Application of the Bilateral Hybrid Methods to Solving Initial -Value
Problems for the Volterra Integro-Differential Equations
VAGIF IBRAHIMOV1, GALINA MEHDIYEVA2, MEHRIBAN IMANOVA3,
DAVRON ASLONQULOVICH JURAEV4
1Institute of Control Systems, Department of Computational mathematics,
Baku State University, Baku,
AZERBAIJAN
2Department of Computational Mathematics,
Baku State University, Baku,
AZERBAIJAN
3Science Development Foundation under the President of the
Republic of Azerbaijan, Institute of Control Systems, Baku,
AZERBAIJAN
4Department of Scientific Research, Innovation and Training of Scientific and Pedagogical Staff,
University of Economics and Pedagogy, Karshi,
UZBEKISTAN
Abstract: - The many problems of natural sciences are reduced to solving integro-differential equations with
variable boundaries. It is known that Vito Volterra, for the study of the memory of Earth, has constructed the
integro-differential equations. As is known, there is a class of analytical and numerical methods for solving the
Volterra integro-differential equation. Among them, the numerical methods are the most popular. For solving
this equation Volterra himself used the quadrature methods. How known in solving the initial-value problem
for the Volterra integro-differential equations, increases the volume of calculations, when moving from one
point to another, which is the main disadvantage of the quadrature methods. Here the method is exempt from
the specified shortcomings and has found the maximum value for the order of accuracy and also the necessary
conditions imposed on the coefficients of the constructed methods. The results received here are the
development of Dahlquist's results. Using Dahlquist’s theory in solving initial-value problem for the Volterra
integro-differential equation engaged the known scientists as P.Linz, J.R.Sobka, A.Feldstein, A.A.Makroglou,
V.R.Ibrahimov, M.N.Imanova, O.S.Budnikova, M.V.Bulatova, I.G.Buova and ets. The scientists taking into
account the direct connection between the initial value problem for both ODEs and the Volterra integro-
differential equations, the scientists tried to modify methods, that are used in solving ODEs and applied them to
solve Integro-differential equations. Here, proved that some modifications of the methods, which are usually
applied to solve initial-value problems for ODEs, can be adapted for solving the Volterra integro-differential
equations.
Here, for this aim, it is suggested to use a multistep method with the new properties. In this case, a question
arises, how one can determine the validity of calculated values. For this purpose, it is proposed here to use
bilateral methods. As is known for the calculation of the validity values of the solution of investigated
problems, usually have used the predictor-corrector method or to use some bounders for the step-size. And to
define the value of the boundaries, one can use the stability region using numerical methods. As was noted
above, for this aim proposed to use bilateral methods. For the illustration advantage of bilateral methods is the
use of very simple methods, which are called Euler's explicit and implicit methods. In the construction of the
bilateral methods it often becomes necessary to define the sign for some coefficients. By taking this into
account, here have defined the sign for some coefficients.
Key-Words: - The Volterra integro-differential equation, multistep and hybrid methods, symmetric and bilateral
methods, degree and stable, advanced methods.
Received: June 12, 2023. Revised: September 6, 2023. Accepted: October 5, 2023. Published: October 20, 2023.
WSEAS TRANSACTIONS on MATHEMATICS
DOI: 10.37394/23206.2023.22.86
Vagif Ibrahimov, Galina Mehdiyeva,
Mehriban Imanova, Davron Aslonqulovich Juraev
E-ISSN: 2224-2880
781
Volume 22, 2023
1 Introduction
As was noted above, the investigation of the many
problems from the different areas of the natural
sciences is reduced to solving the initial value
problem for the Volterra integrodifferential
equations. Note that this problem can be formulated
in different forms. One of the popular presentations
of this problem can be presented as follows [1], [2],
[3], [4], [5], [6], [7], [8]:
.],,[
,)(,))(,,(),(
0
00
0
constXxx
yxydssysxKyxfy x
x
(1)
If the continuous to the totality of arguments
function
),( yxf
and
))(,,( sysxK
are given, then
the problem (1) can be taken as the given. Spouse
that these functions are defined in some closed area
and have the partial derivatives up to p. And also
suppose that the problem (1) has a unique solution,
which is defined on the segment
].,[ 0Xx
It is known that there are some classes of
numerical methods for solving problems (1). It is
also known that the estimation obtained for the
errors of these methods holds for the sufficiently
small step size h. Therefore one of the main results
in these areas is the construction of numerical
methods, results finding by which can be accepted
as the reliable information for the selected results.
To solve such problems, Chapligin proposed his
own two-sided or bilateral method, which now is
called the bilateral analytical approximate method.
The bilateral method was not developed because its
advantage has not been proven. By using that, here
to suggest a way for the construction of the
numerical bilateral methods. And also to give some
comparisons between bilateral methods and bilateral
formulas, [9], [10], [11], [12], [13], [14], [15]. For
the illustration of these let us consider the
construction of bilateral methods.
§1.The bilateral methods and their application
to solve some specific task.
As is known, the multistep method with constant
coefficients is one of the popular methods for
solving problem (1), which can presented as
follows:
kNnyhy k
iini
k
iini
,...,1,0,
00
, (2)
here the mesh-point
m
x
is defined as:
)1,...,1,0(
1
Nihxx ii
and
h0
is the
step-size.
If method (2) is applied to solving the problem
(1), then receive:
k
i
k
i
k
iiniiniini hfhy
0 0 0
(3)
here
,))(,,()(
0
dssysxKx in
x
x
inin
but
x
x
dssysxKx
0
))(,,()(
and the values
).,...,2,1,0(),( kNmyxff mmm
),...,1,0( ki
in
can be found as the values of
the solution of the following problem:
.0)()),(,,()( 0
xxyxxKx
(4)
In this way, the solving of the problem (1), has
reduced to solving the initial-value problem for
ODEs of the first order. Vito Volterra himself
suggested the quadrature methods for solving
problems (1). If here, has used the method for
solving the problem (4), then by using that in the
method (3), receive the following, [16], [17], [18],
[19], [20], [21], [22]:
).,...,2,1,0(
),,(
0
)(
00
kNn
yxxKhfhy ininjn
k
i
k
ij
j
i
k
iini
k
iini
(5)
Noted that for the application of method (2),
often arises the necessity to define the order of
accuracy for using multistep methods which can be
defined as the following form:
Definition1. Integer variable
p
- is called the
degree for the method (2) if the following holds:
k
i
p
ii hhOihxyhihxy
0
1.0),())()((
(6)
From the asymptotic equality (6), receive that the
degree for the method (2) equal to
p
. Dahlquist
proves that if the method (2) is stable,
,0
k
then
]2/[2 kp
for each value of
k
, there exist stable
methods with the degree
.2]2/[2
max kp
Definition 2. Method (2) is called stable if the roots
of the polynomial
0
1
1...)(
k
k
k
k
lie in the unit
circle on the boundary of which there are no
multiple roots. For the construction stable methods
with the degree
2]2/[2 kp
, Prof. V.Ibrahimov
has investigated advanced methods, which can be
received from the (2) in the case
,0
k
and
.0,0... 11 skskkk
He has constructed the concept method with the
degree
2 kp
for the
3k
and
1s
. In the, [23],
has constructed a stable method with the degree
5p
. Thus receive that advanced methods are more
WSEAS TRANSACTIONS on MATHEMATICS
DOI: 10.37394/23206.2023.22.86
Vagif Ibrahimov, Galina Mehdiyeva,
Mehriban Imanova, Davron Aslonqulovich Juraev
E-ISSN: 2224-2880
782
Volume 22, 2023
promising. As is known one of the popular methods
of type (2) is the Simpson method, which can be
received from method (2) in the case
2k
.
Dahlquist result shows that in this case, the stable
method with the degree
4p
is the Simpson
method. In this case
2k
constructed method with
degree
4p
is unique.
And now let us consider the case
1k
. The
popular methods in this case are the explicit and
implicit Euler methods and trapezoidal rules.
Explicit Euler method for solving problem (1) can
be presented as:
2/)),,(),,(( 11 nnnnnnnnn yxxKyxxKhhfyy
. (7)
But the implicit Euler method for solving
problem (1) can be presented as follows:
),,( 11111 nnnnnn yxxhKhfyy
. (8)
It is known that methods (7) and (8) correspond
to the following Euler’s methods:
nn
n
eyhyy
1
;
1
1
nn
n
iyhyy
. (9)
As is known the local traction error for these
methods can be written as following, respectively:
)(2/),(2/ 3232 hOyhhOyh nn
.
Hence it follows that methods (7) and (8) the
bilateral, so as
i
nn
e
nyxyy 111 )(
if
.0)(
xy
And if
,0)(
xy
then
e
nn
i
nyxyy 111 )(
.
)( 1n
xy
is the exact value of the solution of
problem (1) at the point
1n
x
.
It is not difficult to prove that the value
2/)( 11
iin
e
nn yyy
will be exact than the
e
n
y1
and
i
n
y1
.
1n
y
will be same with the value
finding by the trapezoidal rule.
In our case, the simple, step-by-step algorithm
for the application of Euler’s methods to solving any
problems can be presented as follows:
Step 1 Input (initial values);
Step 2 For
1j
step 1
1n
do steps 3-5;
Step 3 Calculation
e
j
y1
by the method (7);
Step 4 Calculation
ij
y1
by the method (8);
Step 5 Calculation
2/)( 111
e
j
ijj yyy
; Print
);;( 111 j
e
j
ijyyy
end.
Step 6 STOP.
It is not difficult to believe that the method
constructed in the above-mentioned way is bilateral.
Let us consider the following couple of methods:
)(3/3/,2 543
1
)1(
2hOyhyhRyhyy IV
nnnnnn
(10)
).(3/3/
)11(,3/)(2
543
12
)2(
2
hOyhyhR
yyyhyy
IV
nnn
nnnnn
Here the
n
R
denote the local truncation error of
these methods. It is obvious that the value
2/)( )2(
2
)1(
22 nnn yyy
will be more accurate than
the values
)1(
2n
y
and
)2(
2n
y
. The receiving method is
the same as the Simpson method. Note that the
methods can be used to determine the limit for the
value
2n
y
.
And now let us consider the application of the
methods (10) and (11) to solving equation (1). In
this case, receive:
.6/)),,(),,(
),,(),,(
),,(2(
3/)(2
));,,(
),,((2
2
111112
222
12
)2(
2
112
1111
)1(
2
nnnnnn
nnnnnn
nnn
nnnnn
nnn
nnnnnn
yxxKyxxK
yxxKyxxK
yxxKh
fffhyy
yxxK
yxxKhhfyy
It is easy to verify that, half sum for the values
)1(
2n
y
and
)2(
2n
y
can be presented as follows:
.6/)),,(
),,(),,(4
),,(4),,(2(
3/)4(
2111
112222
122
nnn
nnnnnn
nnnnnn
nnnnn
yxxK
yxxKyxxK
yxxKyxxKh
fffhyy
This is one of the modifications of Simpson’s
method for solving Volterra integro-differential
equations. From here one can see that the
presentation of multistep methods for solving initial-
value problems for Volterra integro-differential
equations is not unique. In the application of this
method some difficulties in the calculation of the
values
2n
y
, which participates in the explicit
method designed for the calculation of this value.
For this aim one can use the values calculated by the
midpoint method of (10), [19], [20], [21], [22], [23],
[24], [25], [26], [27], [28].
However, by using the Simpson method,
determining the boundary of the error is impossible.
Hence, the use of bilateral methods, which is
possible to find an error on any point. In the theory
of numerical methods for solving integro-
differential equations, often symmetrical multistep
methods, therefore the following section is devoted
to the study of this question.
§2. Application of symmetrical methods to
solving problem (1).
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The concept of symmetry is discovered in nature,
for that is not a mathematical term. And then this
term was more commonly used in astronomy. As is
known, a planet in our galaxy is symmetrical to the
plane of Earth. In the theory of multistep methods,
the notion of symmetry was used by Dahlquist.
However, in the theory of quadrature formulas, the
concept of symmetry was used in the study of Gauss
and Chebyshev methods. As is known the Gauss
nodes and the coefficients are symmetrical. We
believe that one of such well-known methods is the
midpoint method and another is the trapezoidal
rules. Note that these methods do not satisfy the
Dahlquist requirement, but experts have always
accepted these methods as symmetrical, which can
be presented as:
.2/)(; 112/11
nnnnnnn yyhyyyhyy
Some authors give advantages of using
symmetrical methods, which can be defined in the
following form:
Definition 3. (Dahlquist), [12]. Stable method (2)
is called symmetrical if the following holds:
2 kp
and
),...,1,0(; ki
ikiiki
.
Taking into account these conditions, we get that
k
-is the even number. Hence, the amount of the
mesh-points in the multistep method will be odd. In
this case, by using the necessary condition of
convergence, receive that
0... 011
kk
. Hence
0
2/
k
. In
this case,
2k
the symmetrical method can be
written as:
3/)4( 122 nnnnn yyyhyy
, (12)
which is unique and has the degree
4p
or
kp 2
. Note that the method receiving as the
results of half sum of equalities (7) and (8) can be
taken as symmetrical. Noted for the application of
these methods to solve the problem (1), it is
necessary to find the values of the coefficients
),...,1,0,(,, )( kij
j
iii
. A similar
investigation was carried out by some authors, [29],
[30], [31], [32], [33], [34], [35], [36], [37], [38],
[39], [40], [41], [42], [43], [44]. Here, suppose that
the method (2) is given. In this case, the application
of method (5) to solving problem (1), must be
known as the values of the coefficients
),...,1,0,(
)( kij
j
i
. For this aim, one can use
the following linear system of algebraic equations:
),...,1,0( ki
k
ij i
j
i
. (13)
Noted that amount of the solutions in this system
more than one. Therefore users have the often to
choose different methods.
Note that in the generalization of the midpoint
method, one can receive the hybrid methods. Hybrid
methods in a general form can be presented as the
following:
).,..,1,0;1(
0 0
kiyhy i
k
i
k
iiniini i
(14)
The methods used in the previous section cannot
be received from the method (13) as the partial case.
As was noted above, the midpoint is stable has the
degree
2p
, and is explicit. Let us note that all the
hybrid methods of type (14) will be explicit. It does
not follow from here that, the hybrid methods can be
applied in direct form to solve any problems. Let us
show that it’s not right. For this consider the
following method:
,6/32/1
,6/32/1,2/)(
1
01 10
nnnn yyhyy
(15)
In the construction of this method have used two
hybrid points, the method is stable and has the
degree
4p
. Let us consider the application of the
method (15) to solving an initial-value problem for
ODEs, this can be presented as follows:
,2/)),(),(( 1100
1
nnnnnn yxfyxfhyy
, here
))(,()( xyxfxy
.
It is obvious that in the application of this
method are arises to calculation of the values
6/)33( n
y
and
6/)33( n
y
, which are not easy. As is
known the following method:
)16(,6/)4( 2/111 nnnnn fffhyy
also has the degree
4p
and is called the
Simpson method.
Remark 1.
Let us note that method (16) has been received
from Simpson’s method, which resembles the
Runge-Kutta method and is the one step. It is known
that one-step methods have some advantages. For
example, easily be applied to solving various
problems. All the methods have their advantages
and disadvantages. Taking into account the above
mentioned, here I wanted to reduce multistep
methods to the one-step methods. Runge-Kutta and
(16) are not the same. Methods Runge-Kutta are
explicit, but method (16) is implicit. Note that if
method (16) and method Runge (constructed by
Runge, which has the degree
4p
then these
methods will be the same. Therefore, the properties
WSEAS TRANSACTIONS on MATHEMATICS
DOI: 10.37394/23206.2023.22.86
Vagif Ibrahimov, Galina Mehdiyeva,
Mehriban Imanova, Davron Aslonqulovich Juraev
E-ISSN: 2224-2880
784
Volume 22, 2023
of the methods greatly depend on the problem,
which must be solved.
receive any difficulty related to calculation of
values
0
receive any difficult related to
calculation of values
),..,2,1,0( kiy i
in
. As
was noted above, if method (14) is applied to the
calculation of definite integral, then it does not
cause any difficulty. But, when applied to solve the
problem (1) for the case
0),( yxf
, arises some
difficulty, which can be solved in different ways.
For example, in the application of the method (15)
difficulties related to the calculation of the values
0
n
y
1
n
y
, which can be calculated by using
the described algorithm.
It is not difficult to show that methods
constructed at the intersection of methods (2) and
(14) are more exact. For this let us consider the
following linear method:
),,..,2,1,0
,1(,
0 0 0
ki
yhyhy i
k
i
k
i
k
iiniiniini i
(17)
which was constructed on the intersection of the
multistep and the hybrid methods. Note that the
method of (16) is more exact than the others. For
Example, if method (16) is stable, then there are
stable methods of type (16) with the degree
33 kp
and if method (14) is stable then there are
in the class of methods (14), stable methods with the
degree
22 kp
.
§3. On some advantages of the advanced
methods and their application.
In, [36], has proved that some of the advantages
of the advanced multistep methods is that if they are
stable, then in this class methods exist the stable
advanced methods, which are more accurate than
others. At first have been constructed a lot of
multistep methods and then constructed advanced as
new methods for comparison with the known and
multistep methods. Note that advanced methods can
be presented as multistep methods. For the
illustration of this let us consider to
following method:
mk
i
k
iiniini mkNnmyhy
0 0
).,..,1,0,0(
(18)
If comprise methods (2) and (18), then receive
that methods (2) and (18) can be taken as the same
only for the case
0m
. But, from the above-given
condition we get that,
0m
. As follows from here,
the class of methods (18) is the independent field of
research. Formally one can say that by using the
selection coefficient
k
0
k
, one can receive
method (18) from method (2). Let's show that it's
not. For this it is enough to recall Dahlquist's
condition, which can presented as the following
suppose that the method (2) is convergence, then its
coefficients must satisfy the following conditions,
[16], [17], [18], [19], [20], [21], [22], [23], [45]:
A. Coefficients
),..,1,0(, ki
ii
are real
numbers
0
k
.
B. The characteristic polynomials:
k
i
i
i
0
)(
and
k
i
i
i
0
)(
have no common factor different from constant.
C. The condition
0)1(
holds and
1p
(
p
-is the degree).
As was noted above V.Ibrahimov constructed and
investigated the method (16) and, therefore received
similar conditions for the coefficients, which can be
presented as:
A. The coefficients
),...,1,0(, ki
ii
are real
numbers and
.0
mk
B. The characteristic polynomials:
mk
i
i
i
0
)(
and
k
i
i
i
0
)(
have no common factor different from constant.
C. The conditions
1,0)1( p
are hold (
p
-
is the degree of the method (14)).
By using the condition A, receive that
0
k
. It is
not difficult to understand that one of the important
questions in the investigation of the define the
maximum value of the degree for the investigated
methods. One can prove that p≤2k (degree for the
method (2) and method with the degree p=2k is
unique, but
mkp 2
(degree for the method (13)
and method with the degree
mkp 2
-is unique, if
method (2) is stable then
2]2/[2 kp
and there
are stable methods of type (2) with the degree
2]2/[2
max kp
for each
k
. But if method (13) is
stable and has the degree of
p
, then
)3(1 mkmkp
. By using these estimations
one can compare methods (2) and (13) in full form.
As is known all the methods have their advantages
and disadvantages. The stable advanced method is
more exact than the corresponding methods of type
(2). But in using that there arises some difficulty
with the calculation of the values of the solution
investigated problems at the next points. To
demonstrate this let us consider the following
method:
.12/)85( 211
nnnnn yyyhyy
(19)
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One of the base properties of this method is the use
of the values of the desired function on the
subsequent points.
Let us apply this method to solving the problem
(15). In this case receive:
.12/)),,(),,(4
),,(4),,(),,(2
),,(2(12/)85(
222112
11121
211
nnnnnn
nnnnnnnnn
nnnnnnnn
yxxfyxxK
yxxKyxxKyxxK
yxxKhfffhyy
(20)
Thus receive the nonlinear equation for finding the
value
,
1n
y
to fined which is not easy. Therefore
here recommended to use the following step-by-step
algorithm (Algorithm 2):
Step 1. Input (initial values);
Step 2. For
1j
step 1 to n-1 do step 3-10;
Step 3. Calculation
e
j
y1
by the method (9);
Step 4. Calculation
ij
y1
by the method (9);
Step 5. Calculation
2/)( 111
e
j
ijj yyy
;
Step 6. Calculation
1
2j
y
by the method (9);
Step 7. Calculation
2
2j
y
by the method (9);
Step 8. Calculation
1
2j
y
by the method (10);
Step 9. Calculation
2
2j
y
by the method (11);
Step 10. Calculation
2/)( 2
2
1
22 jjj yyy
;
Print
);; 11
1
j
e
j
j
iyyy
end.
Step 11. Stop.
Note that method (18) is stable, has the degree
3p
, and rises to the class of one-step and
multistep methods. It is easy to understand using the
method (18). It is enough to know the one value of
the desired solutions at the previous point. For the
sake of objectivity, note that for the application of
the method (18), it is needed to have some
information about the values of the solution at the
current and next points. To find these values by the
required exactness, one can use the above-described
way. Note that if the trapezoid method to used in the
application of the method (18), then some difficult
liberation from which is even more difficult. Thus in
the application of the method (18) to solve some
problems, it is needed to use any method for
calculating the values
mkn
y
and
)1,..,1,0(
mjjy mkn
the results of which
receive the block method. The step-by-step
algorithm, which is presented above is also a block
method. If It is necessary one can increase the
accuracy above calculated values. In our case, it is
desirable to use the Simpson method, which is
presented by the method (12). This method when
applied to solving problem (1) can be presented as
follows:
.6/)),,(2
),,(4),,(4),,(
),,((3/)4(
222
1121112
1212
nnn
nnnnnnnnn
nnnnnnnn
yxxK
yxxKyxxKyxxK
yxxkhfffhyy
(21)
By using method (21) with the above-presented
method (20), one can solve the problem (1) with
high precision. Note that in the application of the
method (21), some difficulties are related to solving
the nonlinear algebraic equations. However, this
problem can be solved with the predictor-corrector
method. Each of these methods has its advantages
and disadvantages. If you compare the methods
described above, then get that the methods described
above using step-by-step algorithms can be taken as
better.
Remark 2. As is known in recent times scientists
tried to construct simple methods for solving some
problems. The above-constructed algorithms belong
to the class of simple methods. Let us show that
there are simple methods with a high order of
accuracy that differ from the above-mentioned
methods. Let us remember the midpoint roll, which
in the application to solving the problem (1) can be
presented as:
.2/)),,(,,((
))2/(,2/(
2/12/12/12/12/11
1
nnnnnn
nnnn
yxxKyxxKh
hxyhxhfyy
(22)
This method is stable, has a degree
2p
, and is
explicit. By simple comparison, that method (21) is
better than the above investigated. Note that the
method (22) also has some disadvantages, for
example calculating the value
2/1n
y
. Some similar
investigations were carried out by some authors in
solving different problems, [42], [43], [44], [45],
[46], [47], [48]. Note the method (22) reminds us
hybrid methods that more accurate than the known.
There is a lot of work dedicated to the study of hbrid
methods, [49], [50], [51], [52], [53], [54], [55].
2 Numerical Results
For the demonstration receiving here result, let us
consider the application of the above-presented
algorithms to solve the following simple examples:
.10,0)0(
,))(exp(2)exp(1
0
22
xy
dttyxtxxyy x
(23)
The exact solution of this example:
xxy )(
.
WSEAS TRANSACTIONS on MATHEMATICS
DOI: 10.37394/23206.2023.22.86
Vagif Ibrahimov, Galina Mehdiyeva,
Mehriban Imanova, Davron Aslonqulovich Juraev
E-ISSN: 2224-2880
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Volume 22, 2023
There are many methods for solving this
problem, [54], [55], [56], [57], [58], [59], [60], [61],
[62], [63], [64].
However, here we profer to useabove proposed
methods.
First applied Algorithm 1 and 2 to solve problem
(23), results for which are tabulated in Table 1.
Table 1. Maximum error for the above-presented
algorithm 1 and algorithm 2.
Step
size
Value
of
x
Maximal error
for the
Algorithm 1
Maximal error
for the
Algorithm 2
h=0,05
1
0.28 E -03
0.1 E -05
The receiving results are corresponded to the
theoretical. Now let us consider solving the
following problem:
.20,1)0(
,)()exp(1(
0
1
xy
dssyaxayy x
(24)
The exact solution for which can presented as:
).exp()( xxy
For this aim to use algorithm1 and methods (15),
(18).
Note that depending on the methods applied to
the calculation of the value
2n
y
, method (18) can
change its properties. For example,
1
n
y
if
n
y
to
use the formula:
2/)3( 112 nnnn ffhyy
in
the method of (18), then in the resulting of which
the method will be A- stable, [48].
All results are tabulated in Table 2.
Table 2. Results for step-size h=0.01.
x
Method(18)
+Tr
Method(18)+
1
E
Method (18)+
2
E
0.1
4.5E-09
4.5E-09
1.8E-06
0.3
1.2E-09
1.2E-09
5.1E-07
0.5
2.1E-09
2.1E-09
8.6E-07
0.8
3.5E-09
3.5E-09
1.4E-06
1.0
1.1E-07
1.1E-07
4.4E-05
Here have used denotation Method (18)+
2
E
,
Method(18)+
1
E
, Method(18)+Tr.
Method (18)+
2
E
-is the predictor-corrector
method. Here the predictor method has used the
explicit method, but as corrector method, used
method (18). Here
1
E
-is the implicit Euler method,
2
E
-is the explicit Euler method, but the Tr-
Trapezodial rule.
And now let us the initial value in problem (23)
to use method (15) and the Trapezoidal rule, but as
the predictor method has used the explicit Euler
method. Results tabulated in Table 3.
Table 3. Results for h=0.01
x
1
1
Method
(15)
Trapezoidal
rule
Method
(15)
Trapezoidal
rule
1.1
1.5E-7
4.6E-4
2.0E-8
5.9E-5
1.4
8.9E-7
6.2E-4
5.8E-8
4.4 E-5
1.7
2.1E-6
8.4E-4
7.5E-8
3.2 E-5
2.0
4.1E-6
1.1E-3
7.9E-8
2.4 E-5
Results received for the implicit method taken as the
predictor method have been tabulated in Table 4.
Table 4. Results for h=0.01.
x
1
1
Method
(15)
Algorithm1
Method
(15)
Algorithm1
1.1
1.0E-8
6.0E-5
1.2E-9
7.2E-6
1.4
5.7E-8
8.1E-5
3.5E-9
5.3 E-6
1.7
1.3E-7
1.0E-4
4.6E-9
3.5 E-6
2.0
2.6E-7
1.4E-4
4.9E-9
2.9 E-6
But now, let us as the predictor method with the
following midpoint rule.
).2/(
1hxyhyy iii
Table 5. Results received for the values h=0.05.
x
5
5
Method
(15)
Trapezoidal
rule
Method
(15)
Trapezoidal
rule
1.1
2.3E-3
5.3E-1
5.1E-8
1.2E-5
1.4
4.4E-2
2.4E-0
4.3E-8
2.9 E-6
1.7
3.4E-1
1.1E-1
1.6E-8
6.5E-7
2.0
2.2E-1
4.9E-1
5.2E-9
1.4E-7
By the results tabulated here, one can take the
midpoint rule as the better, which is related to the
using value,
i
y
calculated by the method (15). Here
tabulated the results received by the method (15)
with the degree
4p
, by the trapezoidal rule
2p
, and the algorithms 1 and 2. By the
comparison of the results tabulated here, we see that
these results correspond to the theoretical.
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DOI: 10.37394/23206.2023.22.86
Vagif Ibrahimov, Galina Mehdiyeva,
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3 Conclusion
Here, have shown that numerical methods
constructed for solving initial-value problems for
ODEs can be applied to solving initial-value
problems for the Volterra integro-differential
equations. We apply this approach to constructing
bilateral numerical methods to solving initial-value
problems for the Volterra integro-differential
equations. For this aim have used simple numerical
methods and have proven that the results received
by the bilateral methods are better. This idea has
been applied to the construction of two numerical
bilateral methods and has shown how one can
construct a similar method. As is known, bilateral
(or two-sided) methods for solving initial-value
problems for the Volterra integro-differential
equations, can be said to be almost uninvestigated.
Therefore, in this area, the new results obtained here
are of interest to many specialists from different
areas of modern sciences. In the work, [45], a two-
sided method is constructed for solving the ODEs,
but the work, [53], has constructed the bilateral
methods to solve the Volterra integral equations by
using the Runge-Kutta methods. The receiving
theoretical result has been demonstrated in simple
examples. We hope that this method will find its
followers. Therefore, they can be considered
promising. It is known that one of the promising
methods is the advanced method. By taking into
account the results tabulated in Table 3, Table 4,
and Table 5, the methods with the fractional step
size give good results. These methods are reminded
of the hybrid methods. Therefore, these methods are
promising.
Acknowledgment:
The authors wish to express their thanks to
academicians Telman Aliyev and Ali Abbasov for
their suggestions to investigate the computational
aspects of our problem and for their frequent
valuable suggestions. This work was supported by
the Science Development Foundation under the
President of the Republic of Azerbaijan- Grant
AEF-MCG-2022-1(42)-12/4/1-M-4
Thanks to the anonymous reviewers for their
valuable comments, which improved the estimation
of receiving here results.
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Contribution of Individual Authors to the
Creation of a Scientific Article (Ghostwriting
Policy)
Corresponding author V.R. Ibrahimov was
responsible for used methods, research,contribution
of the concept; G.Y.Mehdiyeva -data interpretation
and analysis,M.N.Imanova-illustration of received
results; D.A.Jurayev-data interpretation and
analysis.
Sources of Funding for Research Presented in a
Scientific Article or Scientific Article Itself
This work was supported by the Science
Development Foundation under the President of the
Republic of Azerbaijan- Grant AEF-MCG-2022-
1(42)-12/4/1-M-4
Conflict of Interest
The authors have no conflict of interest to declare.
Creative Commons Attribution License 4.0
(Attribution 4.0 International, CC BY 4.0)
This article is published under the terms of the
Creative Commons Attribution License 4.0
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