About One Multistep Multiderivative Method of Predictor-Corrector

Type Constructed for Solving İnitial-Value Problem for ODE of Second

Order

M. N. IMANOVA1,2, V. R. IBRAHIMOV2,3

1Science Development Foundation of the Republic of Azerbaijan,

Baku AZ1025,

AZERBAIJAN

2Institute of Control System of the Ministry of Science and Education,

Baku Az1141,

AZERBAIJAN

3Computational Mathematics,

Baku State University,

Baku AZ1148,

AZERBAIJAN

Abstract: - Considering the wide application of the initial-value problem for Ordinary Differential Equations

second-order with a special structure, here for solving this problem constructed the special Multistep

Multiderivative Methods. Many scientists studied this problem , but the most distinguishing is the Ştörmer. To

solve this problem here is proposed to use the Multistep Second derivative Method with a special

structure. This method has been generalized by many authors, which is called as the linear Multistep

Multiderivative Methods with the constant coefficients. Many authors shave shown that the Multistep Second

derivative Method can be applied to solve the initial-value problem for ODEs of the first order. Euler himself

using his famous method discovered that, in his method when moving from one point to another local

truncation errors add up, the results of which reach a very large value. To solve this problem, he suggested

using more accurate methods. For this aim, Euler proposed calculating the next term in the Taylor series of the

solutions of the investigated problem. Developing this idea and papulation of the Multistep Multiderivative

Methods here to solve the named problem it is suggested to use MultistepThriedderivative Methods, taking into

account that methods of this type are more accurate. For the demonstration above, receiving results here have

constructed some concrete methods. Also by using some of Dahlquist’s and Ibrahimov’s results for Multistep

Methods with the maximum order of accuracy were compared. Proven that the MultistepThriedderivative

Methods are more accurate than the others. By using model problems have illustrated some results received

here.

Key-Words: - Initial-value problem for ODEs, Stability and Degree, Multistep Multiderivative Methods

(MMM), Local Truncation Error, Störmer Method, Multistep Secondderivative Methods

(MSM).

Received: March 16, 2024. Revised: August 19, 2024. Accepted: September 11, 2024. Published: October 9, 2024.

1 Introduction

As was noted above, here considering the

construction of more exact stable methods and

their application them to solve the initial-value

problem for the ODEs of the second order, which

can presented as follows:

.,')(',)(

)),('),(,()(

00000 Xxxyxyyxy

xyxyxFxy







(1)

As is known, Newton’s second law of motion

leads to systems of second-order differential

equations. Hence, the problem (1) was studied by

many specialists. The problem in fundamental form

was investigated by Ştörmer using the numerical

method, which is popular as a Störmer method.

Suppose that the problem (1) has the unique

continuous solution defined in the segment [

Xx ,

0

].

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The function

),,( zyxF

is defined in some closed

set which has the continuous partial derivatives to

the totality of arguments to some p, inclusively. As

was noted above, the aim of this work is to construct

some simple numerical method for finding the

values of the solution of problem (1) at the mesh

points defined as the

hNihxx ii 

0),,..,1,0(

1

is the step-size.

Note that usually the approximate values of the

solution at the point

i

x

are denotes by the

i

y

, but

the corresponding exact values by the

),..,1,0()( Nixy i

. For solving problem (1) have

constructed approximate methods, such as

analytical, analitico-numerical and numerical

methods. Among them, the most popular are

numerical methods, the application of which is

associated with the development of computer

technology. As is known the initial-value problem

for ODE of the highest order by using the method of

undetermined coefficients can be reduced to a

system of ODEs of the first order. In this case, to

solve this system one can apply the following

method:

.0;,..,1,0;'

00  





 k

ikini

k

iini kNnyhy



(2)

This method has been invastigated by many

authors, [1], [2], [3], [4], [5], [6], [7], [8], [9], [10],

[11], [12], [13], [14]. Noted that method (2)

fundamentally has investigated by Dahlquist (see for

example [3]). Some authors for solving the problem

(1), have used the following method:

,0;,..,2,1,0

;'

0 0

2

0



    







k

i

k

iiniini

k

iini

kNn

Fhyhy





(3)

which is generalized of the method (2) and the

Ştörmer method. Noted that the Ştörmer method in

our case can be presented as the followers, [1], [2],

[5], [6], [10], [15], [16], [17], [18], [19], [20], [21],

[22], [23], [24], [25], [26], [27], [28], [29]:

.0;,..,2,1,0;

0

2

0 





 k

ikini

k

iini kNnFhy



(4)

By taking into account that the method (4) is a

partial case of the method (3), one can consider the

method (4) as the some parts of the method (3).

Therefore one might think that if method (3) is

stable, then the Ştörmer method is also stable. We

will show later that this is not so. Here the goal

research is in the application of the Multistep

Thirdderivative Methods to solve problem (1), given

that these methods are more accurate.

§1. The Multistep Thirdderivative Methods and

its application to solve problem (1).

The Multistep Thirdderivative Methods in one

version can presented as follows:

.0;,..,1,0

;'

0 0

3

0

2

0









    











k

i

k

iini

k

iiniini

k

iini

kNn

ylhyhyhy





(5)

If applied this method to solve the problem (1),

then received the following:

.,..1,0;

'

0

3

0 0

2

0

kNnglh

Fhyhy

k

iini

k

i

k

iiniini

k

iini







 





 









(6)

Obviously, a method (6) is a partial case of a

method (5).

Here, function

dxyyxdFyyxg /)',,()',,( 

.

İn other words, the function of

)',,( yyxg

is defined

as the first full derivative from the

function

)',,( yyxF

. The methods, which

are pesented here, takes as given if are known of the

values of k and the coefficients

),..,1,0(,,, kiliiii 



. Therefore, let us consider

the definition of the values of the coefficients

),..,2,1,0(,,, kiliiii 



. To do this, it is proposed

to use the method of undetermined coefficients.

Typically the use of this method is accompanied by

the use of a function decomposition Taylor series. İn

our case one can suggest using the following Taylor

series:

)3,2,1,0(,0),(

)!(

...)(

!2

)()()(

1)(

)(

)2(

2

)1()()(















jhhOy

jp

h

xy

h

xhyxyhxy

jpp

x

jp

jjjj

,

(7)

here,

).()(

)0( xyxy 

By using Taylor series (7) in the following

equality

,0),())(

)()(')((

13

0

2

















hhOihxylh

ihxyhihxyhihxy

p

i

k

iiii



(8)

receive the following asymptotic equality:

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.0),()

!

.

)!1()!2()!3(

(...

)()

!2

()(')()(

1)(

)(

12

0

3

0 0 0

2



























  



  

hhOy

p

i

p

i

p

i

l

p

i

h

xy

i

ihxyihxy

pp

xi

p

i

p

i

p

k

ii

p

k

i

k

i

k

iiiiiii





(9)

Note that for the comparison of the Multistep

Methods usually are used the conceptions of the

degree and stability.

Definition1. The integer value p is called as the

degree for the method (6), if the asymptotic relation

(8) is held.

Definition2. The method (6) is called as stable, if

the roots of the polynomial

01

1

1...)(



 

k

k

,

located in the unit circle on the boundary of which

there are no multiple roots.

Noted that methods (2) and (3)

were investigated by many authors, [27], [28], [29],

[30], [31], [32], [33], [34].

As follows from the above mentioned the

conception of stability is defined by using the values

of the coefficients

),..,1,0( ki

i



. Therefore, one

can say that with the help of the selection of

coefficients

),..,2,1,0( ki

i



it is possible to

construct stable and instable methods of types (2)-

(5). We later will show that this is not always true

for the methods (3) and (4). The conception of

stability and degree very impotant for the

comparison of above mentioned Multistep Methods.

By using these conceptions, let us compare the

methods (2) and (3). As is known, stability is the

necessary and sufficient condition for the

convergence of the Multitsep Multiderivative

Methods (see for example [1]). Therefore let us

compare stable methods of types (2),(3), and (5).

Dahlquist proves that in a class of method (2) there

are stable methods with the degree

2]2/[2  kp

, if

0

k



and if

0

k



then

there are stable methods with the degree

kp 

and

there are stable methods with a degree

max

p

for all

the values of order k. And now let us investigate the

method (3). Let the method(3) have the degree of p,

in the case:

0... 01  



kk

and

0

k



.

Then in the class of method (3), there are stable

methods with the degree

22  kp

and for all the

k, there are stable methods with the

degree

22

max  kp

, if

0 kk



and method

(3) is stable then

kp 2

. In the case

),..,1,0(0 ki

i



, there is not stable method in

the class of (3). Therefore in this case, the

following definition of stability.

Definition3. Method (3) in the

case

),..,1,0(0 ki

i



called as the stable, if the

roots of the polynomial

)(



located in the unit

circle on the boundary of which there is not

multiply root, except double root

1



.

Let us suppous that method (3) is stable for the

case

),..,1,0(0 ki

i



, then in class (4), there

are methods with the degree

2]2/[2

max  kp

for

all the order of k. And now let us to consider

the investigation of a method (5). The maximum

order for the stable methods of type (5), one can

find by using the following theorems. İn first let us

consider the case, when

0... 01  



kk

and

0... 01  



kk

. In this case method (5)

is called stable if the roots of the polynomial located

in unit circle on the boundary of which, there is not

multiplied root in addition to the three-fold root

1



.

If the stable method from the above mentioned

class has the degree p, then in the receiving class

methods there are stable methods with the degree

2]2/[2

max  kp

. And now to consider the case,

when

0... 01  



kk

. Let us consider

the following theorem.

Theorem. If method (5) is stable, and has the

degree of p, then in the class of method (5), there

are methods with the degree

43  kp

. If

,0 kkk l



, then in the receiving class of

methods, there are stable methods with the degree

13  kp

.

As is known, one of the main problems in the

study of Multistep Methods is the ways of finding

the coefficients of the method (5), so in the next

paragraph we will consider finding the coefficients

of the method (5).

§2. About some ways for the definition of the

values of the coefficients in the method (5).

Let’s consider finding coefficients for the

construction a Multistep Thirdderivative Methods

having appropriate precision. For this aim from the

asymptotic equality of (9) one can receive the

following linear system of algebraic equations:

).0(

!)!1()!2()!3(

;..;

!2

)(;;0

00

123

0 0 0 0 0

2





























    





    

ki

k

i

p

k

ii

p

i

p

i

p

k

i

k

i

k

i

k

i

k

iiiiiii

p

i

p

i

p

i

l

p

i

ii





(10)

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The amount of the unknowns in this system of

(10) is equal to 4k+4 and the amount of the

equations is equal to p+1. By taking into account

that the system (10)-is the homogenues, therefore if

441  kp

, then it follows from here that the

solution of system (10) will be a unit for the case

p=4k+2. Noted that some of the authors suggested

replacing the system of (10) with the following, in

the case

),...1,0(0 kili

:





























p

l

m

c

0

1

,)1(ln)()(

;)1())(ln(

);1(2)1()1(2)1();1()1(;0)1(







(11)

here



 





 

1

0

00 0

,...2,1,0,)1)...(1(

!

1

,)(;)(;)(

iduiuuu

i

cl

k

i

k

i

k

i



Let us note that the system of algebraic

equations (10) and (11) are equivalent, so these

systems are the necessary and sufficient conditions

in order for these methods to have a degree p. As is

known, every necessary and sufficient condition can

be taken as a definition for some conception.

And now let us consider construction some

examples for the case k=3. In this case, by solving

the system (10) one can construct the following

methods:

.45360/)979285372272099(

27216/)4285

166592270710781(

3/)(

123

2

123

213

nnnn

n

nnn

nnnn

FFFFh

y

yyyh

yyyy





















(12)

The local trancation error for the method (12)

can be presented as follows:

.0),(156800/3 10)9(9  hhOyhR nn

As is known for the application of implicit

methods one can use the predictor-corrector

methods. For this aim usually are used implicit

method as the

corrector, but as the predictor methods are used

explicit method. And let us note that usually a

predictor method selected the explicit method with

the maximum degree. applyingthis theory to our

case, the following method can be taken as the

predictor method:

).(1680/

2/)271089(

2/)18910887(

2/)459702245(

10)9(9

1

2

21

213

hOyh

FFFh

yyyh

yyyy

n

nnn

nnnn

























(13)

It is known, that the predictor method can be

taken as unstable. Noted that there are numerous

works dedicated to the investigation of the methods

(2)-(5), [35], [36], [37], [38], [39], [40], [41], [42],

[43], [44].

In the application of methods (12) and (13) to

solve some problems arises nesserty calculation of

the values

21  nn yandy

, since

n

y

- can be

determined from the initial value condition. One can

be suggested for finding the value

1n

y

by using the

initial-values

n

y

and

n

y

. But in this case the rate of

approximation will be low. By using this, here is

recommended to use some sequence of methods, as

the simple algorithm.

It is obvious that

000 ,yandyy 

-are known.

Let us consider the following algorithm.

Step 1. Input

);;,,,( 0000 hyyyx 

Step 2. For

0:i

step 1 to N-1 do begin

;

3

2

3/2 iii yhyy 









;4/)),,(3

),,((

;4/)3(:

;2/)(:

3/23/23/2

1

3/21

3/23/2



































iii

iiiii

iiii

yyxF

yyxFhyy

yyhyy

print

),( 11  

ii yy

;

Step 3:

);,,(

3

2

;1: 3/2 jjjjj yyxhFyyij 









 

3/)(: 3232  





 jjjj yyhyy

(Trapezoidal

method);

;4/)),,(3),,((

;4/)3(:

3/23/23/21

3/21

























jjjjjjjj

jjjj

yyxFyyxFhyy

yyhyy

Print

;2;1:);,( 11 steptogoiiyy jj 



Step 4: Stop.

Here have used the Hybrid method with

a simple structure. The degree for this method can

be defined as the

3p

.

Note that these methods can be taken as the one-

step methods and can applied to the calculation of

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definite integrals, by the one and the same form.

For this aim, one can use the following method:

,12/)(512/)( 1

2

11  







 nnnnnn FFhyyhyy

(14)

here

).,..,2,1,0(),,,( NiyyxFF iiii 





Note that in using the method (14), the question

of calculating the first derivatives of the original

solutions also arises. How does it follow from here,

that in the application of method (14), it becomes

necessary to calculate of the

values

)1,( 

nnjyandyjj

. If

)0( 

jyj

are known, then by using method (14) one can

calculate the values of the solution of the problem

(1). Noted that method (14) also has the degree

4p

.

It is not difficult to understand that for the

receiving best results, one can use the above

proposed methods (12) and (13).

Each method has its advantages and

disadvantages, so here the proposed methods also

have some advantages and disadvantages. However,

depending on the problem being selected one of the

above-suggested methods. Method (6) is more

accurate and easily adapts to solve the problem (1).

Taking into account that method (6) is more

accurate, one can use the stable explicit methods as

type (6).

2 Numerical Methods

Let us consider the following model problem:

.)0(,1)0(),()( 2









 yyxyxy

(15)

The exact solution for which can be presented

as:

).exp()( xxy





For the simplicity to solve this problem, let us

use the following known method:

.12/)10(2 12

2

12 nnnnnn yyyhyyy 









 

(16)

The receiving results are tabulated in Table 1.

Table 1. Results for the step-size

01.0h

x

1



1



5



5



0.20

0.60

1.00

5.66E-13

6.23E-12

2.12E-11

4.92E-13

4.17E-12

1.10E-11

1.28E-08

4.31E-07

5.76E-06

6.5E-09

8.13E-08

6.11E-07

Now let us consider the case, when



is

constant, but

h

-receive different values.

Table 2. Results obtained for the case

1



and

.01.0;05.0;1.0h

Variable

x

Step size

1.0h

Step size

05.0h

Step size

01.0h

0.20

4.45E-09

3.8E-10

7.32E-13

0.60

5.29E-08

3.45E-09

5.68E-12

1.00

1.2E-07

7.51E-09

1.2E-11

In order to describe the properties of the

solutions of problem (16) in an accessible form

decided here, take into account the value of the

parameter



and the arguments

x

, which are

tabulated in Table 1 and Table 2.

Noted that, above-given method can be applied

to solve Volterra integral and Volterra

integrodifferential equations, [45], [46], [47], [48],

[49]

For this aim let us consider the following.

Example:

 

 

.1,0,)0(

;1)0(,)1())(1(1(

0

22







 

xy

ydssyaxyay x





(17)

The exact solution for this example:

).exp()( xxy





From the example 2 it follows the example 1 for

the case

.1a

Receiving results are tabulated in Table 3.

Table 3. Error for method (15) at

01.0;1  ha

x

1



1



5



5



0.20

0.60

1.00

2.04E-12

1.7E-12

4.1E-11

2.16E-12

2.05E-12

5.53E-11

2.53E-08

1.17E-08

2.86E-06

3.26E-08

1.17E-08

1.97E-07

The results are corresponding theoretical.

By results receive that method can be taken as

the normal.

3 Conclusion

There are some classes of methods constructed to

solve problems (1). This problem is usually

investigated in two forms. One of them investigates

problem (1) in the given form and the others

follows:

Xxxyxyyxyyxfy 









 00000 ,)(,)(),,(

.

(18)

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Solving as many problems are reduced to

solving the problem (18), therefore the problem (18)

is called the special presentation of the ODEs of

second order. Many known specialists have

investigated the problem (18). Have proven that

Multistep Secondderivative Method (3)-(6) can

apply to solve problem (1), with one and the same

success. But these methods have different

properties. The problem (18) is intersected with the

above-investigated problems. Therefore, by using

the above-presented methods one can solve problem

(1) and all the partial cases of this problem. For the

sake of objectivity, let us note that all the above-

noted methods can be applied to solve problems

(18). In this case, there is a need to calculate values

)1( 

mym

at every step. On the other side the

function

),( yxf

independent from the

)(xy

.

Therefore, there is no need for calculation

values

)0( 

iyi

.

Ştörmer by using this property, suggested for

the calculated the values

)0( 

iyi

of the solution

of problem (14), by using method (4). Note that the

definition of the conception of stability for method

(4) can use the definition3.

Let us note that method (14) has the degree of

p=4 and the method used in the algorithm, has the

degree p=3 and is stable. To increase the accuracy

of which one can use in the above-constructed

method is the half sum of the following methods:

,4/)3( 3/111  





 nnnn yyhyy

(19)

.4/)3(

ˆ3/21  





 nnnn yyhyy

(20)

Method (20) was used in the algorithm. By

simple comparison of the method (14) and half sum

of the method (19) and (20) receive that, the use of

methods (19) and (20) is preferable. One by using

the method of (6) can be construct the stable method

with the high order. By using the higher accuracy of

the method (12), one can recommend an application

to solve some problems by using the method (13) as

the predictor method. Method (20) and method (13)

are representatives of different classes of methods.

Methods (19) and (20) are usually called as the

hybrid or fractional step method. Here given one

way to construct numerical methods with the new

properties. We hope that this method described here

will find its followers.

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.

Finally, the authors thank the reviewers for their

comments, which improved the content of this

article.

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All authors, M. N. Imanova and V. R. Ibrahimov

contributed equally.

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Scientific Article or Scientific Article Itself

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interest.

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