The New Way to Solve Physical Problems Described by ODE of the
Second Order with the Special Structure
V. R. IBRAHIMOV
Institute of Control Systems named after Academician A.Huseynov, Baku AZ1141,
Department of Computational Mathematics, Baku State University, Baku AZ1148, Baku,
AZERBAIJAN
M. N. IMANOVA
Science Development Foundation under the President of the Republic of Azerbaijan Baku, Baku,
AZERBAIJAN
Abstract- In the last decade, many researchers have studied extensively theoretical and practical problems of
natural sciences using ODEs as a means to analyze and understand them. Specifically, second-order ODEs with
special complex structures provide the necessary tools to construct mathematical models for several physical -
and other- processes such as the Schturm-Liouville, Schrölinger, Population, etc. As a result, it is of great
importance to construct special stable methods of a higher order as a means to solve differential equations. One
of the most important efficiency methods for solving these problems is the Stёrmer-Verlet method which
consists of hybrid methods with constant coefficients. In this paper, we expand on recent studies that prove that
the hybrid methods are more precise than the Stёrmer-Verlet method while investigating the convergence
variable. This paper aims to prove the existence of a new, stable hybrid method using a special structure of
degree(p)=3k+2, where k is the order of the multistep methods. Lastly, we also provide a detailed mathematical
explanation of how to construct stable methods on the intersection of multistep and hybrid methods having a
degree(p)≤3k+3.
Key-Words: Hybrid method of Stёrmer type, multistep second derivative method, stability and degree,
relationship between order and degree.
Received: May 22, 2022. Revised: January 29, 2023. Accepted: February 24, 2023. Published: March 9, 2023.
1 Introduction
As is known, some theoretical and applied problems
are reduced to solve the initial value problem for
ODE of the second order, which can be presented as:
.,)(,)(),,,()( 00000 XxxyxyyxyyyxFxy
(1)
There are wide classes of numerical methods for
solving this problem. For solving problem (1), here
proposed to use the multistep multiderivative
methods with constant coefficients of the hybrid
type. Let us assume that problem (1) has a unique
solution for a continuous total arguments function
that has defined or some closed area. Some known
scientists who investigated the solution using ODE
participated in the problem(1). For the illustration of
this, let us consider the following generalization of
the known Schrödinger and Sehturm-Liouville
equations (see [1], [2], [3). Note that the
Schrölinger, and Schturm-Liouville problems are
usually formulated using the boundary-value
problem for the abovementioned equation (see for
example, which can be reduced to the solution of the
initial value problem for the ODE of the second
order (see for example [4], [5], [6]). The problem (1)
has been investigated by many authors by using the
one-step or multistep methods (see for example [7],
[8], [9]). For solving the problem (1), here proposed
to use some generalization of the Stёrmer-Verlet
methods, the construction of which has used the
hybrid method with the special structure. To
determine the maximum value of the order of
accuracy for the proposed methods, have used the
way of unknown coefficients and the theory of
nonlinear systems of algebraic equations (see for
example [10], [11], [12], [13], [14], [15]).
As is known, the Stёrmer-Verlet method can be
applied to solving the initial-value problem for the
above-mentioned second order with the special
structure. For the illustration of this, consider the
construction of an effective method for solving
named problems.
Suppose that the right-hand side of the ODE in the
problem (1) has been presented as .
In this case, the problem (1) can be written as the
following:
. (2)
The solution to this problem can be presented as:
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(3)
By using this equality one can write the following:
hx
x
hx
x
hx
x
hx
x
dssfhdssfsxxyhxxxyxyhxy
dssfhdssfsxxyhxxxyxyhxy
0 0
0 0
.)()()()())(()()(
,)()()()())(()()(
0000
0000
(4)
Here, the solution of the understudy is constructed in
such a form that allows one to construct the Stёrmer-
Verlet method using a minimum number of
operations.
Applying any quadrature method to the calculation
of definite integrals involved in the above equality
and comparing them, we obtain the following
numerical method for solving the problem, which
can be presented in the following form:
)5(.
0 0 0
2
k
i
k
i
k
iiniiniini fhyhy
The results of the investigation of method (5)
received some connection (see for example, [7], [8],
[9], [10], [11]).
For using method (4) it is necessary to construct
some algorithm for the calculation of the values
,...)2,1,(mym
. For this aim, let us consider the
following:
x
hx
hx
x
dssfxyhxy
dssfxyhxy
.)()()(
;)()()(
(6)
By using above mentioned, for the calculation of the
value
one can be recommended the following:
This method can be received from the known
multistep method, which is applied to solve ODE of
the first order, by the change with the.
Noted that for solving the problem (2), let us use the
points
,,, 000 hxxhx
or the points
hxxhx ,,
. For the construction methods
similar to the Stёrmer method let us consider the
following equalities:
hx
x
hx
x
hx
hx
hx
hx
dssfhdssfsxxyhxyhxy
dssfhdssfsxxyhhxyhxy
0
.)()()()()(
,)()()(2)()(
(8)
By using (3) and (4) or the equalities (8), receive:
x
hx
hx
x
x
hx
hx
hx
dssfh
dssfhdssfsxdssfsxhxyxyhxy
.)(
)()()()()()()(2)(
(9)
If put in the equality of (9),
then receive:
.)()()(
)()(2
2
1
2
1
21
kn
kn
kn
kn
kn
kn
x
x
x
x
kn
x
x
knknknkn
dssfhdssfsx
dssfsxyyy
(10)
To use some quadrature formula for the calculation
of definite integrals participated in equation (10),
receive
k
i0
f
(11)
Here the coefficients of are
calculated by the values of the coefficients of the
quadrature formula which has been applied to the
calculation of the definite integrals participating in
the equality of (10). By the generalization of the
linear part of the method (11), one can receive the
following:
)12(.
0 0
2
k
i
k
iiniini fhy
By the comparison of methods (5) and (12), method
(12) can be obtained from method (5) as the partial
case. As is known, one of the basic conceptions for
comparison of numerical methods is in regards to
the degree and its stability, which can be defined in
the following way (see for example [7], [8], [9],
[10]).
Definition 1. The integer value is called the degree
for the method of (5) if the following asymptotic
equality takes place:
k
i0
. (13)
If
),,..2,1,0(0 ki
i
then the asymptotic equality
presented as:
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.0)()()(( 22
0
hhOihxyhihxy p
i
k
ii
(14)
Definition 2. Method of (12) is called as stable if the
roots of the polynomial
01
1
1...)(
k
kk
k
located in the
united circle, on the boundary of which there are no
multiple roots.
From here we receive that method (12) is an
independent object for the investigation. Note that
method (12) was investigated by many authors (see
for example, [4], [5], [6], [7], [8], [9], [10], [11],
[12], [13]). And have defined the conditions which
must satisfy its coefficients for its convergence. The
relation between the exact and numerical solution of
problem (1) is investigated in the next paragraph.
2 Application of Hybrid Methods to
Solve the problem (1).
Usually, finding numerical solutions to problem (1)
has used methods which are called multistep
methods with constant coefficients or the finite
difference method. For construction, more exact
methods here have proposed to use the forward-
jumping and hybrid methods, so these methods have
some advantages. For the construction methods of
type (12), for solving the following problem:
.)(,)()),(,()( 000 yxyyxyxyxfxy
(15)
Let us consider the finding of the coefficients in the
method (12). For this aim, consider the
approximation of definite integrals participating in
the equality of (8). As is known there are some
classes of methods for the calculation of definite
integrals. Here proposed to use the following
method, which can compare with the equality of
(11):
).,...,1,0,1(,
)()()()()(
00
22
21
1
1
kifhfh
dssfhdssfsxdssfsx
k
iiini
k
iini
x
x
x
x
x
x
knkn
i
kn
n
kn
n
kn
n
(16)
By using these equalities in the asymptotic equality
of (12) one can receive:
k
i
k
i
k
iiiniiniini kifhfhy i
0 0 0
22 ).,...,1,0;1(,
(17)
Note that for the calculation of definite integrals, one
can use the method (17). In this case
follows from the formula (17) the known
quadrature methods. In other cases, the method (17)
receives the new method, the properties of which are
depending on the values of coefficients of the
formula (14) and from the values of.
.
If applied the method of unknown coefficients is to
determine the values of the coefficients which are
participated in the formula (12), then receive
methods which are usually called the finite-
difference method. To determine the values of the
coefficients, a nonlinear system of algebraic
equations is obtained. For the construction of the
named system, let us use the following Taylor
series:
By using these equalities in the asymptotic equality
of (12) one can receive:
(18)
It follows from here that, if the method of (12) has
the degree of
p
, then by the comparison of the
asymptotic equalities and receive that the following
must satisfy (see for example, [16], [17], [18], [19],
[20], [21]):
.
(19)
By taking into account that the systems
or,
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are
independent, receive that for satisfying the equality
of (19) the following system must have the solution
(see for example, [13], [18], [19], [20], [21], [22],
[23], [24], [25], [26], [27], [28], [29], [30]) :
.1,...,3,2;0)
)!2(!
(;0;0
0
2
00
pj
j
i
j
i
ik
ii
i
jj
k
ii
k
ii
(20)
By finding the coefficients of method (12) we
receive the linear system of algebraic equations. The
error for method (12) can be estimated by the error
of the quadrature methods. For the sake of
objectivity, let us note that in the application of the
method (12) to solving initial-value problems some
errors.
From the stability of method (12), it follows that all
the errors arising in using method (19) will be
bounded. Therefore, let us investigate the system of
. In this system, the amount of the unknowns is
equal to , but the amount of the equations is
equal to . It is not difficult to prove that the
linear system has a unique solution for the case
. But this equality for the stable methods of type
(11) may be written as: . And also,
the constant must satisfy the condition . This
condition follows from equality (11). As a result, it
is proved the following lemma:
Lemma. If method (12) has the degree of, then its
coefficients must satisfy the system of (20) and vice
versa. If the coefficients of method (12) satisfy
condition, then method will have the
degrees of, which satisfies the condition
for the stable and the condition
for other methods.
And now let us consider the investigation of the
method (17). For the investigation method (17), here
offer one way for finding the values of the
coefficients
).,...,2,1,0(,,, ki
iiii
For this aim, one can be used
the above-presented Taylor series with the
following:
here
By repeating the above using description for the
finding of the coefficients,
, receive the following nonlinear system
of algebraic equations:
k
i0
k
i0
; (21)
And now consider the explanation of the condition
Note that by using the first two equations of
system (20) or (21) receive that is the double
root of the polynomial. Note that this condition
is necessary for convergence of the method (17),
therefore can be written as:
By taking this into account in the system of (21)
receive that the amount of the unknowns equal to
, but the amount of the equations equal to
and the received system will be linear
nonhomogeneous which will have a unique solution
in case and . From
here it follows that, if
.
System (21) is different from the system (20) as
system (21) is nonlinear. Moreover, system (21)
follows the system (20) in the case of
.
Here, also by using the properties of the first two
equations receive that the amount of the equation
can be taken as and in this case, the
homogeneous system (21) becomes the non-
homogeneous system. From here it follows receive
that the system (21) can have the solution by which
will construct methods with the degree.
And by the construction of the concrete stable
methods with the degree reserve that
method (17) can have the degree. From
here we obtain the following theorem.
Theorem: Method (17) has the degree of
p
,
satisfying its coefficients
the nonlinear system of (21) is necessary and
sufficient. If method (17) is stable, then there are
methods with the degree.
It is not difficult to prove that if there exist methods
of type (5) with the degree. But if
method (5) in this case has the degree and is stable
then the degree for method (5) satisfies the
condition::
Method (17) coefficients the nonlinear system of
(21) which is necessary and sufficient. If method
(17) is stable, then there are methods with this
degree.
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By comparison of all the above-described
advantages and disadvantages properties of the
suggested methods, receive that the methods of type
(17) have some advantages. Therefore, they can be
taken as the perspective. Note that, [31], [32], [33],
[34], [35], [36] obtained very interesting results to
solve some practical problems. We would like to
note that, [37], [38] obtained very investing results,
which are related to the construction of linear and
nonlinear models. Similar studies are found in recent
bibliography, [39], [40], [41], [42], [43], [44], [45],
[46], [47], [48], [49], [50], [51], [52], [53], [54]. In
the following sections, we compare the results
received here, with the existing results derived from
the recent literature.
3 Numerical Results
To compare the results obtained here with the recent
academic literature, let us consider defining the
numerical solution of the following simple
examples:
,10,)0(,1)0(,)()1())(1(
0
32
xyydssyaxayay x
(22)
the exact solution can be represented:
).exp()( xxy
By using that the right-hand side of the example
does not depend on
)(xy
, to find the solution of this
example one can use the following method:
,4,4/3,9/)44(2 111
2
12
pyyyhyyy nnnnnn
(23)
Which was applied to solving our example in the
case, when
1a
and
1a
. To compare the results
obtained, here have used the following method of
Numerov:
)24(.4,12/)10(2 211
2
12
pyyyhyyy nnnnnn
The results received by method (23) are presented
in table 1. Similarly, results received by method (2
are presented in table 2.
Table 1. Results received by the method (23) for
10;5;1;01.0;1
ha
Variab
le
x
1
1
5
5
10
10
0.20
2.04
2.16
2.53
3.26
8.86
1.34
E-12
E-12
E-08
E-08
E-07
E-06
0.60
1.7E-
11
2.05
E-11
3.17
E-08
1.17
E-08
1.79
E-04
6.22
E-06
1.00
4.1E-
11
5.53
E-11
2.86
E-06
1.97
E-07
2.2E-
02
1.84
E-05
Table 2. Results received by the method (24) for
10;5;1;01.0;1
ha
Variab
le
x
1
1
5
5
10
10
0.20
5.66
E-13
4.92
E-13
1.28
E-08
6.5E-
09
1.49
E-06
4.16
E-07
0.60
6.23
E-12
4.17
E-12
4.31
E-07
8.13
E-08
3.04
E-04
2.52
E-05
1.00
2.12
E-11
1.10
E-11
5.76
E-06
6.11
E-07
2.88
E-02
1.37
E-03
Here we have used several methods for solving
problems (1)-(2). The methods (23)-(24) have the
same degree and are stable. Comparing the above
results, we find that the hybrid methods are
promising.
4 Remark
Method (17) can be taken as more effective than the
others so the stable method of type (17) has the
degree
33 kp
. But in this case,
,0
i
the
maximum value for the stable method of type (17)
satisfies the condition
,22 kp
which has the
same as the degree for the stable method of type (5).
This means that the stable method of type (5) will be
implicit, but the stable method with the maximum
degree of type (17) will be explicit. Consequently,
methods of type (17) have some advantages. To
further explain this statement, let us consider the
following methods:
,5/10,24/)5145(2 111
2
12
nnnnnn yyyhyyy
(25)
.42/13,5655/)1323581323(
1740/)198719(2
111
2
12
2
12
nnn
nnnnnn
yyyh
yyyhyyy
(26)
By the simple comparison of these methods with the
known, receive that, the hybrid methods can be
compared with the Gauss method because they have
some similarities. For example, they can be taken as
symmetries. Note that hybrid methods are more
exact than the Gauss method. Methods (25) and (26)
are stable and have the
6p
and
8p
,
consecutively.
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5 Conclusion
In this research article, we have investigated some of
the generalizations of the Stёrmer -Verlet method
and considered the application of these to solve the
initial-value problem for ODE of the second order
with the special structure. From our research one can
receive many known equations such as Schrōdinger,
Sehturm-Liouville, and others. Moreover, it is a fact
there are several classes of methods constructed for
solving the initial value problem for ODEs of the
second order. Analytically, one of the effective
methods for solving the problem (1) is method (5),
as it is more accurate than the others. Based on our
findings, we have proven that one of the most
efficient methods for solving problems (2) and (15)
is the hybrid method of Stёrmer -Verlet type, which
can be received from (17) as the partial case.
By using the Dahlquist laws, method (5) is more
exact than the others. But here, we have shown that
the hybrid method is more exact than the method of
(5). By taking into account this, here for solving
these problems with a high order of accuracy, we
suggested investigating hybrid methods, which are
constructed by using the formula (17). In our
research, we have proved that if the method defined
by the formula (17) is stable and has a specific
degree of p, then the degree will satisfy the
necessary condition. Specifically, it was shown that
the hybrid methods are more exact than the others,
therefore these methods are perspective.
Furthermore, it comes naturally that one can prove
that the stable hybrid methods with the maximum
degree can be considered as optimal. Moreover,
since these methods are stable, they have maximum
accuracy and a simple structure. It is worth noticing
that in the application of these methods, researchers
usually use some of the existing methods for the
calculation of the values of the solution at the hybrid
points. In addition, we note that depending on the
properties of the solution of the investigated
problem, the methods with fractional step-size can
give results, which are worse than the results
obtained by the non-fractional methods.
Consenquately, it must be considered that it is not
difficult to obtain methods with the fractional step
size that are also a part of the class of hybrid
methods. This means that the results of this paper
have a significant advantage in solving many applied
physical problems including the above-mentioned
problems of Schrölinger, Sturm-Liouville, and
others.
Acknowledgment:
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We would like to thank academicians Telman Aliye
and Ali Abbas for their interest in our work. Lastly,
we would like to express our thanks and gratitude
to the Science Development Foundation under the
President of the Republic: of Azerbaijan - Grant No
EIF-MQM-ETS- 2020-1(35)-08/01/1-M-01 (for
Vagif Ibrahimov). Lastly, we are also grateful to all
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_US
We would like to express our thanks and gratitude
to the Science Development Foundation under the
President of the Republic: of Azerbaijan - Grant No
EIF-MQM-ETS- 2020-1(35)-08/01/1-M-01 (for
Vagif Ibrahimov)."
