D1
n
1X
k1 .x
n
!
ti /k2 .ti
!
tj /ru.tj /
i D1
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Worst case error for integro-differential equations
Z
n
!
1X
k2 .x
C
n
y/rk1 .y
tj /u.tj /d y
j D1
n
!
1 X!
k2 .x
n
ti /rk1 .ti
tj /u.tj /
i D1
C
1
n
n
X
Z
!
k2 .x
!
y/r k2 .y
tj /ru.tj /d y
j D1
n
1 X!
k2 .x
n
!
ti /r k2 .ti
!
tj /ru.tj /
i D1
DW I C II C III C IV:
Therefore we have
ˇ
n Z
1 X ˇˇ
kI C IIk D sup
ˇ k1 .x
ˇ D
x2D n
y/k1 .y
tj /d y
j D1
n
1X
k1 .x
n
ˇ
ˇ
ˇ
tj /ˇ
ˇ
ti /k1 .ti
i D1
ˇ Z
n
1 X ˇˇ
k .x
C sup
ˇ
ˇ D 1
x2D n
!
y/k2 .y
tj /d y
j D1
n
!
ti /k2 .ti
1X
k1 .x
n
i D1
DW sup
x2D
1
n
n
X
ji j C sup
x2D
j D1
1
n
n
X
ˇ
ˇ
ˇ
tj / :ru.tj /ˇ
ˇ
!
jiij
j D1
and
kIII C IVk D sup
x2D
n ˇZ
1 X ˇˇ !
ˇ k2 .x
n
D
y/:rk1 .y
tj /d y
j D1
n
1 X!
k2 .x
n
ti /:rk1 .ti
i D1
ˇ Z
n
!
1 X ˇˇ
C sup
k2 .x
ˇ
ˇ
D
x2D n
!
y/:r k2 .y
ˇ
ˇ
tj /ˇˇ
tj /d y
j D1
n
1 X!
k2 .x
n
i D1
!
ti /:r k2 .ti
ˇ
ˇ
ˇ
tj / :ru.tj /ˇ
ˇ
!
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D. Rostamy, M. Jabbari and M. Gadirian
n
n
1X
1X
DW sup
jiiij C sup
jivj;
x2D n
x2D n
j D1
j D1
where
Z
X
iD
k1 .x
h2Zd
¹0º
hz0 .mod n/
Z X
X
D
D
h2Zd
¹0º
hz0 .mod n/
X
D
y/k1 .y
X
h2Zd
¹0º
hz0 .mod n/
kO1 .l/e 2 i l:x
ii D
h2Zd
dy
2 i.hCl/:y
dy
2 i.hCl/:tj
;
!
y/k2 .y
tj /:ru.tj /e
2 i h:y
dy
D
X
¹0º
hz0 .mod n/
l2Zd
X
X
D
tj /e
2 i h:y
tj /e
l2Zd
h2Zd ¹0º
hz0 .mod n/
D
k1 .y
Z
kO1 .l/e 2 i l:x kO1 .h C l/e
k1 .x
X
k1 .y
D
Z
X
dy
l2Zd
X
X
y/
kO1 .l/e 2 il:.x
h2Zd ¹0º l2Zd
hz0 .mod n/
D
2 i h:y
tj /e
D
kO1 .l/e 2 i l:x
!
k2 .y
Z
tj /:ru.tj /e
2 i.hCl/:y
dy
D
kO1 .l/e 2 i l:x kO3 .h C l/e
2 i.hCl/:tj
;
h2Zd ¹0º l2Zd
hz0 .mod n/
!
where we define k3 .y tj / D k2 .y tj / ru.tj /,
Z
X
!
iii D
k2 .x y/:rk1 .y tj /e 2 i h:y d y
h2Zd ¹0º
hz0 .mod n/
D
X
h2Zd
¹0º
hz0 .mod n/
D
X
D
Z X !
kO2 .l/e 2 i l:.x
D
y/
:rk1 .y
2 i h:y
dy
l2Zd
Z
X!
2 il:x
O
k2 .l/e
:
rk1 .y
h2Zd ¹0º l2Zd
hz0 .mod n/
tj /e
tj /e
2 i.hCl/:y
dy
D
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315
Worst case error for integro-differential equations
X!
O 1 .h C l/e
kO2 .l/e 2 i l:x :rk
X
D
h2Zd
¹0º
hz0 .mod n/
¹0º
hz0 .mod n/
D
D
¹0º
hz0 .mod n/
h2Zd
¹0º
hz0 .mod n/
y/
!
:r k2 .y
h2Zd
¹0º
hz0 .mod n/
!
!
where k4 D r k2 .y
2 i h:y
dy
tj /:ru.tj /e
2 i h:y
tj /:ru.tj /e
2 i.hCl/:y
dy
D
l2Zd
2 i.hCl/:tj
;
l2Zd
tj /ru.tj /. Therefore we write
k.K
j D1
X
j D1
n ˇ
X
ˇ
1
ˇ
C sup
ˇ
n x2D
j D1
n ˇ
X
ˇ
1
ˇ
C sup
ˇ
n x2D
j D1
X
X
kO1 .l/kO1 .h C l/e 2 i l:x e
ˇ
ˇ
ˇ
2 i.hCl/:tj ˇ
h2Zd ¹0º l2Zd
hz0 .mod n/
n ˇ
X
ˇ
1
ˇ
C sup
ˇ
n x2D
X
h2Zd
¹0º
hz0 .mod n/
X
X
kO1 .l/kO3 .h C l/e
2 i l:x
e
ˇ
ˇ
ˇ
2 i.hCl/:tj ˇ
l2Zd
X!
O 1 .h C l/e 2 i l:x e
kO2 .l/:rk
ˇ
ˇ
ˇ
2 i.hCl/:tj ˇ
h2Zd ¹0º l2Zd
hz0 .mod n/
X
X! !
kO2 .l/:kO4 .h C l/e 2 i l:x e
ˇ
ˇ
ˇ
2 i.hCl/:tj ˇ
h2Zd ¹0º l2Zd
hz0 .mod n/
!
X
jkO1 .l/jjkO1 .h C l/j C jkO1 .l/jjkO4 .h C l/j
h2Zd ¹0º l2Zd
hz0 .mod n/
dy
l2Zd
!
X!
kO2 .l/e 2 il:x :kO4 .h C l/e
X
Kn /Kn k
n ˇ
X
ˇ
1
ˇ
sup
ˇ
n x2D
tj /:ru.tj /e
Z
X!
!
kO2 .l/e 2 i l:x :
r k2 .y
X
D
!
y/:r k2 .y
Z X !
kO2 .l/e 2 i l:.x
h2Zd
D
!
k2 .x
D
h2Zd
X
;
l2Zd
Z
X
iv D
2 i.hCl/:tj
!
!
O 1 .h C l/j C jkO2 .l/ !
C jkO2 .l/ rk
k4 .h C l/j
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D. Rostamy, M. Jabbari and M. Gadirian
!
!
!
kk1 k2H C kk1 kH kk3 k C kk2 kH;1 krk1 kH;1 C jk2 kH;1 kk4 kH;1 Sn;d .z/
!
!
kk1 k2H C kk1 kH kk2 kH krukH;1 C kk2 kH;1 krk1 kH;1
!
!
C kk2 kH;1 kr k2 kH;1 krukH;1 Sn;d .z/:
!
If k1 ; k2 2 Æ and krukH;1 1, then we have
Kn /Kn k .ˇ 2 C ˇ 2 C ˇ 2 C ˇ 2 /Sn;d .z/ D 4ˇ 2 Sn;d .z/:
k.K
On the other hand, we write
n D k.I
1
K/
kk.K
Kn /Kn k 4ˇ 2 Sn;d .z/;
so we have n < 1. Hence we write
Sn;d .z/ <
1
:
4ˇ 2
Also, we observe that
k.I
Kn /
1
k
1 C k.I
Kn /
1 C 2ˇ
Qd
1 k.kk k C kk k /
1 H
3 H
1
1
1
j D1
1 C 2.˛/j
Qd
j D1
1 C 2.˛/j
1=2
n
1=2
4ˇ 2 Sn;d .z/
d
Y
1=2
1 C 2ˇ
1 C 2.˛/j
:
2
4ˇ Sn;d .z/
(B.1)
j D1
On the other hand, we conclude that
ku
un ksup k.I
Kn /
1
kk.K
Kn /uksup
d
1=2
.1 C 2ˇ/ Y
1
C
2.˛/
k.K
j
1 4ˇ 2 Sn;d .z/
Kn /uksup
j D1
d
1=2
.1 C 2ˇ/ Y
1 C 2.˛/j
2
1 4ˇ Sn;d .z/
j D1
k.I K/ 1 kkgkH kk1 kH C ˇ Sn;d .z/
1=2
Q
.1 C 2ˇ/ jdD1 1 C 2.˛/j
.ˇ C ˇ/Sn;d .z/:
1 4ˇ 2 Sn;d .z/
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317
Worst case error for integro-differential equations
C
Appendix 3. Proof of Proposition 3.1
y/ D k1 as a constant function such that
Again we consider k1 .x
e 1 k ;
kk1 kH D k1 ˇ and kK
!
also, we assume that k1 .x y/ and k2 .x y/ are separable kernels. Then, we
have
Z
Z
!
!
a1 ru.x/ D g.x/ C 1
k1 .x y/u.y/d y C 2
k2 .x y/ ru.y/d y;
D
D
or we write
!
a1 ru.x/ D g.x/ C 1 ˛1 .x/
Z
D
Z
b1 .y/u.y/d y C 2 ˛2 .x/
D
b2 .y/ ru.y/d y;
D g.x/ C 1 c1 C 2 c2 :
We find c1 and c2 by integration of the above equation, thus we have
Z
u.x/ D g.x/d x C .1 c1 C 2 c2 /x C d1 :
Hence we have
Z
Z
g.x/ C 1 c1 C 2 c2 D g.x/ C 1
k1
D
Z
C 2
D
g.y/d y C .1 c1 C 2 c2 /y C d1 d y
g.y/ C 1 c1 C 2 c2 d y;
therefore
c2 D
Z Z
1 c1
C d1
1 k1
g.y/d yd y C
2
1 2 K21 22
D
Z
C 2
g.y/d y C 1 c1 C d2 c1 1 c1
1
2
D
D l.c1 /:
Then we have
Z
u.y/ D g.y/d y C 1 c1 y
C
1
2
Z Z
1 k1 y
1 c1
C d1
2
1 c1 ;
g.y/d yd y C
1 2 K21 22
D
Z
C 2
g.y/d y C 1 c1 C d2
c1
D
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D. Rostamy, M. Jabbari and M. Gadirian
and
Z
un .y/ D
g.y/d y C 1 c1 y
C
2
n Z
1X
1 k1 y
g.y/d y
n
ti
"
1
1 2 K21
22
i D1
C
1 c1
C d1
2
n
2 X
C
g.ti / C 2 1 c1 C d2
n
#
!
c1
1 c1
i D1
and so
u
un D
Z Z
1
1 2 K21
2
22
1 2 k1 y
g.y/d yd y
D
n
Z
C 2
n Z
1X
g.y/d y
n
ti
i D1
!
g.y/d y
D
1X
g.ti /
n
:
i D1
Hence we have
en;d .An / sup kS.g; k1 ; k2 /
An .g; k1 ; k2 /ksup
kgkH 1
D
1 k1
1
k 1 1
2
Z Z
g.y/d yd y
D
n Z
1X
g.y/d y
n
ti
i D1
!
wor-int
C en;d
.t1 ; : : : ; tn / ;
so we have
DF
en;d .An / ‚
Z Z
1 k1
1
k 1 1
2
1
D
…„
ƒ
n Z
1X
g.y/d yd y
g.y/d y
n
ti
i D1
!
wor-int
C en;d
.t1 ; : : : ; tn / :
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319
Worst case error for integro-differential equations
On the other hand, by using (2.10) and (2.11), we have
en;d .An / 1 k1
k 1 1
2
1
1
max F C
2ı.˛/1
n˛
12
;
d
1 Y
.1 C 2ı.˛/w˛ j /
n
FC
! 12 !
1
:
j D1
e n un D !
We know that if K
a1 ru
g, then we have
e n 1 .!
un D K
a1 ru
fn u D !
Also, we have K
a1 ru
g/:
g and then
fn
uDK
1
e n u D .K
en
Moreover, if we write K
.!
a1 ru
g/:
e C Ku
e or
K/u
e n 1 .K
en
uDK
e CK
e n 1 Ku;
e
K/u
then we have
un
e n 1 Ku
e
uDK
e n 1 .K
en
K
e n 1 Ku
e D
K
e
K/u
e n 1 .K
en
K
e
K/u;
or
un
e
e n 1 .K
uDK
e n /u:
K
(C.1)
Therefore we conclude the following inequality:
!
kS.g; k1 ; k2 /
!
An .g; k1 ; k2 /k D ku
fn
un k kK
1
e
kk.K
e n /uksup : (C.2)
K
We recall that
e
n WD kK
1
therefore we write
e n 1k kK
e
kk.K
e n /K
e n k < 1;
K
e 1 kkK
e nk
1 C kK
:
1 n
(C.3)
On the other hand, we have
e n k 1 kk1 ksup C 2 kk3 ksup
kK
.1 kk1 kH C 2 kk3 kH /
d
Y
.1 C 2.˛/j /:
j D1
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(C.4)
320
D. Rostamy, M. Jabbari and M. Gadirian
Hence we write the following inequality:
ku
e n 1 kk.K
e
un ksup kK
e n /uksup
K
e 1 k kK
e nk
1 C kK
e
k.K
1 n
e
1 C kK
e n /uksup
K
1=2
Q
C 2 kk3 kH / jdD1 1 C 2.˛/j
e 1 k k.K
e K
e n /K
e nk
kK
1 k. kk k
1 1 H
1
e K
e n /uksup :
k.K
Moreover, we know that
Z
Z
!
e K
e n /u.x/ D 1
.K
k1 .x; y/u.y/d y C 2
k2 .x; y/ ru.y/d y
D
D
D
n
n
2 X !
1 X
k1 .x ti /u.ti / C
k2 .x ti / ru.ti /
n
n
i D1
i D1
X
X
UO 1x .h/ 2
UO 2x .h/
1
h2Zd ¹0º
hz0 .mod n/
where U1x D k1 .x
UO 1x
h2Zd ¹0º
hz0 .mod n/
!
y/u.y/, U2x D k2 .x y/:ru.y/,
Z
D
k1 .x y/u.y/e 2 ih:y d y
D
Z X
2 il:.x y/
O
k1 .l/e
u.y/e
D
D
kO1 .l/e 2 i l:x
X
Z
u.y/e
2 i.hCl/:y
dy
D
l2Zd
D
dy
l2Zd
X
D
2 i h:y
kO1 .l/u.h
O C l/e 2 il:x ;
l2Zd
and
UO 2x D
D
Z
!
k2 .x
D
y/ ru.y/e
Z X !
kO2 .l/e 2 i l:.x
D
2 i h:y
y/
dy
ru.y/e
2 ih:y
dy
l2Zd
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321
Worst case error for integro-differential equations
Z
X!
2 il:x
O
D
k2 .l/e
ru.y/e
l2Zd
D
2 i.hCl/y
dy
G
X!
kO2 .l/e 2 il:x u.h
O C l/2 i.h C l/:
l2Zd
Thus, we have
e
k.K
ˇ
ˇ
e
K n /uksup D sup ˇˇ1
x2D
ˇ
ˇ
D sup ˇˇ1
x2D
X
X
UO 1x .h/ C 2
h2Zd ¹0º
hz0 .mod n/
X
h2Zd ¹0º
hz0 .mod n/
X
ˇ
ˇ
O
U2x .h/ˇˇ
kO1 .l/u.h
O C l/e 2 il:x
h2Zd ¹0º l2Zd
hz0 .mod n/
X
C 2
ˇ
X!
ˇ
2 il:x
O
k2 .l/u.h
O C l/e
2 i.h C l/ˇˇ
h2Zd ¹0º l2Zd
hz0 .mod n/
DI0
‚
1
X
…„
X
ƒ
jkO1 .l/jju.h
O C l/j
h2Zd ¹0º l2Zd
hz0 .mod n/
X
C 2
X !
jkO2 .l/u.h
O C l/2 i.h C l/j :
h2Zd ¹0º l2Zd
hz0 .mod n/
ƒ‚
„
Therefore we compute
"
X
O
I0 1
jk1 .l/j
l2Zd
X
ju.h
O C l/j r˛ .; h C l/
¹0º
hz0 .mod n/
X
h2Zd ¹0º
hz0 .mod n/
1
l2Zd
1=2
2
h2Zd
X
…
DII 0
jkO1 .l/j2
1
r˛ .; h C l/
1=2 #
1=2
ju.h
O C l/j r˛ .; h C l/
X
2
h2Zd ¹0º
hz0 .mod n/
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D. Rostamy, M. Jabbari and M. Gadirian
X
X
l2Zd
D 1
X
h2Zd ¹0º
hz0 .mod n/
jkO1 .l/jkukH
X
l2Zd
D 1 kukH
1
r˛ .; h C l/
l2Zd
X
1=2
X
h2Zd ¹0º
hz0 .mod n/
1
r˛ .; h C l/
1=2
jkO1 .l/j2 r˛ .; l/
l2Zd
X
l2Zd
X
h2Zd ¹0º
hz0 .mod n/
1
r˛ .; l/ r˛ .; h C l/
1=2
D 1 kukH kk1 kH Sn;d .z/
e 1 kk!
1 kK
a1 ru gkH kk1 kH Sn;d .z/
1 ˇSn;d .z/:
Also, we have
X !
kkO2 .l/k1 ku.h
O C l/2 i.h C l/k1
X
II0 2
h2Zd ¹0º l2Zd
hz0 .mod n/
"
2
X
!
kkO2 .l/k1
1=2
ku.h
O C l/2 i.h C l/k21 r˛ .; h C l/
X
h2Zd ¹0º
hz0 .mod n/
l2Zd
X
h2Zd ¹0º
hz0 .mod n/
2 krukH;1
1
r˛ .; h C l/
1=2 #
X !
1=2
kkO2 .l/k21 r˛ .; h/
l2Zd
X
l2Zd
X
h2Zd ¹0º
hz0 .mod n/
1
r˛ .; h/r˛ .; h C l/
1=2
!
2 krukH;1 kk2 kH;1 Sn;d .z/
2 ˇkrukH;1 Sn;d .z/:
(C.5)
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323
Worst case error for integro-differential equations
In (C.5), we assume krukH;1 1. Therefore we have
e
k.K
e n /uksup .1 C 2 /ˇSn;d .z/:
K
(C.6)
Using a similar argument, we obtain
e
.K
e n /K
e nu D K
eK
e nu K
e nK
e nu
K
Z
n
1 X
D 1
k1 .x y/
k1 .y
n
G
tj /u.tj /
j D1
!
n
2 X !
C
k2 .y tj /ru.tj / d y
n
j D1
Z
n
!
1 X
C 2
rk1 .y tj /u.tj /
k2 .x y/
n
D
j D1
!
n
2 X !
r k2 .y tj /ru.tj / d y
C
n
j D1
1
n
n
X
k1 .x
ti /
i D1
n
1 X
k1 .ti
n
tj /u.tj /
j D1
n
2 X !
k2 .ti
C
n
!
tj /ru.tj /
j D1
2
n
n
X
!
k2 .x
ti /
i D1
n
1 X
rk1 .ti
n
tj /u.tj /
j D1
n
2 X !
r k2 .ti
C
n
!
tj /ru.tj /
j D1
Z
n
1 X
D
1
k1 .x
n
D
y/k1 .y
tj /u.tj /d y
j D1
n
1 X
k1 .x
n
!
ti /k1 .ti
tj /u.tj /
i D1
C
Z
n
2 X
1
k1 .x
n
D
!
y/k2 .y
tj /ru.tj /d y
j D1
n
1 X
k1 .x
n
!
ti /k2 .ti
!
tj /ru.tj /
i D1
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D. Rostamy, M. Jabbari and M. Gadirian
Z
n
!
1 X
2
k2 .x
C
n
G
y/rk1 .y
tj /u.tj /d y
j D1
n
2 X !
k2 .x
n
!
ti /rk1 .ti
tj /u.tj /
i D1
C
Z
n
!
2 X
2
k2 .x
n
G
!
y/r k2 .y
tj /ru.tj /d y
j D1
n
2 X !
k2 .x
n
!
ti /r k2 .ti
!
tj /ru.tj / ;
i D1
DW I C II C III C IV:
We write
I C II D
Z
n
1 X
1
k1 .x
n
D
y/k1 .y
tj /d y
j D1
n
1 X
k1 .x
n
!
ti /k1 .ti
tj / u.tj /
i D1
C
Z
n
2 X
1
k1 .x
n
G
!
y/k2 .y
tj /d y
j D1
n
1 X
k1 .x
n
!
ti /k2 .ti
!
tj / :ru.tj /;
i D1
ˇ Z
n
1 X ˇˇ
k1 .x
kI C IIk D sup
ˇ1
ˇ
D
x2D n
y/k1 .y
tj /d y
j D1
n
1 X
k1 .x
n
ti /k1 .ti
ˇ
ˇ
ˇ
tj /ˇ
ˇ
!
y/k2 .y
tj /d y
i D1
ˇ
Z
n
2 X ˇˇ
k1 .x
C sup
ˇ 1
ˇ
D
x2D n
j D1
n
1 X
k1 .x
n
i D1
DW sup
x2D
1
n
n
X
j D1
ji j C sup
x2D
2
n
n
X
!
ti /k2 .ti
ˇ
ˇ
ˇ
tj / :ru.tj /ˇ
ˇ
!
jiij
j D1
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325
Worst case error for integro-differential equations
and
ˇ Z
n
!
1 X ˇˇ
kIII C IVk D sup
k2 .x
ˇ2
ˇ
D
x2D n
y/:rk1 .y
tj /d y
j D1
n
2 X !
k2 .x
n
ti /:rk1 .ti
ˇ
ˇ
ˇ
tj /ˇ
ˇ
!
y/:r k2 .y
tj /d y
i D1
ˇ
Z
n
!
2 X ˇˇ
C sup
k2 .x
ˇ 2
ˇ
D
x2D n
j D1
n
2 X !
k2 .x
n
ˇ
ˇ
ˇ
tj / :ru.tj /ˇ
ˇ
!
!
ti /:r k2 .ti
i D1
DW sup
x2D
1
n
n
X
jiiij C sup
x2D
j D1
n
X
2
n
jivj
j D1
where
n
Z
i D 1
D
k1 .x
y/k1 .y
X
1
¹0º
hz0 .mod n/
X
1
h2Zd
¹0º
hz0 .mod n/
D
X
1
h2Zd
D
Z
k1 .x
ti /k1 .ti
tj /
y/k1 .y
tj /e
2 ih:y
dy
D
Z X
D
kO1 .l/e 2 i l:.x
y/
k1 .y
2 i h:y
tj /e
dy
l2Zd
X
¹0º
hz0 .mod n/
l2Zd
X
X
1
!
i D1
h2Zd
D
tj /d y
D
1X
k1 .x
n
kO1 .l/e 2 il:x
Z
k1 .y
tj /e
2 i.hCl/:y
dy
D
kO1 .l/e 2 il:x kO1 .h C l/e
2 i.hCl/tj
;
h2Zd ¹0º l2Zd
hz0 .mod n/
Z
ii D 1
D
1
!
k1 .x y/k2 .y tj /ru.tj /d y
D
n
!
1X
k1 .x ti /k2 .ti
n
!
tj /:ru.tj /
i D1
X
h2Zd ¹0º
hz0 .mod n/
Z
k1 .x
!
y/k2 .y
tj /:ru.tj /e
2 i h:y
dy
D
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D
D. Rostamy, M. Jabbari and M. Gadirian
Z X
X
1
h2Zd
¹0º
hz0 .mod n/
D
X
1
¹0º
hz0 .mod n/
l2Zd
X
X
1
y/
!
k2 .y
tj /:ru.tj /e
2 i h:y
dy
l2Zd
X
h2Zd
D
D
kO1 .l/e 2 i l:.x
kO1 .l/e 2 il:x
!
k2 .y
Z
tj /:ru.tj /e
2 i.hCl/:y
dy
D
kO1 .l/e 2 il:x kO3 .h C l/e
2 i.hCl/:tj
;
h2Zd ¹0º l2Zd
hz0 .mod n/
iii D 2
D
n
!
k2 .x
Z
y/:rk1 .y
Z
X
2
h2Zd
¹0º
hz0 .mod n/
D
h2Zd
¹0º
hz0 .mod n/
D
h2Zd
¹0º
hz0 .mod n/
!
k2 .x
Z
iv D 2
D
2
2
2 i h:y
tj /e
dy
y/
rk1 .y
tj /e
2 i h:y
dy
tj /e
2 i.hCl/:y
dy
D
2 i.hCl/:tj
;
l2Zd
!
y/:r k2 .y
tj /:ru.tj /d y
D
1 X!
k2 .x
n
i D1
Z
X
h2Zd ¹0º
hz0 .mod n/
D
tj /
l2Zd
l2Z d
n
D
y/:rk1 .y
X!
O 1 .h C l/e
kO2 .l/e 2 il:x :rk
X
2
ti /:rk1 .ti
D
Z
X!
2 i l:x
O
k2 .l/e
:rk1 .y
X
2
!
k2 .x
Z X !
kO2 .l/e 2 i l:.x
X
2
!
i D1
h2Zd ¹0º
hz0 .mod n/
D
tj /d y
D
1 X!
k2 .x
n
X
h2Zd
¹0º
hz0 .mod n/
!
ti /:r k2 .ti
!
k2 .x
!
tj /ru.tj /
!
y/:r k2 .y
tj /:ru.tj /e
2 i h:y
dy
D
Z X !
kO2 .l/e 2 i l:.x
D
y/
!
:r k2 .y tj /:ru.tj /e
2 i h:y
l2Zd
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dy
327
Worst case error for integro-differential equations
D
Z
X!
!
2 il:x
O
k2 .l/e
:
r k2 .y
X
2
h2Zd
l2Zd
¹0º
hz0 .mod n/
D
!
X!
kO2 .l/e 2 il:x : kO 4 .h C l/e
X
2
tj /ru.tj /e
2 i.hCl/:y
dy
D
2 i.hCl/tj
:
h2Zd ¹0º l2Zd
hz0 .mod n/
Therefore we conclude
e
k.K
e n /K
e nk
K
n ˇ
X
ˇ
21
ˇ
sup
ˇ
n x2D
X
j D1
j D1
n ˇ
X
ˇ
1 2
ˇ
C
sup
ˇ
n x2D
j D1
n ˇ
X
ˇ
22
ˇ
sup
ˇ
n x2D
j D1
X
kO1 .l/kO1 .h C l/e 2 i l:x e
ˇ
ˇ
ˇ
2 i.hCl/:tj ˇ
h2Zd ¹0º l2Zd
hz0 .mod n/
n ˇ
X
ˇ
1 2
ˇ
C
sup
ˇ
n x2D
C
X
X
X
X
kO1 .l/kO3 .h C l/e 2 i l:x e
ˇ
ˇ
ˇ
2 i.hCl/:tj ˇ
h2Zd ¹0º l2Zd
hz0 .mod n/
X
X!
O 1 .h C l/e 2 il:x e
kO2 .l/:rk
ˇ
ˇ
ˇ
2 i.hCl/:tj ˇ
h2Zd ¹0º l2Zd
hz0 .mod n/
X
X! !
kO2 .l/: kO 4 .h C l/e 2 il:x e
ˇ
ˇ
ˇ
2 i.hCl/:tj ˇ
h2Zd ¹0º l2Zd
hz0 .mod n/
21 jkO1 .l/jjkO1 .h C l/j C 1 2 jkO1 .l/jjkO3 .h C l/j
h2Zd ¹0º l2Zd
hz0 .mod n/
!
!
!
O 1 .h C l/j C 2 jkO2 .l/ kO 4 .h C l/j
C 1 2 jkO2 .l/ rk
2
!
!
21 kk C1 k2H C 1 2 kk1 kH kk2 kH;1 C 1 2 kk2 kH;1 krk1 kH;1
!
! C 22 jk2 kH kr k2 kH Sn;d .z/;
thus we have
e
k.K
e n /K
e n k .21 ˇ 2 C 21 2 ˇ 2 C 22 ˇ 2 /Sn;d .z/;
K
and we write
e
n WD kK
1
e
k k.K
e n /K
e n k .1 C 2 /2 ˇ 2 Sn;d .z/:
K
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328
D. Rostamy, M. Jabbari and M. Gadirian
1
Also, if we put n < 1, then it is sufficient to demand that Sn;d .z/ < . C
2 2.
1
2/ ˇ
Hence we write the following inequality:
Q
e 1 k.1 kk1 kH C 2 kk3 kH / d .1 C 2.˛/j /
1 C kK
j D1
1
en k kK
1 n
Q
1 C .1 C 2 /ˇ jdD1 .1 C 2.˛/j /
1 ˇ 2 .1 C 2 /2 Sn;d .z/
1
d
Y
1 C ˇ.1 C 2 /
.1 C 2.˛/j /:
ˇ 2 .1 C 2 /2 Sn;d .z/
j D1
On the other hand, from (C.1) we have
ku
e n 1 k k.K
e
un ksup kK
e n /uksup
K
1
d
Y
1 C ˇ.1 C 2 /
e
.1 C 2.˛/j /kK
ˇ 2 .1 C 2 /2 Sn;d .z/
1
1 C ˇ.1 C 2 /
ˇ 2 .1 C 2 /2 Sn;d .z/
j D1
d
Y
k k!
a1 ru gkH kk1 kH
!
C 2 krukH;1 kk2 kH;1 Sn;d .z/
e
.1 C 2.˛/j / 1 kK
j D1
e n /uksup
K
.1 C ˇ.1 C 2 //
1
ˇ 2 .1
1
Qd
C
j D1 .1 C 2.˛/j /
.1 2 /2 Sn;d .z/
C 2 /ˇSn;d .z/:
Acknowledgments. The authors express their gratitude to Professor Joe Stephen
form University of Waikato for many helpful remarks and suggestions. During the
preparation of the paper, we received helpful suggestions from Dr. Ali Abkar from
Imam Khomeini International University. Therefore, the authors are very grateful
to them for valuable comments.
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Received December 19, 2012; accepted September 23, 2013.
Author information
Davoud Rostamy, Department of Mathematics,
Imam Khomeini International University, Qazvin, Iran.
E-mail: [email protected]
Mohammad Jabbari, Department of Mathematics,
Imam Khomeini International University, Qazvin, Iran.
Mahshid Gadirian, Department of Mathematics,
Imam Khomeini International University, Qazvin, Iran.
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