On Some Ways to Increase the Exactness of the Calculating Values of
the Required Solutions for Some Mathematical Problems
M. N. IMANOVA1,2, V. R. IBRAHIMOV2,3
1Science Development Foundation of the Republic of Azerbaijan,
Baku AZ1025,
AZERBAIJAN
2Institute of Control System named after Academician A. Huseynov,
Baku Az1141,
AZERBAIJAN
3Computational Mathematics,
Baku State University,
Baku AZ1148,
AZERBAIJAN
Abstract: - The expansion of the application of computational methods for solving many mathematical
problems from various fields of natural knowledge does not raise any doubts. One of the promising directions
in contemporary sciences is considered to be in areas that are at the intersection of different sciences. Solving
such problems is more difficult because different laws from different areas are used. It should be noted that at
the intersection of these sciences, there are problems, which can come down to solving ordinary differential
equations. Therefore, studies of differential equations have always been considered promising. Based on this,
the application of some methods for solving initial problems for first-order ODEs is investigated. For this
purpose, scientists studied a numerical solution to the initial problem of the ODE. Here, we have reviewed the
study of linear Multistep Methods with constant coefficients. With its help, the order of accuracy of the
calculated values is determined. In addition, determines how much accuracy values increase when using
Richardson extrapolation methods and also when using linear combinations of various methods. To construct
an innovative method is proposed here using advanced methods. It is shown that using these methods it is
possible that A-stable methods can be taken as innovative.
Key-Words: - Ordinary Differential Equation (ODE), Local truncation Error, Multistep Method (MM),
Richardson Extrapolation (RE), Stability and Degree (S,D), Initial Value Problem, Advanced
Methods, Multistep Secondderivative Methods (MSM).
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1 Introduction
We were seriously engaged in the study of Ordinary
Differential Equations after familiarizing ourselves
with the work: Constantin Carotheodory, “Calculus
of variations and partial differential equations of the
first order”.
Let us note that many well-known scientists
were engaged in the search for a solution to the
initial problem of the ODE. They constructed some
classes of methods having different properties. Thus
creating the opportunity for a wide selection of
numerical methods. For this purpose, scientists
defined some conceptions for their comparisons. For
this purpose, scientists have found some conception
by which one can define the boundaries for all the
errors received in using methods with constant
coefficients (see for example, [1], [2], [3], [4], [5],
[6], [7], [8], [9], [10], [11]). For the compassion of
the known methods let us consider investigating the
following problem:
,,)()),(,( 000 Tttztztztz
(1)
which usually is called the initial-value problem for
ODEs of the first order. For the construction
numerical methods with the new properties, let us
impose some restrictions on the solution of problem
(1) and also on the function
),( zt
. Let the
solution to the problem (1) be a continuous function
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defined on the segment
],[ 0Tt
. We mean that a
continuous function
),( zt
is defined in a certain
limited domain and, inclusive, has partial
derivatives up to
p
. To find a numerical solution to
the problem (1), we divide the segment
],[ 0Tt
into
N
parts using grid points
,..)2,1,0(
1
ihtt ii
. And also, here indicate
exact values of the solution of the problem (1) by
)( i
tz
, but by the
i
z
-the corresponding approximate
values of the function
)(tz
at the point
i
t
.
Note that one of the popular classes of
numerical methods is the class of multi-step
methods, which can be depicted as follows:
m
j
m
jinjjnj mNnbhza
0 0
,,..,1,0,
(2)
here
).,...,1,0(),( Njzt jjj
Such methods have been studied in the works of
many authors,(see for example [3], [4], [5], [12],
[13], [14], [15], [16], [17], [18], [19], [20], [21],
[22], [23], [24]). But fundamentally has been
investigated by Dahlquist.
From the outside, [4] has proven that if (2) is a
stable method and has a degree
p
, then
2]2/[2 mp
Conception stability and degree,
which have been used can be presented as follows:
Definition 1. An integer value is called a power for
method (2) if the identity is satisfied:
.0,)())()((
0
1
hhOihxhihxz
m
i
p
i
(3)
Definition 2. Note that (2) is called a stable
method if the roots of the characteristic polynomial
01
1
1...)(
k
k
k
k
are located
in the unit circle, which does not have multiple roots
on the boundary. This conception is given in [3] and
called the “dispersion”. But, [7], have used the
concept of “stability”. By using results receiving in
[4] one can be noted that if method (2) is stable,
then
2
max kp
. The scientists for the calculation
of the values of the solution problems (1), have
suggested some ways. One of these ways is the
known Richardson extrapolation, which in the
application to method (2) can be constructed by
using the local truncation error.
§1. Some ways to increase of the exactness of the
receiving results by using the known methods.
Based on method (2), we assume that in order to
construct ways to improve the accuracy of
calculated values, that method (2) has a degree
p
.
It is known that the local truncation error for method
(2) can be represented as follows:
).0(),(2)1(1 shOzch spp
n
p
(4)
And now suppose that step-size
h
to change by the
kh
. Then formula (4) can be written as:
).0(),(2111 shOzhck spp
kn
pp
(5)
To illustration of Richardson’s extrapolation, let us
multiply local truncation error (4) by
, but local
truncation error (5) by
1
. Then after summing
(4) and (5) receive:
).())1(( 2)1(11 spp
n
pp hOZkch
(6)
In usually the values for the
k
has been taken as the
22/1 kork
. But here the known cases are
generalized by the constant of
k
.
As is known some authors noted that by using
Richardson extrapolation one can be construct more
exact methods. Let’s show that this is not so. To
increase the accuracy of the method it is enough that
satisfies the following equation:
.0)1( 1 p
k
(7)
For this case s=1, here is some constant
participated in asymptotic relation (5).
It is obvious that the solution of this equation
will be a real number. Thus, after using the value of
, one can construct a method with constant
coefficients. Therefore, as a result, the resulting
method must obey the laws from the [4]. Note that
the method does not change its structure. In this way
receive that, one gets that the function to which the
multistep method is applied is changed. Because of
this, the calculated values for solving our problem
by Richardson extrapolation are more accurate:
.,..,2,1,0
,)1( )()(
mNn
zzz kh
knn
hmnmn
(8)
For the
2/1
required values can be presented
as:
.2/)( )2/( 2
)( hmn
hmnmn zzz
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If method (2) is stable and has the degree of
p
,
then receive that
2]2/[2 ksp
is holds. Now
let’s look at finding the values of the solution to
problem (1) using a linear combination of some
methods. Let's use the following Euler methods to
illustrate the advantages of this path
).,();,(
ˆ1111 nnnnnnnn zthzzzthzz
(9)
It is not difficult to prove that the Local
Truncation Error for these methods can be presented
as follows:
.0
),(2/),(2/
ˆ3232
h
hOzhRhOzhR nnnn
(10)
As follows from this, the half-sum of these local
truncation errors will be smaller than the errors have
defiant by the asymptotic equality (10). Indeed this
is so, for this let's consider the half-sum above the
given methods (9), and then we get the following
method:
,2/)),(
),((2/)
ˆ
(
11
11
nn
nnnnn
zt
zthzzz
(11)
which is the known Trapezoidal role. This method
has the degree
2p
, but methods (9) have, the
degree
1p
. Consequently, the Trapezoidal rule is
more exact.
It is easy to define that calculation
1n
z
is more
difficult, than the calculation of the value
1
ˆn
z
. For
the correction of this disadvantage, let us to define
the value
1n
Z
by the following method:
)),,(,( 11 nnnnnn zthzthzz
(12)
which is explicit and does not arise any difficulty in
the calculation of the value
1n
z
by this formula.
It is easy to show that, method of (12) can presented
as the follows:
).,( 111 nnnn zthzz
(13)
Thus, by the described above-mentioned
method, receive some predictor-corrector method.
The predictor and corrector methods have one and
the same degree, which is equal to 1 (one). As was
shown above, by using half sum of the values of
problem (1) calculated within using predictor and
corrector methods, receive the new method, which
is more exact than the predictor and corrector
methods. In our case, the one-step method of (9) has
constructed the one-step method, but in using
Richardson extrapolation method remains the same.
Thus, in using Richardson extrapolation receives the
new function to calculate which applied the using
method. We get that when using the Richardson
extrapolation each time one can increase the
exactness of calculated values at the mesh points.
However, in using linear combinations of some
multistep methods the degree of exactness must
obey the laws from [5]. In this case, if one can
receive the results with a higher degree by using a
linear combination of some multistep methods, then
receive that or the method used has a low order of
accuracy or the number of terms in the resulting
method increases. And now let us consider the
following methods:
1
0 0
)2(
1
0 0
)1(
ˆ
,
k
j
k
jjnjjnjkn
k
j
k
jjnjjnjkn
hzz
hzz
(14)
with the local truncation errors
,0),(
),(
1
)1(
2
)2(
1
)1(
1
)1(
2
1
hhOzcR
hOzcR
sp
p
nn
sp
p
nn
here
),min( 21 sss
.
To get the best results it is enough to use the
solution of the following equation:
,0)1( 21 cc
in the following expression
.)1( )2()1( knknkn zzz
(15)
Note that based on formula (15), the resulting
value of the solution to problem (1) will be more
accurate than the values of
)1( kn
z
and
)2( kn
z
It is easy
to understand that the values
)1( kn
z
and
)2( kn
Z
are
calculated with the order of accuracy of
p
, then
value calculated by the formula (15) will be has the
order of accuracy of
1p
. Considering that in this
process has used the values, which have been
calculated by using a stable method, we get that this
scheme gives a positive result. By the above
described, one can construct a very simple way to
increase the accuracy of the approximate values as
the solution of the problem (1). For this aim let us
consider the following section.
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§2. On some ways for the increasing of exactness
of the numerical method (2).
Noted that by the equality of (15) receive that, for
more accurate results, here have used linear
interpolation. Currently, to improve the accuracy of
the solution values to the problem (1), it is proposed
to use methods with second derivatives, which are
represented as follows:
k
i
k
i
k
iiniiniini nZhZhZ
0 0 0
2,...).2,1,0(
(16)
It easy to understand that
zt
tz )(
.
In the finding, the values of the computational work
are increased, which depends on the calculation of
the values
),(),( ztandzt zt
at the mesh points
)0( Nmtm
. With this in mind, it is proposed
here to use an extended (or jumping) method, which
in some simple form can be represented as follows:
mk
i
k
iininiini nmzthz
0 0
,..).2,1,0;0(),(
(17)
Similar to the described above, studied by many
authors, [25], [26], [27], [28], [29], [30], [31], [32],
[33], [34], [35], [36],[37].
Method (17) was fundamentally studied by in
the work [19], [20].who came to the conclusion that
if method (17) is stable, then in class (17) there are
methods with degree
1 mkp
for
mk 3
.
Obviously, based on Dahlquist's law, we can
conclude that if method (2) is stable, then in the
class of method (2) there are stable methods with
degree
2]2/[2 kp
(here
mk
). By a usual
comparison, we find that stable methods like (17)
are more accurate than (2). As an example, in class
(2) there are stable methods of degree
2]2/[2 kp
for all
k
. Therefore, for
3k
there is a stable method with degree
)1(4 maxmax kpp
. But in (17) there is a
stable method with degree
5p
for
3k
,
represented as:
,57/)245710(
19/)811(
321
12
nnnn
nnn
h
zzz
(18)
Obviously, this method is stable and also has a local
truncation error of degree
5p
, written as
).(
3420
11 7)6( )(
6hOzhR n
xn
It is noted that the main disadvantage of these
methods is finding the values of the solution to
problem (1) in neighboring grid nodes. To solve this
problem, we use a predictor-corrector similar to the
methods. Let's look at a method like:
12/)51623(1223 nnnnn hzz
. (19)
By using method (19) in the formula (18), receive
the following method:
.57/)12/)51623(
,(24
5710(19/)811(
12
232
112
nnn
nnn
nnnnn
h
zth
hzzz
(20)
And now let us change
3n
z
participation in (18) by
the following:
3/)27( 1213 nnnnn hzz
. (21)
In this case, receive the next methods:
57/)3/)27(
,(57/)24
5710(19/)811(
12
132
112
nnn
nnn
nnnnn
h
zth
hzzz
(22)
Obviously, this method is stable and also
implicit. Method (20) is A-stable, but the method
(22) is stable. To ensure the accuracy of the results,
consider the following section.
2 Numerical Results
To illustrate the results, we provide relevant
examples:
.20,1)0(,
tzzz
exact solution for which can be presented as:
)exp()( ttz
. To solve this example let us to us
the following couple methods
,2
ˆ
3
ˆ12 nnnn hzzz
(23)
,12/)58
ˆ
(122 nnnnn hzz
(24)
,2/)3(
ˆ112 nnnn hzz
(25)
,12/)58
ˆ
(122 nnnnn hzz
(26)
),46(98
ˆ112 nnnnn hzzz
(27)
,24/)318
ˆ
9( 1212 nnnnn hzz
(28)
.9/)
ˆ
8( 222 nnn zzz
(29)
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It is known that the local truncation error for
methods (24) and (26) is represented as:
).(24/53 hOzhR nn
The receiving results for method (24),(26), and
(27) are tabulated in the following tables:
Table 1. Results for the
05.0h
and
.1
i
t
Method
(24)
Method
(26)
Exact value
Method
(29)
0.
1
0.
5
2.
0
1.1051988
6
1.9313402
2
more
1.1051702
5
1.6487264
6
7.7698516
8
1.10551709
1
1.64872122
7.76790016
1.1051712
0
1.6487827
3
7.7678623
2
Table 2. Results for the
1.0h
and
1
i
t
Method
(24)
Method
(26)
Exact
value
Method
(29)
0.
2
0.
9
2.
0
1.2216634
3.3829898
8
more
1.2214040
8
2.4599552
2
7.3962984
1
1.2214126
6
2.4596023
2
7.3890552
2
1.2214126
6
24560233
7.3890553
27679016
2
As follows from the Table 2, receive that the
results received by the method (29) are
unacceptable. For the corrected this situation, let’s
consider the case, when
0
the solution is
decreasing.
Table 3 . Results for the
05.0h
and
.1
i
t
Method
(24)
Method
(26)
Exact
value
Method
(29)
0.
1
0.
5
2.
0
0.9448141
8
0.4168961
0
more
0.9448374
3
0.6065349
6
0.1353487
4
0.9483745
0.6065309
8
0.1353353
0
0.9463966
0
0.6065325
1
0.1353304
4
Table 4. Results for the
1.0h
and
1
i
t
Method
(24)
Method
(26)
Exact
value
Method
(29)
0.
2
0.
9
2.
0
0.818473
40
0.337485
07
more
0.818722
19
0.367995
56
0.135498
88
0.8187307
6
0.3678795
0
0.1355335
38
0.8187310
7
0.3687812
28
0.1355330
5
By the results of the Table 1, Table 3, Table 4,
the results received by the method (29) can be
considered as the better.
3 Conclusion
Here are given some ways which usually are used
for the increased accuracy of the receiving results by
using stable Multistep Methods. Shown that by the
selection of predictor methods in the predictor-
corrector methods one can receive the method,
which behaves like an unstable method. Note that in
the predictor-corrector method of (25)-(26), the
predictor method is stable, but in the predictor-
corrector method of (23) and (24) the predictor
method is unstable. Obviously, the predictor-
corrector method is convergent if the corrector
method is stable. Here, the predictor method was
used as a separate unstable method, so the results
obtained by method (24) are unacceptable.
However, the second predictor method is
convergence (the predictor-corrector method is
convergence because the predictor-corrector
methods are robust). It is known that in the linear
combination constructed using methods (27)-(29),
unstable methods are involved as predictors, but
because of this, the results are better. Similar results
are obtained by using Richardson extrapolation.
Thus we receive that using linear combinations
gives the best results. However, the selected
appropriate methods are very important. The reason
for the increased accuracy of Richardson
extrapolation and the linear combination of various
methods is also explored here. Note that for the
increased accuracy of the calculated values of
solution of the investigated problem, one can use the
bilateral methods, this method can be taken as the
better, so by using the bilateral methods one can
define the availability of the receiving results. We
would like to note that here used references with we
have also encountered in other popular works. And
have given some information about our new articles.
Note that this method is interesting and very simple,
so we hope that the methods described above will be
very useful for a circle of readers and researchers.
Acknowledgments:
The authors thank Academician T. Aliyev, and
Academician A. Abbasov for their useful Valuable
advice. This work was partially supported by the
Science Development Foundation of the Republic of
Azerbaijan - grant AEF-MCG-2022-1(42)-12/4/1-
M-4.
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Finally, the authors thank the reviewers for their
comments, which improved the content of this
article.
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WSEAS TRANSACTIONS on MATHEMATICS
DOI: 10.37394/23206.2024.23.45
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WSEAS TRANSACTIONS on MATHEMATICS
DOI: 10.37394/23206.2024.23.45
M. N. Imanova, V. R. Ibrahimov
E-ISSN: 2224-2880
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Volume 23, 2024
