The block-by-block method with Romberg quadrature for the solution of nonlinear volterra integral equations on large intervals
We investigate the numerical solutions of nonlinear Volterra integral equations by the block-by-block method especially useful for the solution of integral equations on large-size intervals. A convergence theorem is proved showing that the method has at least sixth order of convergence. Finally, the...
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irk-123456789-1644482020-02-10T01:28:15Z The block-by-block method with Romberg quadrature for the solution of nonlinear volterra integral equations on large intervals Katani, R. Shahmorad, S. Статті We investigate the numerical solutions of nonlinear Volterra integral equations by the block-by-block method especially useful for the solution of integral equations on large-size intervals. A convergence theorem is proved showing that the method has at least sixth order of convergence. Finally, the performance of the method is illustrated by some numerical examples. Дослiджено чисельний розв’язок нелiнiйних iнтегральних рiвнянь Вольтерра поблочним методом, який є особливо корисним при розв’язуваннi iнтегральних рiвнянь на великих iнтервалах. Доведено теорему про збiжнiсть, яка показує, що цей метод має щонайменше шостий порядок збiжностi. Дiю методу проiлюстровано на кiлькох числових прикладах. 2012 Article The block-by-block method with Romberg quadrature for the solution of nonlinear volterra integral equations on large intervals / R. Katani, S. Shahmorad // Український математичний журнал. — 2012. — Т. 64, № 7. — С. 919-931. — Бібліогр.: 14 назв. — англ. 1027-3190 http://dspace.nbuv.gov.ua/handle/123456789/164448 517.9 en Український математичний журнал Український математичний журнал |
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Статті Статті Katani, R. Shahmorad, S. The block-by-block method with Romberg quadrature for the solution of nonlinear volterra integral equations on large intervals Український математичний журнал |
| description |
We investigate the numerical solutions of nonlinear Volterra integral equations by the block-by-block method especially useful for the solution of integral equations on large-size intervals. A convergence theorem is proved showing that the method has at least sixth order of convergence. Finally, the performance of the method is illustrated by some numerical examples. |
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Katani, R. Shahmorad, S. |
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Katani, R. Shahmorad, S. |
| author_sort |
Katani, R. |
| title |
The block-by-block method with Romberg quadrature for the solution of nonlinear volterra integral equations on large intervals |
| title_short |
The block-by-block method with Romberg quadrature for the solution of nonlinear volterra integral equations on large intervals |
| title_full |
The block-by-block method with Romberg quadrature for the solution of nonlinear volterra integral equations on large intervals |
| title_fullStr |
The block-by-block method with Romberg quadrature for the solution of nonlinear volterra integral equations on large intervals |
| title_full_unstemmed |
The block-by-block method with Romberg quadrature for the solution of nonlinear volterra integral equations on large intervals |
| title_sort |
block-by-block method with romberg quadrature for the solution of nonlinear volterra integral equations on large intervals |
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Український математичний журнал |
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2012 |
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Статті |
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http://dspace.nbuv.gov.ua/handle/123456789/164448 |
| citation_txt |
The block-by-block method with Romberg quadrature for the solution of nonlinear volterra integral equations on large intervals / R. Katani, S. Shahmorad // Український математичний журнал. — 2012. — Т. 64, № 7. — С. 919-931. — Бібліогр.: 14 назв. — англ. |
| series |
Український математичний журнал |
| work_keys_str_mv |
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| first_indexed |
2025-07-14T17:00:22Z |
| last_indexed |
2025-07-14T17:00:22Z |
| _version_ |
1837642466431336448 |
| fulltext |
UDC 517.9
R. Katani, S. Shahmorad (Univ. Tabriz, Iran)
THE BLOCK BY BLOCK METHOD WITH ROMBERG QUADRATURE
FOR SOLVING NONLINEAR VOLTERRA INTEGRAL EQUATIONS
ON THE LARGE INTERVALS
ПОБЛОЧНИЙ МЕТОД IЗ КВАДРАТУРОЮ РОМБЕРГА ДЛЯ РОЗВ’ЯЗУВАННЯ
НЕЛIНIЙНИХ IНТЕГРАЛЬНИХ РIВНЯНЬ ВОЛЬТЕРРА
НА ВЕЛИКИХ IНТЕРВАЛАХ
We investigate the numerical solution of nonlinear Volterra integral equations by block by block method, which is useful
specially for solving integral equations on large-size intervals. A convergence theorem is proved that shows that the method
has at least sixth order of convergence. Finally, the performance of the method is illustrated by some numerical examples.
Дослiджено чисельний розв’язок нелiнiйних iнтегральних рiвнянь Вольтерра поблочним методом, який є особливо
корисним при розв’язуваннi iнтегральних рiвнянь на великих iнтервалах. Доведено теорему про збiжнiсть, яка
показує, що цей метод має щонайменше шостий порядок збiжностi. Дiю методу проiлюстровано на кiлькох числових
прикладах.
1. Introduction. Consider nonlinear Volterra Integral Equations (VIEs) of the form
f(x) = g(x) +
x∫
0
k(x, s, f(s))ds, 0 ≤ x ≤ X, (1.1)
where g and k are continuous respectively on [0,∞], and D = {(x, s, f)|0 < s < x < R,−∞ <
< f <∞}. Suppose
(i) ∀X > 0 and ψ : [0, X] → R continuous, k(x, s, ψ(s)) is continuous for s ∈ (0, x) and∣∣∣∣∫ x
0
k(x, s, ψ(s))ds
∣∣∣∣ <∞ for x ∈ [0, X];
(ii) ∃ q(x, s) continuous on 0 < s < x <∞ satisfying
x∫
0
q(x, s)ds <∞, x ∈ (0,∞),
and all X > 0, as positive δ −→ 0,
x+δ∫
x
q(x+ δ, δ)ds −→ 0, x ∈ [0, X],
uniformly in x, such that
|k(x, s, f1)− k(x, s, f2)| ≤ q(x, s) |f1 − f2| ∀d > 0
and f1, f2 ∈ Sd = {f ∈ R| |f | < d}.
Under these conditions, there exists a constant α > 0 such that on [0, α], Eq. (1.1) has a unique
solution. Furthermore, if there exist B > 0, such that for |f(x)| ≤ B on the all intervals of the form
[0, β] (β > 0), the Eq. (1.1) has a unique solution, then Eq. (1.1) has a unique solution on [0,∞) [5].
c© R. KATANI, S. SHAHMORAD, 2012
ISSN 1027-3190. Укр. мат. журн., 2012, т. 64, № 7 919
920 R. KATANI, S. SHAHMORAD
Numerical methods to approximate solution of Eq. (1.1) have been extensively studied in lite-
rature [2 – 4, 7, 8]. Except some low order methods, such as the trapezoidal rule, the other methods
that are based on numerical integration require one or more starting values which must be found
by an other method. A different class of approximating formulas for (1.1) is based on extensions of
the Runge-Kutta methods which have been studied in details by Pouzet [10, 11]. The Runge-Kutta
methods are self-starting, but tend to be complicated and inefficient and hence its practical use is
limited.
There is another approach which uses numerical quadrature, and its computations are arranged
in such a way that several values of the unknown function are obtained at the same time. This is
generally called a block by block method and it is what we are concerned about. The block by block
approach was first suggested by Young [13] in connection with product integration techniques. In fact,
a block by block method is an extrapolation procedure which has advantages of being self-starting
and producing a block of values at the same time [2, 3]. Linz [7] described a two blocks method and
used it for solving nonlinear VIE of the second kind. Also AL-Asdi [1] used two and three blocks
for solving Hammerstien VIE of the second kind and then Saify [12] used two, three and four blocks
for solving a system of linear VIE of the second kind. In 2010, the authors used a block by block
method for solving system of nonlinear VIEs [6]. In this paper, we extend this method by using
Romberg quadrature rule to get a desired order of the error. In addition to the general advantages of
the block by block methods such as having no need to start values, simple structure for application
and computing several values of the unknown function at the same time, the presented method has
the following advantages.
1. Most of the available methods for solving (1.1) are based on expansion of solution, for example
the Taylor and Chebyshev expansion methods, the Tau method, the Adomian and homotopy methods
and so on. These methods are efficient only for the intervals with small length (say [0, 1] or [−1, 1])
and they are useless for the large intervals. The method of this paper is one of the most suitable
methods for the large intervals. In the final section of this paper, we will compare numerical results
between HPM (Homotopy Perturbation Method) [14], ADM (Adomian Decomposition Method) [4]
and the given block by block method (Tables 1 and 3).
2. For the given step size h, the order of convergence for this method is at least h6 while it is h4
by using Simpson rule [7].
3. By increasing number of blocks, the order of convergence increases in such a way that it would
be at least h8 and h10 respectively for 8 and 16 blocks.
4. At the first step of Romberg rule, the Simpson rule can be used instead of trapezoidal rule for
increasing order of convergence.
5. Compared to known methods, the computation time for this method is low.
The rest of the paper is organized as follows. In Section 2, the general process is presented. In
Section 3, the method for the large interval is described. The sixth-order convergence is proved in
Section 4. Finally, the paper is closed by giving numerical experiments in order to test reliability of
the method in Section 5.
2. The general process. Let 0 = x0 < x1 < . . . < xN = X be a partition of [0, X] with the step
size h, such that xi = x0 + ih, i = 1, 2, . . . , N, and let Fi be the approximate value of f(x) at the
ISSN 1027-3190. Укр. мат. журн., 2012, т. 64, № 7
THE BLOCK BY BLOCK METHOD WITH ROMBERG QUADRATURE FOR SOLVING . . . 921
mesh point x = xi and F0 = g(x0). To simplify formulation, let the number of blocks to be 4 (for
8, 16, . . . blocks the process will be similar).
Putting x = x4m+p in (1.1), we have
F4m+p ' f(x4m+p) = g(x4m+p) +
x4m+p∫
0
k(x4m+p, s, f(s))ds = g(x4m+p)+
+
x4m∫
0
k(x4m+p, s, f(s))ds+
x4m+p∫
x4m
k(x4m+p, s, f(s))ds, m = 0, 1, . . . , N/4− 1, p = 1, 2, 3, 4.
(2.1)
Assume that F0, F1, . . . , F4m are known, then the first integral can be approximated by standard
quadrature rules and the second one can be estimated by Romberg quadrature rule at the points x4m,
x4m+1, x4m+2, x4m+3 and x4m+4. Therefore we obtain a system containing four simultaneous equa-
tions that is solved for a block of four values of F. To simplify notation, we set ki := k(x4m+p, xi, Fi)
and use the trapezoidal rule for
∫ xu
xv
k(x4m+p, s, f(s))ds, thus we obtain
T (0)
u,v :=
xu−v
2
[kv + ku] ,
T (1)
u,v :=
1
2
T (0)
u,v +
xu−v
2
ku+v
2
,
T (2)
u,v :=
1
2
T (1)
u,v +
xu−v
4
[
ku+3v
4
+ k3u+v
4
]
,
T (3)
u,v :=
1
2
T (2)
u,v +
xu−v
8
[
ku+7v
8
+ k3u+5v
8
+ k5u+3v
8
+ k7u+v
8
]
.
It is easy to get
x4m+p∫
x4m
k(x4m+p, s, f(s))ds '
64
45
T
(2)
4m+p,4m −
20
45
T
(1)
4m+p,4m +
1
45
T
(0)
4m+p,4m =
=
xp
90
[
7(k4m + k4m+p) + 12k4m+p/2 + 32(k4m+p/4 + k4m+3p/4)
]
, p = 1, 2, 3, 4, (2.2)
by using Romberg rule. If
ip
4
, i = 1, 2, 3, are not integers, the points x
4m+
ip
4
will not belong to the
mesh points and F
4m+
ip
4
will be unknown which leads to a difficulty in computing (2.2). In this case,
we use the Lagrange interpolation polynomial at the points x4m, x4m+1, x4m+2, x4m+3 and x4m+4
to approximate F
4m+
ip
4
, i.e.,
F4m+ ip
4
≈ P
(
x4m +
ip
4
h
)
=
4∑
j=0
Lj
(
ip
4
)
F4m+j , i = 1, 2, 3,
ISSN 1027-3190. Укр. мат. журн., 2012, т. 64, № 7
922 R. KATANI, S. SHAHMORAD
where
Lj
(
ip
4
)
:=
4∏
ii=0
ii 6=j
ip/4− ii
j − ii
.
Then we find
F
4m+
3
2
≈ −5
128
F4m +
15
32
F4m+1 +
45
64
F4m+2 −
5
32
F4m+3 +
3
128
F4m+4,
F
4m+
3
4
≈ 195
2048
F4m +
585
512
F4m+1 −
351
1024
F4m+2 +
65
512
F4m+3 −
45
2048
F4m+4,
F
4m+
9
4
≈ 35
2048
F4m −
63
512
F4m+1 +
945
1024
F4m+2 +
105
512
F4m+3 −
45
2048
F4m+4, (2.3)
F
4m+
1
2
≈ 35
128
F4m +
35
32
F4m+1 −
35
64
F4m+2 +
7
32
F4m+3 −
5
128
F4m+4,
F
4m+
1
4
≈ 1155
2048
F4m +
385
512
F4m+1 −
495
1024
F4m+2 +
105
512
F4m+3 −
77
2048
F4m+4.
The first integral in (2.1) can be approximated by similar Romberg rule without any difficulty.
If 4m is a multiple of 8, then by using 3-stage Romberg quadrature rule we define
A :=
x4m∫
0
k(x4m+p, s, f(s))ds ≈ (4096T
(3)
4m,0 − 1344T
(2)
4m,0 + 84T
(1)
4m,0 − T
(0)
4m,0)/2835 =
=
x4m
2835
[
108.5(k0 + k4m) + 218k2m + 176(km + k3m) + 512(km/2 + k7m/2 + k3m/2 + k5m/2)
]
,
(2.4)
otherwise
A :=
x4m∫
0
k(x4m+p, s, f(s))ds =
x4∫
0
k(x4m+p, s, f(s))ds+
x4m∫
x4
k(x4m+p, s, f(s))ds ≈
≈ (64T
(2)
4,0 − 20T
(1)
4,0 + T
(0)
4,0 )/45 + (4096T
(3)
4m,4 − 1344T
(2)
4m,4 + 84T
(1)
4m,4 − T
(0)
4m,4)/2835 =
=
x4
90
[7(k0 + k4) + 12k2 + 32(k1 + k3)] +
x4m − x4
2835
[
217
2
(k4 + k4m) + 218k2m+2 +
+176(km+3 + k3m+1) + 512
(
km+7
2
+ k7m+1
2
+ k3m+5
2
+ k5m+3
2
)]
. (2.5)
Consequently, substituting in (2.1), yields for p = 4
F4m+4 = g(x4m+4) +A+
x4
90
[7(k4m + k4m+4) + 12k4m+2 + 32(k4m+1 + k4m+3)] , (2.6)
ISSN 1027-3190. Укр. мат. журн., 2012, т. 64, № 7
THE BLOCK BY BLOCK METHOD WITH ROMBERG QUADRATURE FOR SOLVING . . . 923
and p = 1, p = 3
F4m+p = g(x4m+p) +A+
xp
90
[
7(k4m + k4m+p) + 12k
(
x4m+p, x4m+p/2,P
(
x4m +
p
2
h
))
+
+32
(
k
(
x4m+p, x4m+p/4,P
(
x4m +
p
4
h
))
+ k(x4m+p, x4m+3p/4,P
(
x4m +
3p
4
h)
))]
. (2.7)
For p = 2 need not use the Lagrange interpolation for F4m+ p
2
, so
F4m+2 = g(x4m+2) +A+
x2
90
[
7(k4m + k4m+2) + 12k4m+1+
+32
(
k
(
x4m+2, x4m+1/2,P
(
x4m +
1
2
h
))
+ k
(
x4m+2, x4m+3/2,P
(
x4m +
3
2
h
)))]
. (2.8)
Therefore in each step (for different values of m) (2.6), (2.7) and (2.8) for p = 1, 2, 3, 4, forms a
system of equations with four unknowns F4m+1, F4m+2, F4m+3 and F4m+4, which will be linear and
nonlinear respectively for linear and nonlinear integral equations. The linear case the system can be
solved via a direct method, but in the nonlinear case the system may be solved by iterative methods
or by a suitable software package such as Maple.
3. Large intervals. We noticed that the method of previous section gives four values of the
unknown function in each step, but these values are computed approximately and we use some of
them for the next steps. For instant F0, F1, . . . , F4 for m = 1 are applied from previous step to
approximate the first integral in (2.1). In order to reduce the effect of accumulated errors, we need
more accurate approximation for this integral in evaluating the unknown function at the points near
to the end of interval, when x changes in a large interval or when the step size is very small. Thus
for approximating the integral
∫ xu
xv
k(x4m+p, s, f(s))ds, under above strategy, we define
T (4)
u,v :=
1
2
T (3)
u,v+
+
xu−v
16
[
ku+15v
16
+ k3u+13v
16
+ k5u+11v
16
+ k7u+9v
16
+ k9u+7v
16
+ k11u+5v
16
+ k13u+3v
16
+ k15u+v
16
]
then by using Romberg quadrature rule with 2, 3 and 4 stages, we have, respectively,
xu∫
xv
k(x4m+p, s, f(s))ds ≈
64
45
T (2)
u,v −
20
45
T (1)
u,v +
1
45
T (0)
u,v =
=
u− v
90
h
[
7(kv + ku) + 12ku+v
2
+ 32
(
ku+3v
4
+ k3u+v
4
)]
, (3.1)
xu∫
xv
k(x4m+p, s, f(s))ds ≈ (4096T (3)
u,v − 1344T (2)
u,v + 84T (1)
u,v − T (0)
u,v )/2835 =
ISSN 1027-3190. Укр. мат. журн., 2012, т. 64, № 7
924 R. KATANI, S. SHAHMORAD
=
u− v
2835
h
[
108.5(kv + ku) + 218ku+v
2
+ 176
(
ku+3v
4
+ k3u+v
4
)
+
+ 512
(
ku+7v
8
+ k7u+v
8
+ k3u+5v
8
+ k5u+3v
8
)]
(3.2)
and
xu∫
xv
k(x4m+p, s, f(s))ds ≈
≈ (1048576T (4)
u,v − 348160T (3)
u,v + 22848T (2)
u,v − 340T (1)
u,v + T (0)
u,v )/722925 =
=
u− v
722925
h
[
13779.5(kv + ku) + 27559ku+v
2
+ 27728
(
ku+3v
4
+ k3u+v
4
)
+
+22016
(
ku+7v
8
+ k7u+v
8
+ k3u+5v
8
+ k5u+3v
8
)
+ 65536
(
ku+15v
16
+ k15u+v
16
+
+ k3u+13v
16
+ k13u+3v
16
+ k5u+11v
16
+ k11u+5v
16
+ k7u+9v
16
+ k9u+7v
16
)]
. (3.3)
Now, if 4m is multiple of 16, then we use the 4-stages Romberg rule (3.3) to approximate the first
integral in (2.1) and if remaining of
4m
16
is 4, then we write
A :=
x4m∫
0
k(x4m+p, s, f(s))ds =
x4∫
0
k(x4m+p, s, f(s))ds+
x4m∫
x4
k(x4m+p, s, f(s))ds
and approximate the first integral by (3.1) and the second one by (3.3). Similarly, if remaining of
4m
16
is 8, then we write
A :=
x4m∫
0
k(x4m+p, s, f(s))ds =
x8∫
0
k(x4m+p, s, f(s))ds+
x4m∫
x8
k(x4m+p, s, f(s))ds
and use (3.2) and (3.3) respectively for the first and second integrals. Otherwise we write
A :=
x4m∫
0
k(x4m+p, s, f(s))ds =
=
x4∫
0
k(x4m+p, s, f(s))ds+
x12∫
x4
k(x4m+p, s, f(s))ds+
x4m∫
x12
k(x4m+p, s, f(s))ds
and approximate the integrals respectively by (3.1), (3.2) and (3.3).
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THE BLOCK BY BLOCK METHOD WITH ROMBERG QUADRATURE FOR SOLVING . . . 925
Therefore, having larger interval or very small step size, one can use more accurate integration
rules to approximate the first integral in (2.1). We conclude that, this method has not any restriction
on the given intervals.
4. Convergence analysis.
Theorem 4.1. The approximation method given by the system (2.6), (2.7) and (2.8) is con-
vergent and its order of convergence is at least 6 for the functions k and f with at least six order
derivatives.
Proof. It can be written from (2.4) and (2.5)
A :=
x4m∫
0
k(x4m+p, s, f(s))ds ≈ h
4m∑
i=0
wik(x4m+p, xi, Fi).
Let p = 1 or p = 3 (for other values of p the process is similar), then from (2.1) and (2.7) we obtain
|ε4m+p| := |f(x4m+p)− F4m+p| =
=
∣∣∣∣∣∣
x4m∫
0
k(x4m+p, s, f(s))ds+
x4m+p∫
x4m
k(x4m+p, s, f(s))ds −
−h
4m∑
i=0
wik(x4m+p, xi, Fi)−
xp
90
[
7k4m + 7k4m+p + 12k
(
x4m+p, x4m+p/2,P
(
x4m +
p
2
h
))
+
+32k
(
x4m+p, x4m+p/4,P
(
x4m +
p
4
h
))
+32k
(
x4m+p, x4m+3p/4,P
(
x4m +
3p
4
h
))]∣∣∣∣∣ .
By adding and diminishing the terms
h
4m∑
i=0
wik(x4m+p, xi, f(xi)),
7xp
90
k (x4m+p, x4m, f(x4m)) , . . .
. . . ,
32xp
90
k(x4m+p, x
4m+
3p
4
,
4∑
j=0
Lj
(
3p
4
)
f(x4m+j))
and using (ii) for k(x, s, f(s)), one obtains
|ε4m+p| ≤ h
4m∑
i=0
wiq(x4m+p, xi)|εi|+
+
7xp
90
q(x4m+p, x4m)|ε4m|+
7xp
90
q(x4m+p, x4m+p)|ε4m+p|+
+
12xp
90
q(x4m+4, x4m+p/2)
∣∣∣∣∣∣P
(
x4m +
p
2
h
)
−
4∑
j=0
Lj
(p
2
)
f(x4m+j))
∣∣∣∣∣∣+
ISSN 1027-3190. Укр. мат. журн., 2012, т. 64, № 7
926 R. KATANI, S. SHAHMORAD
+
32xp
90
q(x4m+p, x4m+p/4)
∣∣∣∣∣∣P
(
x4m +
p
4
h
)
−
4∑
j=0
Lj
(p
4
)
f(x4m+j))
∣∣∣∣∣∣+
+
32xp
90
q(x4m+p, x4m+3p/4)
∣∣∣∣∣∣P
(
x4m +
3p
4
h
)
−
4∑
j=0
Lj
(
3p
4
)
f(x4m+j))
∣∣∣∣∣∣+R1 +R2,
where R1 and R2 are errors of the numerical integrations. Since q(x, s) was supposed continuous, it
is bounded on [0, X]. Therefore
|ε4m+p| ≤ h
4m∑
i=0
wili |εi|+
7p
90
hl4m |ε4m|+
7p
90
hl4m+p |ε4m+p|+
+
12p
90
hl
4m+
p
2
max
j
{
Lj
(p
2
)} 4∑
j=0
|ε4m+j |+
+
32p
90
hl
4m+
p
4
max
j
{
Lj
(p
4
)} 4∑
j=0
|ε4m+j |+
+
32p
90
hl
4m+
3p
4
max
j
{
Lj
(
3p
4
)} 4∑
j=0
|ε4m+j |+R1 +R2 ≤
≤ hc
4m∑
i=0
|εi|+ hc1 |ε4m+1|+ hc2 |ε4m+2|+ hc3 |ε4m+3|+ hc4 |ε4m+4| .
Without lose of generality, let ‖εj‖∞ = maxj=1,2,3,4 |ε4m+j | = |ε4m+p| , then it is easy to get
|ε4m+p| ≤ hc′
4m+p−1∑
i=0
|εi|+ hc′′ |ε4m+p|+R1 +R2, (4.1)
where c′ and c′′ are constants or equivalently
‖εj‖∞ ≤
hc′
1− hc′′
4m+p−1∑
i=0
|εi|+
R1 +R2
1− hc′′
.
Then from the Gronwall inequality [8], we conclude that
‖εj‖∞ ≤
R1 +R2
1− hc′′
e
c′
1−hc′′ xn ,
which implies
‖εj‖∞ ≤
R1 +R2
1− hc′′
e
c′
1−hc′′ xn →
as h→0
0.
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THE BLOCK BY BLOCK METHOD WITH ROMBERG QUADRATURE FOR SOLVING . . . 927
For the functions k and f with at least sixth order derivatives, the orders for R1 and R2 will be at
least O(h6) and so
‖εj‖ = O(h6).
5. Numerical results. In this section, some examples are given to illustrate convergence and error
bound of the presented method. These examples have chosen from [9] and the results computed by
programming in Maple 10.
I. Equation with exponential nonlinearity
f(x) +A
x∫
a
eλf(s)ds = Bx+ C, 0 ≤ x ≤ X,
with the exact solution
f(x) =
−1
λ
ln[Aλ(x− a) + e−Cλ], B = 0,
−1
λ
ln
[
A
B
+
(
e−λf0 − A
B
)
eλB(a−x)
]
, f0 = aB + C, B 6= 0.
II. Equation with power-low nonlinearity
f(x) +A
x∫
a
f2(s)ds = Bx+ C, 0 ≤ x ≤ X,
with the exact solution
f(x) =
K
(K + fa)e
2AK(x−a) + fa −K
(K + fa)e2AK(x−a) − fa +K
, K =
√
B
A
, fa = aB + C, AB > 0,
C
AC(x− a) + 1
, AB = 0,
K tan
[
AK(a− x) + arctan
fa
K
]
, K =
√
−B
A
, fa = aB + C, AB < 0.
III. A linear equation with trigonometric kernel
f(x)−A
x∫
a
cos (λx)
cos (λs)
f(s)ds = g(x), 0 ≤ x ≤ X,
with the exact solution
f(x) = g(x) +A
x∫
a
eA(x−s)
cos (λx)
cos (λs)
g(s)ds.
The results in Tables 1 – 5, show the absolute error |f(xi)− Fi|, for i = 1, 2, . . . , N, at the selected
grid points, where f(xi) and Fi are the exact and corresponding approximate solutions at x = xi.
ISSN 1027-3190. Укр. мат. журн., 2012, т. 64, № 7
928 R. KATANI, S. SHAHMORAD
In Tables 1 and 3 the results of the HPM [14], ADM [4] and block by block method are compared,
where m denotes number of iterations for HPM and ADM and N denotes number of the mesh points
for the block by block method.
The results of Examples 1 and 2 (see Tables 1 and 3) show that the HPM and ADM behave worse
than the presented method even in the interval [0, 1]. For the Example 1, HPM and ADM does not
work respectively for more than 3 and 5 iterations (probably because of exponential nonlinearity).
Table 3, shows that the above mentioned behavior occur again for the large intervals (sayX = 10)
by the HPM and ADM, although it does not occur for [0, 1].
The last rows of Tables 1 – 5 compar the computing time for the HPM, ADM and block by block
method which for the last one is less than two others, where programming for all methods have been
done using Maple package.
Remark 5.1. Let E(h) = Chq be error of the block by block method, where C and q are
respectively a constant and order of the error, then q =
ln(E(h)/C)
ln(h)
.
By computing the order q from this formula for the reported errors in Tables 1 – 5, we conclude
that q ≥ 6. This result confirms the claim that was stated in introduction and was proved in Section 4.
For example, we get q = 7.68 in Table 2 for xi =
1
20
(X = 1, N = 20), error = 7.664e−10,
h = 0.05, and q = 9 in Table 5 for xi = 10 (X = 10, N = 1000), error = 1.604e−18, h = 0.01.
Table 1. Numerical results of example I (λ = 1/2, A = 4, B = 3, C = 1/8, a = 0)
xi
HPM ADM block by block
X = 1 X = 10 X = 1 X = 10 X = 1 X = 10
N = 40 m = 3 m = 3 m = 5 m = 3
4X/N 9.037e−01 7.561e+00 2.725e−05 6.812e−01 9.709e−10 1.222e−04
8X/N 1.770e+00 4.875e+01 1.255e−05 1.158e+01 5.386e−10 3.729e−05
12X/N 2.621e+00 2.442e+01 1.275e−04 8.093e+01 5.840e−10 1.477e−05
16X/N 3.473e+00 1.131e+03 6.470e−04 4.047e+04 5.415e−10 7.495e−06
20X/N 4.341e+00 5.115e+03 2.252e−03 1.867e+03 2.286e−10 1.150e−05
24X/N 5.238e+00 2.298e+04 6.187e−03 8.439e+03 2.285e−09 1.848e−05
28X/N 6.180e+00 1.031e+05 1.446e−02 3.790e+04 2.105e−09 8.723e−04
32X/N 7.182e+00 4.620e+05 3.004e−02 1.699e+05 6.388e−10 7.938e−04
36X/N 8.265e+00 2.076e+06 5.706e−02 7.615e+05 5.781e−10 3.607e−04
X 9.459e+00 9.280e+06 1.010e−01 3.410e+06 5.781e−10 3.607e−04
time 1.703′′ 3.375′′ 172.921′′ 0.609′′ 1.078′′
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THE BLOCK BY BLOCK METHOD WITH ROMBERG QUADRATURE FOR SOLVING . . . 929
Table 2. Numerical results of example I (λ = 10, A = 1/10, B = 0, C = 1/100, a = 0)
xi
X = 1 X = 10 X = 30
N = 20 N = 60 N = 20 N = 60 N = 60 N = 100
X/N 7.664e−10 1.469e−12 3.907e−05 3.904e−07 3.907e−05 5.377e−06
2X/N 4.450e−10 8.644e−13 2.130e−05 2.194e−07 2.130e−05 2.961e−06
5X/N 5.842e−10 1.224e−12 5.199e−05 3.792e−07 5.199e−05 6.261e−06
7X/N 5.472e−10 1.199e−12 4.022e−05 3.203e−07 4.022e−05 5.023e−06
X/2 5.471e−11 1.966e−13 1.551e−05 1.956e−07 1.791e−05 6.662e−05
(N − 7)X/N 3.182e−10 2.498e−11 2.841e−05 1.693e−06 2.825e−04 2.686e−04
(N − 5)X/N 3.001e−10 2.453e−11 2.502e−05 1.624e−06 2.719e−04 2.631e−04
(N − 2)X/N 1.769e−10 1.342e−11 2.349e−06 1.888e−06 1.336e−04 5.289e−04
X 1.737e−10 1.317e−11 2.144e−06 1.806e−06 1.293e−04 5.188e−04
time 3.891′′ 5.343′′ 3.813′′ 6.297′′ 6.172′′ 8.188′′
Table 3. Numerical results of example II (A = 1/2, B = 2, C = 1, a = 0)
xi
HPM ADM block by block
X = 1 X = 10 X = 1 X = 10 X = 1 X = 10
m = 10 m = 10 m = 9 m = 9 N = 60 N = 100
X/N 7.218e−05 3.128e−03 0 1.000e−09 4.518e−12 2.709e−07
5X/N 2.095e−03 1.666e−01 1.000e−09 3.000e−09 4.912e−13 3.524e−08
10X/N 9.995e−03 1.481e+00 2.000e−09 1.050e−05 3.559e−12 4.986e−08
15X/N 2.655e−02 8.021e+00 1.000e−09 1.431e−03 5.771e−12 1.492e−07
25X/N 1.002e−01 1.121e+70 1.000e−09 2.824e+67 4.888e−12 8.361e−08
X/2 1.666e−01 3.078e+340 4.000e−09 1.000e+320 2.543e−13 2.160e−06
(N − 25)X/N 2.606e−01 1.283e+507 1.900e−08 2.308e+491 6.600e−12 1.794e−06
(N − 15)X/N 5.631e−01 3.875e+559 4.030e−07 5.031e+551 7.977e−13 1.167e−06
(N − 10)X/N 7.924e−01 3.072e+583 1.431e−06 3.508e+576 1.561e−11 1.046e−06
(N − 5)X/N 1.092e+00 1.916e+606 4.510e−06 1.075e+600 8.741e−11 8.601e−07
X 1.481e+00 9.205e+627 1.050e−05 2.021e+622 4.768e−11 5.353e−06
time 75.235′′ 125.858′′ 24.609′′ 26.329′′ 0.329′′ 0.626′′
ISSN 1027-3190. Укр. мат. журн., 2012, т. 64, № 7
930 R. KATANI, S. SHAHMORAD
Table 4. Numerical results of example II (A = 1/10, B = 0, C = 1, a = 0)
xi
X = 1 X = 10 X = 30
N = 100 N = 1000 N = 100 N = 300 N = 100 N = 300
X/N 4.444e−18 4.494e−24 3.977e−12 5.922e−15 2.287e−09 3.977e−12
10X/N 2.430e−18 2.642e−24 9.448e−13 2.666e−15 1.938e−10 9.448e−13
20X/N 9.411e−20 9.993e−27 5.535e−13 3.706e−16 5.214e−10 5.535e−13
30X/N 2.033e−18 2.596e−24 9.575e−14 1.450e−15 1.104e−10 9.575e−14
40X/N 7.571e−20 9.316e−27 2.554e−13 1.929e−16 2.029e−09 2.554e−13
X/2 1.618e−18 3.712e−21 2.146e−11 1.875e−11 1.567e−07 1.065e−07
(N − 40)X/N 3.002e−19 2.375e−18 7.913e−12 2.938e−09 1.414e−08 6.862e−06
(N − 30)X/N 1.726e−18 2.850e−18 1.833e−10 1.923e−09 4.355e−07 2.524e−06
(N − 20)X/N 1.722e−20 3.420e−18 6.947e−11 3.895e−09 3.494e−09 5.948e−06
(N − 10)X/N 1.409e−18 3.370e−18 7.000e−10 7.443e−09 3.929e−07 1.345e−06
X 2.661e−18 4.031e−18 7.436e−09 5.407e−09 1.343e−05 6.929e−06
time 0.64′′ 6.484′′ 0.672′′ 2.046′′ 0.702′′ 2.046′′
Table 5. Numerical results of example III (A = 1/100, λ = 2, g(x) = cos (2x), a = 0)
xi
X = 1 X = 10 X = 30
N = 100 N = 1000 N = 100 N = 1000 N = 100 N = 1000
X/N 5.869e−17 1.291e−22 1.869e−09 5.869e−17 3.072e−06 3.463e−13
10X/N 5.887e−16 2.045e−23 1.576e−08 5.887e−16 4.132e−06 1.256e−12
20X/N 1.539e−18 5.205e−27 6.651e−11 1.539e−18 3.372e−07 5.532e−15
30X/N 1.864e−15 1.218e−22 1.821e−09 1.864e−15 1.773e−05 5.187e−12
40X/N 3.505e−18 1.728e−26 1.691e−11 3.505e−18 2.031e−07 1.527e−14
X/2 2.756e−15 7.623e−24 1.188e−09 1.744e−16 5.367e−07 6.250e−13
(N − 40)X/N 8.240e−18 1.132e−22 1.313e−10 4.937e−16 7.443e−08 2.100e−13
(N − 30)X/N 2.744e−15 3.420e−21 4.674e−10 1.904e−15 7.642e−07 5.577e−11
(N − 20)X/N 2.820e−18 1.370e−22 1.061e−10 1.711e−15 2.074e−07 2.430e−12
(N − 10)X/N 4.316e−15 3.410e−21 6.186e−09 2.209e−15 2.076e−06 2.528e−12
X 9.055e−18 2.221e−23 6.585e−11 1.604e−18 3.491e−06 1.080e−13
time 0.265′′ 3.077′′ 0.311′′ 3.765′′ 0.312′′ 3.844′′
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THE BLOCK BY BLOCK METHOD WITH ROMBERG QUADRATURE FOR SOLVING . . . 931
6. Conclusion. In this paper, we presented a block by block method for solving Volterra integral
equations on the large intervals with at least 6 order of convergence. The method can be improved by
using accurate Romberg rule or even other suitable integration methods. Numerical results given in
Tables 1 – 5 show high accuracy of the method. The last rows of these tables show that the computing
time of the presented method is less than two other methods (HPM and ADM).
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Received 30.04.11
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