Х А Б А Р Л А Р Ы

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ISSN 1991-346X (Print) ҚАЗАҚСТАН РЕСПУБЛИКАСЫ

ҰЛТТЫҚ ҒЫЛЫМ АКАДЕМИЯСЫНЫҢ

Х А Б А Р Л А Р Ы

ИЗВЕСТИЯ

НАЦИОНАЛЬНОЙ АКАДЕМИИ НАУК РЕСПУБЛИКИ КАЗАХСТАН

N E W S

OF THE NATIONAL ACADEMY OF SCIENCES OF THE REPUBLIC OF KAZAKHSTAN

ФИЗИКА-МАТЕМАТИКА СЕРИЯСЫ

 СЕРИЯ

ФИЗИКО-МАТЕМАТИЧЕСКАЯ



PHYSICO-MATHEMATICAL SERIES

5 (315)

ҚЫРКУЙЕК – ҚАЗАН 2017 Ж.

СЕНТЯБРЬ – ОКТЯБРЬ 2017 Г.

SEPTEMBER – OCTOBER 2017

1963 ЖЫЛДЫҢ ҚАҢТАР АЙЫНАН ШЫҒА БАСТАҒАН ИЗДАЕТСЯ С ЯНВАРЯ 1963 ГОДА

PUBLISHED SINCE JANUARY 1963 ЖЫЛЫНА 6 РЕТ ШЫҒАДЫ

ВЫХОДИТ 6 РАЗ В ГОД PUBLISHED 6 TIMES A YEAR

АЛМАТЫ, ҚР ҰҒА АЛМАТЫ, НАН РК ALMATY, NAS RK

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Известия Национальной академии наук Республики Казахстан

Б а с р е д а к т о р ы

ф.-м.ғ.д., проф., ҚР ҰҒА академигі Ғ.М. Мұтанов Р е д а к ц и я а л қ а с ы:

Жұмаділдаев А.С. проф., академик (Қазақстан) Кальменов Т.Ш. проф., академик (Қазақстан) Жантаев Ж.Ш. проф., корр.-мүшесі (Қазақстан) Өмірбаев У.У. проф. корр.-мүшесі (Қазақстан) Жүсіпов М.А. проф. (Қазақстан)

Жұмабаев Д.С. проф. (Қазақстан) Асанова А.Т. проф. (Қазақстан)

Бошкаев К.А. PhD докторы (Қазақстан) Сұраған Д. корр.-мүшесі (Қазақстан) Quevedo Hernando проф. (Мексика), Джунушалиев В.Д. проф. (Қырғыстан) Вишневский И.Н. проф., академик (Украина) Ковалев А.М. проф., академик (Украина) Михалевич А.А. проф., академик (Белорус) Пашаев А. проф., академик (Əзірбайжан)

Такибаев Н.Ж. проф., академик (Қазақстан), бас ред. орынбасары Тигиняну И. проф., академик (Молдова)

«ҚР ҰҒА Хабарлары. Физика-математикалық сериясы».

ISSN 2518-1726 (Online), ISSN 1991-346X (Print)

Меншіктенуші: «Қазақстан Республикасының Ұлттық ғылым академиясы» РҚБ (Алматы қ.)

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Г л а в н ы й р е д а к т о р

д.ф.-м.н., проф. академик НАН РК Г.М. Мутанов Р е д а к ц и о н н а я к о л л е г и я:

Джумадильдаев А.С. проф., академик (Казахстан) Кальменов Т.Ш. проф., академик (Казахстан) Жантаев Ж.Ш. проф., чл.-корр. (Казахстан) Умирбаев У.У. проф. чл.-корр. (Казахстан) Жусупов М.А. проф. (Казахстан)

Джумабаев Д.С. проф. (Казахстан) Асанова А.Т. проф. (Казахстан) Бошкаев К.А. доктор PhD (Казахстан) Сураган Д. чл.-корр. (Казахстан) Quevedo Hernando проф. (Мексика), Джунушалиев В.Д. проф. (Кыргызстан) Вишневский И.Н. проф., академик (Украина) Ковалев А.М. проф., академик (Украина) Михалевич А.А. проф., академик (Беларусь) Пашаев А. проф., академик (Азербайджан)

Такибаев Н.Ж. проф., академик (Казахстан), зам. гл. ред.

Тигиняну И. проф., академик (Молдова)

«Известия НАН РК. Серия физико-математическая».

Собственник: РОО «Национальная академия наук Республики Казахстан» (г. Алматы)

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Известия Национальной академии наук Республики Казахстан E d i t o r i n c h i e f

doctor of physics and mathematics, professor, academician of NAS RK G.М. Mutanov E d i t o r i a l b o a r d:

Dzhumadildayev А.S. prof., academician (Kazakhstan) Kalmenov Т.Sh. prof., academician (Kazakhstan) Zhantayev Zh.Sh. prof., corr. member. (Kazakhstan) Umirbayev U.U. prof. corr. member. (Kazakhstan) Zhusupov М.А. prof. (Kazakhstan)

Dzhumabayev D.S. prof. (Kazakhstan) Asanova А.Т. prof. (Kazakhstan) Boshkayev K.А. PhD (Kazakhstan) Suragan D. corr. member. (Kazakhstan) Quevedo Hernando prof. (Mexico), Dzhunushaliyev V.D. prof. (Kyrgyzstan) Vishnevskyi I.N. prof., academician (Ukraine) Kovalev А.М. prof., academician (Ukraine) Mikhalevich А.А. prof., academician (Belarus) Pashayev А. prof., academician (Azerbaijan)

Takibayev N.Zh. prof., academician (Kazakhstan), deputy editor in chief.

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News of the National Academy of Sciences of the Republic of Kazakhstan. Physical-mathematical series.

Owner: RPA "National Academy of Sciences of the Republic of Kazakhstan" (Almaty)

The certificate of registration of a periodic printed publication in the Committee of information and archives of the Ministry of culture and information of the Republic of Kazakhstan N 5543-Ж, issued 01.06.2006

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N E W S

OF THE NATIONAL ACADEMY OF SCIENCES OF THE REPUBLIC OF KAZAKHSTAN PHYSICO-MATHEMATICAL SERIES

ISSN 1991-346Х

Volume 5, Number 315 (2017), 5 – 12

A.A. Kulzhumiyeva¹, Zh.A. Sartabanov²

1M. Utemisov West-Kazakhstan State University, Uralsk, Kazakhstan;

2K. Zhubanov Aktobe Regional State University, Aktobe, Kazakhstan E-mail: [email protected], [email protected]

REDUCTION OF LINEAR HOMOGENEOUS D

_e

-SYSTEMS TO THE JORDAN CANONICAL FORM

Abstract. In this note we prove a theorem about reducibility to the canonical form of a linear homogeneous system with differentiation operator on diagonal and multiperiodic matrix constant on the diagonal. On the basis of the results obtained, it is possible to find out the structure of the solutions and investigate the conditions of the existence and uniqueness of the (, , ) - periodic solution of the linear D_e-system of equations. When investigating periodic solutions of linear systems of first order partial differential equations, it becomes necessary to reduce matrices with variable elements to convenient form. In this connection, we note the results of [1-2] and commentaries on them in monographs [3-5]. It is known that the study of the problems of multiperiodic solutions of systems of first order partial De -equations with the same principal part originates in works [6-7]. On their basis, further qualitative studies have been continued in [8-11].

Key words: linear homogeneous system, differentiation operator, Jordan canonical form, multiperiodic matrix, main diagonal, vector-period.

The article is devoted investigation of reduction of a linear De-system of the form

x A x

D

_e

 (  )

(1)

with the differential operator

e t D

_e



 



  ,



to the canonical form

x J x

D

_e

 (  )

, (1_*)

where

  (  ,   )  R

,

t  ( t

₁

, ..., t

_m

)  R  ...  R  R

^m^,





 







 



t

m

t t , ...,

1

is a vector operator,

e  ( 1 , ..., 1 )

–

m

-vector,

,

denotes the scalar product,

  t  e 

,

A (  )

an

n  n

^-

matrix, which satisfies condition

) ( )

( )

( k A C

⁽^e⁾

R

^m

A      

_ ,

 k  Z

^m (2) with multiple vector-periods

k   ( k

₁



₁

, ..., k

_m



_m

)

,

  ( 

₁

, ..., 

_m

)

,

k  ( k

₁

, ..., k

_m

)

from the set of integer vectors

Z

^m.

J (  )

an

n  n

-matrix of the Jordan form possessing the properties of multiperiodicity with the same



period and smoothness

e

in

  R

^m^:

) ( )

( )

( k J C

⁽^e⁾

R

^m

J      

_ ,

 k  Z

^m. (2_*) Variable matrices

A (  )

and

J (  )

are called constants on the diagonal

t  e 

.

Let



_j

(  )

be eigenvalues of the matrix

A (  )

of multiplicity

k

_j,

j  1 , s

, possessing the following properties.

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1⁰. Continuous differentiability:



_j

(  )  C

_⁽^e⁾

( R

^m

)

^,

j  1 , n

^.

2⁰. Periodicity with period

  ( 

₁

, ..., 

_m

)

:



_j

(   k  )  

_j

(  )

,

j  1 , n

,

  R

^m^,

Z

m

k 

.

3⁰. Property of having fixed sign



_j

(  )

for each

j  1 , n

: a)



_j

(  )  0

^,

   R

^m^or

b)



_j

(  )  0

,

   R

^m^or

c)



_j

(  )  0

^,

   R

^m^.

4⁰. Separation of eigenvalues:

a) for

j  l 

_j

(  )  

_l

(  )

^,

   R

^m^or

b) for

j  l 

_j

(  )  

_l

(  )

,

   R

^m^,

i.e. for each value

j

the eigenvalue



_j

(  )

has constant multiplicity

k

_j

 const

^{for all}

R

m

 

^.

5⁰. Each of the sets

Re  

_j

(  ) 

^and

Im  

_j

(  ) 

has properties 1⁰-4⁰. The properties 1⁰-5⁰ are briefly called



-properties of the matrix

A (  )

.

It is obvious that characteristic matrix

 E  A (  )  H (  ,  )

has for all

  R

^m constant rank

n

and its invariant



-polynomials

i

₁

(  ,  )

, …,

i

_n

(  ,  )

such that, starting with the second, they are a divisor of the previous one,

i

₁

(  ,  )

, …,

i

_r

(  ,  )

are polynomials of degree greater than zero with respect to



and

1 ) , ( ...

) ,

1

(   



 

_n

 

r

i

.

Then the characteristic matrix

H (  ,  )

is represented by relations

 ( , ), ..., ( , ), 1 , ..., 1  ( , ) )

, ( ) ,

(   P   diag i

₁

  i   Q  

H 

_r , (3)

where

P (  ,  )

and

Q (  ,  )

are non-singular

n  n

-matrices that



- polynomials are with independent of



determinants

det P (  ,  )  p (  )  0

and

det Q (  ,  )  q (  )  0

.

Companion matrices of invariant polynomials

) ( ...

) ( )

,

(   

ⁿ^j



_j₁

 

ⁿ^j ¹



_jn_j



i

j

 

^

 

,

j  1 , r

,

n

₁

 ...  n

_r

 n

denote by



 









 











( ) ( ) ... ( )

) (

1 ...

0 0

0

...

0 ...

1 0

0

0 ...

0 1

0

) (

1 2

1

*



















j jn

jn jn

j

j j

j

A

^,

j  1 , r

^{. (4)}

It is obvious that the representation (3) can be obtained on the basis of elementary transformations known from theory of



-matrices [12] under which properties of multiperiodicity and continuous

(7)

differentiability in



for matrices participating in relation (3) are preserved. Consequently, these properties also hold for the matrices (4).

In [13], the condition of equivalence of matrix

 E  A (  )

to the matrix with one invariant



^-

polynomials are established and theorem of reducibility of the matrix

A (  )

to the Jordan normal form by multiperiodic continuously differentiable non-singular transformation matrix is proved.

Moreover, system (1) was equivalent to one equation with higher order

D

_e operator with companion matrix of the form (4).

In this article we raise the question about investigating the reducibility of system (1) to the

D

_e- system with the matrix of Jordan normal form, when the matrix (2) with several invariant polynomials satisfies the conditions 1⁰-5⁰.

In other words, in [13] we consider

D

_e-system, which is equivalent to one

D

_e-equation of order

n

, and in this case, by (3), our system (1) breaks up into

r

linear

D

_e-equations of orders

n

₁, …,

n

_r

 n

₁

 n

₂

 ...  n

_r

 n 

. The essence of the problem is to reduce this general

D

_e-system (1) to

D

_e^-

system with matrix

J (  )

of the Jordan canonical normal form, where



-properties of matrix

A (  )

are essential value.

When raising the question, it is obvious that this study is adjacent to the studies [14-19].

To solve the problem posed, we use the true normal form

A

^*

(  )

of matrix

A (  )

, which are related by a similarity relation

) ( ) ( ) ( )

(

¹

*

 L  A  L 

A 

^ . (5)

The relation (5) to be a result of the representation (3), where

 ( ), ..., ( ) 

)

(

₁^* ^*

*

 diag A  A

_r



A 

with diagonal elements of the form (4),

L (  )

is a non-singular continuously differentiable



-periodic matrix:

) ( )

( )

( k L C

⁽^e⁾

R

^m

L      

_ ,

 k  Z

^m. (6) Relations (5)-(6), as well as (3)-(4) are obtained on the basis of methods of the theory of equivalent transformations of polynomial matrices for which smoothness and multiperiodicity of the matrices are saved.

Further, in view of (5), (6) and the change

z L

x  (  )

^,

det L (  )  0

,

L (   k  )  L (  )

^,

k  Z

^m (7) system (1) is reducible to the system

z A z

D

_e



^*

(  )

, (8)

which is equivalent to the system of subsystems

j j j

e

z A z

D 

^*

(  )

, (8j) where

A

^*_j

(  )

has the form (4),

j  1 , r

,

z  ( z

₁

, ..., z

_r

)

.

In the case of the known elementary divisors of matrix

A (  )

the system (1), and, consequently, the system (8) can be reduced to an even simpler form.

Indeed, in view of (2) and



-properties of the matrix

A (  )

, we have full information about its eigenvalues. Hence, it exists a non-singular, really smooth



-periodic matrix of the transformation

)

~ ( 

L

such that

)

~ ( ) ( )

~ ( )

~ (  L

₁

 A  L 

A 

^ , (

5 ~

)

(8)

where

A ~ (  )  diag  A ~

1

(  ), ..., A ~

_l

(  ) 

is the second true form of matrix

A (  )

,

A ~ (  )

have the form (4), in which the coefficients of the degree are non-zero elements of the last row

 ( )  ( )

1 2

( )

²

... ( )

1

      









j

j n jn

j n

j n n

j

    



^ ^ ^,

which are an elementary divisor of the characteristic matrix (3). We write the properties of matrix

L ~ (  )

in the form

) ( )

~ ( )

~ (

₍_e₎ _m

R C L

k

L      

_ ,

k  Z

^m,

L ~ (  )  0

. (

~ 6

) Here, the eigenvalues



_j

(  )

,

j  1 , n

are assumed to be real-valued.

Then, by the relations (

5 ~

), (

~ 6

) and the change

z L

x  ~ (  ) ~

(

7 ~

) system (1) can be represented in the form

z A z

D

_e

~  ~ (  ) ~

, (

~ 8

) which consists from

l

subsystems



A  z

z

D

_e

~  ~ ( ) ~

, (

~ 8

_

) where

  1 , l

^,

~ z



 z ~



, ..., ~ z

n_





1 ,

n

₁

 ...  n

_l

 n

,

~ ( ~ , ..., ~ )

1

z

l

z

z 

.

Next, we should consider the reduction of system (1) to system with Jordan canonical form.

In the case of simple roots of matrices

A

^*_j

(  )

:

) ( )

(   



_ji



_jk ^,

  R

^m^,

( i  k )

of the characteristic equation

 ( )  0

det  E  A

_j

 

,

i , k  1 , n

_j,

j  1 , r

^,

n

₁

 ...  n

_r

 n

it is not difficult to verify that the Vandermonde matrix of the form



 









 











( ) ( ) ... ( )

...

) ( ...

) ( )

(

) ( ...

) ( )

(

1 ...

1 1

) (

1 1

2 1

1

2 2

1

2 1







































j j j

j

j j

n jn n

j n

j

jn j

j

jn j

j

B

j

satisfies the matrix equation

) ( ) ( )

( )

( 

_j



_j



_j



j

B B J

A 

^,

where

(  )  

1

(  ), ...,  (  ) 

jnj

j

diag

J 

and also

 ( ) ( )  0

) ( det

1





 





i k

n ji jk

j

B     

.

Consequently, in this case the system (8j) under conditions (2) and 1⁰-5⁰ is reducible to the Jordan canonical form

j j

j

e

y J y

D  (  )

⁽

9

_j) by non-singular linear transformation

j j

j

B y

z  (  )

^,

det B

_j

(  )  0

,

B

_j

(   k  )  B

_j

(  )  C

_⁽^e⁾

( R

^m

)

^,

k  Z

^m, (

8

^*_j)

(9)

where

j  1 , r

,

  R

^m^.

Then for

  R

^m the transformation

y B

z  (  )

,

det B (  )  0

,

B (   k  )  B (  )  C

_⁽^e⁾

( R

^m

)

,

k  Z

^m, (

8

^*) leads system (8) to the

D

_e-system of Jordan canonical form

y J y

D

_e

 (  )

, (9)

where

B (  )  diag  B

₁

(  ), ..., B

_r

(  ) 

,

J (  )  diag  J

₁

(  ), ..., J

_r

(  ) 

,

y  ( y

₁

, ..., y

_r

)

. In the case of multiple elementary divisors of matrix

A (  )

will be necessary to use its second normal form

A ~ (  )

from system (

8 ~

) and its subsystems (

8 ~

_

) with matrices

A ~

_

(  )

.

To reduce the matrix

A ~

_

(  )

to the Jordan normal form

J

_

 

_

E

_

 I

_ with identity matrix

E

 and first off-diagonal oblique range

I

_ it is necessary to construct matrix

T

_

(  )

with elements

 



 









 















, ,

) (

, ,

) ( )

(

1 1 1 1

1 ) 1

(

i j b C

i j C

t

_i

k j i k ii

i j k

k i j i

ij

 









 

where

C

_i^j is total number of combinations of

i

in a total of

j

^.

The reader will have no difficulty in verifying that [20]

) ( ) ( ) ( )

( 

_



_



_





T T J

A 

and also

det T

_

(  )  1

. Then the change



T  y

z ( ) ~

~ 

leads the system (

~ 8

_

) to the system



J  y

y

D

_e

~  ( ) ~

with a Jordan cage

J

_

 

_

E

_

 I

_^.

Consequently, the change

y T z ( )

~  

⁽

~ 8

^*

)

system (8) leads to the system (9) of the Jordan normal form, where

T (  )  diag  T

₁

(  ), ..., T

_l

(  ) 

is non-singular



-periodic, smooth transformation matrix.

In the case of complex eigenvalues, as can be seen from structures of matrices

T

_

(  )

and

J

_

(  )

, matrices

T (  )

and

J (  )

are complex-valued. In view the condition 5⁰ its real and imaginary parts are distinguished without any special difficulties for all

  R

^m^.

Thus, by transformations (6)-(

~ 6

), (7)- (

7 ~

) and (

8

^*)-(

8 ~

^*

) non-singular linear change

y L

x 

^*

(  )

(

1

^*)

leads the

D

_e-system (1) to the

D

_e-system (9) with Jordan matrix

J (  )

. The matrix

L

^*

(  )

is transformation matrix

L

^*

(  )  L (  ) B (  )

and it has properties

(10)

0 ) (

det L

^*

 

^,

L

^*

(   k  )  L

^*

(  )

,

k  Z

^m. (

1

^*^*) We call system (9) the Jordan canonical

D

_e-system of system (1).

We formulate the main result in the form of the following theorem.

Theorem. Let the matrix

A (  )

possessing the property (2) has eigenvalues



_j

(  )

,

j  1 , n

^,

satisfying the conditions 1⁰-5⁰. Then the system (1) can be reduced to the Jordan canonical

D

_e^-system

(9) by linear transformation (

1

^*)-(

1

^*^*).

As an application of the theorem proved, we consider

D

_e-system of triangular type

 







, ) ( )

(

, ) (

22 21

11

y A

x A

y D

x A

x D

e e



(10)

where

x

is

n

₁-vector-function,

y

is

n

₂-vector-function,

A

₁₁

(  )

^,

A

₂₁

(  )

^and

A

₂₂

(  )

^are

multiperiodic with



-vector-period, smooth in

  R

^m matrices of order

n

₁

 n

₁,

n

₂

 n

₂,

1 2

21

n n

n  

.

We suppose that the block matrix

 

 



 

) ( )

( ) ) (

(

22 21

11



 

A A

O

A A

(11)

satisfies the condition

) ( )

( )

( k A C

⁽^e⁾

R

^m

A      

_ ,

k  Z

^m (

11

^*)

where

O

is zero block. The diagonal blocks

A

₁₁

(  )

^and

A

₂₂

(  )

^have



-properties, therefore, these blocks have Jordan forms

) ( ) ( ) ( )

( 

_j¹



_jj



_j



j

L A L

J 

^ ^,

j  1 , 2

(

12

^*) with non-singular



-periodic and smooth matrices

) ( )

( )

(

_j ⁽^e⁾ ^m

j

k L C R

L      

_ ^,

det L

_j

(  )  0

,

k  Z

^m,

j  1 , 2

. (

12

^*^*) Then by theorem linear non-singular



-periodic, smooth in

  R

^m transformation of form

 





v L y

u L x

) (

, ) (

2 1



(12)

leads system (10) to a linear system

 







v J

u B v D

u J u D

e e

) ( )

(

, ) (

2 1



(13)

with diagonal blocks

J

₁

(  )

and

J

₂

(  )

of the Jordan canonical form, where

) ( ) ( ) ( )

(  L

₂¹

 A

₂₁

 L

₁

 B 

^

is smooth,



-periodic in

  R

^m

n

₂

 n

₁-matrix.

It is obvious that the system (13) has more convenient form in comparison with the system (10) for integration and qualitative investigation.

The system of form (13) can be called the semi-canonical form of the triangular system (10).

Thus, we can give the following corollary to theorem proved.

(11)

Corollary. Let triangular matrix (11) satisfying the condition (

11

^*) has



-properties. Then the system (10) by transformation (12)- (

12

^*)-(

12

^*^*) is reduced to the semi-canonical

D

_e-system (13).

In conclusion, we note that the problem posed of studies we have used the methods of [20].

REFERENCES

[1] Sibuya Y. (1965) Some Global Properties of Matrices of Functions of One Variable // Math. Annal. № 161. P.67-77.

[2] Sibuya Y. (1962) Formal Solutions of a Linear Ordinary Differential Equation of the

n

-th Order at a Turning Point //

Funkcial. Ekvas. № 4. P.115-139.

[3] Vazov V. (1968) Asymptotic decomposition of solutions of ordinary differential equations. M.: Mir. (in Russ.)

[4] Samoilenko A.M. (1987) The elements of mathematical theory of multifrequency oscillations. Invariant tors. M.: Nauka.

(in Russ.)

[5] Lappo-Danilevskyi I.A. (1957) Using functions from matrix to the theory of linear systems of ordinary differential equations. M.: GITTL. (in Russ.)

[6] Kharasahal V.H. (1970) Almost periodic solutions of ordinary differential equations. Alma-Ata: Nauka. (in Russ.) [7] Umbetzhanov D.U. (1979) Almost multiperiodic solutions of partial differential equations. Alma-Ata: Nauka. (in Russ.) [8] Sartabanov Zh.A. (1989) About single method of studying periodic solutions of equations in partial derivatives of special form // News. Physico-mathematical series. № 1. P.42-48. (in Russ.)

[9] Sartabanov Zh.A. (2004) The condition of periodicity solutions of differential systems with multivariate time // News.

Physico-mathematical series. № 5. P.44-48. (in Russ.)

[10] Kulzhumiyeva A.A., Sartabanov Zh.A. (2007) Periodic in multivariate time of solutions of system equations with differential operator according to the direction of vector field // Eurasian Mathem. Journal. № 1. - P. 62-72. (in Russ.)

[11] Kulzhumiyeva A.A. (2008) Research of periodic solutions lead to canonic form of systems with linear differential operator in multivariate time // Eurasian Mathem. Journal. № 2. - P. 69-73. (in Russ.)

[12] Gantmaher F.R. (1966) Matrix theory. M.: Nauka. (in Russ.)

[13] Kulzhumiyeva A.A., Sartabanov Zh.A. (2016) On reducibility of linear De-system with constant coefficients on the diagonal to D_e-system with Jordan matrix in the case of equivalence of its higher order one equation // Bulletin of the Karaganda university. Mathematics series. №4(84). P. 88-93. (in Russ.)

[14] Kulzhumiyeva A.A., Sartabanov Zh.A. (2007) Periodic with variable period solutions of system of differential equations of multivariate time // Mathematical journal. t.7. № 2(24). - P.52-57. (in Russ.)

[15] Kulzhumiyeva A.A., Sartabanov Zh.A. (2007) To the question of periodic solutions in multivariate time of system D- equations // // Bulletin of the Orenburg university. № 3. - P.155-157. (in Russ.)

[16] Kulzhumiyeva A.A., Sartabanov Zh.A. (2007) Periodic with multivariate time solutions of system of the quasi-linear differential equations in partial derivative // International Conference «Analysis and Singularities», dedicated to 70th anniversary of V.I. Arnold. Moscow. P.156-158.

[17] Kulzhumiyeva A.A., Sartabanov Zh.A. (2009) Oscillations in quasi-linear system with operator of the differentiation on diagonals of multivariate time // International Conference «Modern problems of mathematics, mechanics and their applications» dedicated to the 70-th anniversary of rector of MSU academic V.A. Sadovnichy. Moscow. P.203.

[18] Muhambetova B.Zh., Sartabanov Zh.A., Kulzhumiyeva A.A. (2015) Multiperiodic solutions of systems of equations with one quasi-linear differential operator in partial derivatives of the first order // Bulletin of the Karaganda university.

Mathematics series. № 2(78). P. 112-117. (in Russ.)

[19] Kulzhumiyeva A.A., Sartabanov Zh.A. (2017) On multiperiodic integrals of a linear system with the differentiation operator in the direction of the main diagonal in the space of independent variables // Eurasian Mathematical Journal. № 1. v. 8.

P. 67-75.

[20] Kulzhumiyeva A.A., Sartabanov Zh.A. (2013). Periodic solutions of system of differential equations with multivariate time. Uralsk: RIC WKSU. (in Russ.)

А.А. Кульжумиева¹, Ж.А. Сартабанов²

1М. Өтемісов атындағы Батыс-Қазақстан мемлекеттік университеті, Орал, Қазақстан

2Қ.Жұбанов атындағы Ақтөбе өңірлік мемлекеттік университеті, Aқтөбе, Қазақстан

СЫЗЫҚТЫ БІРТЕКТІ De -ЖҮЙЕЛЕРДІ ЖОРДАНДЫҚ КАНОНДЫҚ ТҮРГЕ КЕЛТІРУ

Аннотация. Мақалада көп периодты тұрақты матрицамен жəне диагональ бойынша дифференциалдау операторымен сызықты біртекті жүйенің канондық түрге келтірілуі жөнінде теорема дəлелденген. Алынған нəтижелер негізінде De-сызықты теңдеулер жүйесінің (, , )-периодты шешімінің бар жəне жалғыз болуының шартын зертеп жəне шешімнің құрылымын анықтауға болады. Бірінші ретті дербес туындылы теңдеулердің сызықты жүйелерінің периодты шешімдерін зерттеу кезінде айнымалы элементті матрица-

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ларды ыңғайлы түрге келтірілу қажеттілігі туындайды. Осы байланыста [1-2] жұмыстарының нəтижелерін жəне [3-5] монографияларында оларға түсіндірмелерді ескереміз. Негізгі бөлімі бірдей бірінші ретті дербес туындылы De-теңдеулер жүйесінің көп периодты шешімдерінің сұрақтарын зерттеу [6-7] енбектерінен бастау алатыны белгілі. Олардың негізінде кейбір əрі қарай сапалы зерттеулері [8-11] жұмыстарында жалғастырған.

Кілт сөздер: сызықты біртекті жүйе, дифференциалдық оператор, жордандық канондық түрі, көп периодты матрица, негізгі диагональ, вектор-период.

УДК 35B10

А.А. Кульжумиева¹, Ж.А. Сартабанов²

1Западно-Казахстанский государственный университет им. М. Утемисова, Уральск, Казахстан;

2Актюбинский региональный государственный университет им. К. Жубанова,Актобе, Казахстан ПРИВЕДЕНИЕ ЛИНЕЙНЫХ ОДНОРОДНЫХ De -СИСТЕМ

К ЖОРДАНОВОМУ КАНОНИЧЕСКОМУ ВИДУ

Аннотация. В заметке доказана теорема о приводимости к каноническому виду линейной однородной системы с оператором дифференцирования по диагонали и многопериодической матрицей постоянной на диагонали. На основе полученных результатов можно выяснить структуры решений и исследовать условия существования и единственности (, , ) -периодического решения линейной D_e-системы уравнений. При исследовании периодических решений линейных систем уравнений в частных производных первого порядка возникает необходимость приведения матриц с переменными элементами к удобному виду. В этой связи отметим результаты работ [1-2] и комментарии к ним в монографиях [3-5]. Известно, что исследование вопросов многопериодических решений систем De-уравнений в частных производных первого порядка с одинаковой главной частью берет свое начало в трудах [6-7]. На их основе дальнейшие некоторые качественные исследования продолжены в работах [8-11].

Ключевые слова: линейная однородная система, дифференциальный оператор, жордановый канони-ческий вид, многопериодическая матрица, главная диагональ, вектор-период.

Сведения об авторах:

Кульжумиева Айман Амангельдиевна - кандидат физико-математических наук, Западно-Казахстанский государст- венный университет им. М. Утемисова, [email protected];

Сартабанов Жайшылык Алмаганбетович - доктор физико-математических наук, профессор, Актюбинский региональный государственный университет им. К. Жубанова, [email protected]

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N E W S

OF THE NATIONAL ACADEMY OF SCIENCES OF THE REPUBLIC OF KAZAKHSTAN PHYSICO-MATHEMATICAL SERIES

ISSN 1991-346Х

Volume 5, Number 315 (2017), 13 – 21 UDC 53, 532.133, 621.3.018.72.025.1

N.S. Saidullayeva, K.A Kabylbekov, Kh.A. Ashirbaev, A.O. Kalikulova, D.T. Pazylova South Kazakhstan State University named after M. Auezov, Shymkent

ORGANIZATION OF COMPUTER LAB WORK

"CALCULATION AND VISUALIZATION OF FORCED OSCILLATIONS IN THE PRESENCE OF AN EXTERNAL FORCE" WITH THE USE

OF THE SOFTWARE PACKAGE MATLAB

Abstract. The organization of computer lab work "Calculation and visualization of forced oscillations in the presence of an external force" is proposed with the use of the Matlab software package: a) external force - constant;

b) external force - F = F0 e; c) external force - F = F0e cosβt; d) F = 0 for t <0; F = F0 t / T for 0 <t <T;, F = F0 for t> T. For each of these cases, solutions, calculation programs and visualizations are presented. The results are presented in the form of graphs of the dependence of the acting external force on time and the displacement of the particle from the equilibrium position from time.

Key words: external force, damped oscillation, calculation, visualization, graph.

The capabilities of the Matlab system are huge, and in terms of the speed of tasks, it is ahead of many other similar systems. All these features make the MATLAB system very attractive for use in the educational process in higher education institutions [1].

One of the difficult tasks of introducing the results of the use of information technologies in educational institutions is the insufficient practical ability of teachers to use computer models of physical phenomena to organize computer lab work.

Revitalization, motivation and, ultimately, the effectiveness of training largely depend on the organization of computer laboratory works. We have previously written about the creation and use of models of the organization for the performance of computer laboratory work on the study of various physical phenomena in the educational process [2-20].

This article gives an example of the use of the Matlab system in organizing the computer lab work

"Calculation and visualization of forced oscillations in the presence of an external force" for the performance by the students.

Theme of laboratory work No. 1: Calculation and visualization of forced oscillations in the presence of an external force: Determine the forced oscillations of the system under the influence of the external force F (t), if at the initial instant t = 0 the system is at rest in the equilibrium position (x = 0, x = 0 ) For the following cases:

a) F = const = F0. The system oscillates according to the law



sin



.

2 t t

m

x a  

 ^



Calculation and visualization program

>> w=1;

>> a=2;

>> m=0.1;

>> t=0:0.1:30;

(14)

>> x=a./(m*w.^2).*(w.*t-sin(w.*t));

>> plot(t,x,'k-')

>> grid on

Fig.1. The oscillation of the system under the influence of a constant force

The action of a constant force leads to a shift in the equilibrium position around which oscillations occur.

b) F=F₀ e^^at .

The system oscillates according to the law

 

^

 



  

  e^ t t

a m

x F ^at



 



²⁰ ² ^cos ^sin

>> f0=2; w0=1;

>> m=0.1;

>>f0=2; w0=1;

>>m=0.1;

>> t=0:0.1:4;

Fig.2. The force acting on the system

0 5 10 15 20 25 30

0 100 200 300 400 500 600 700

t, s

X, m

X=F(t)

0 0.5 1 1.5 2 2.5 3 3.5 4

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

t, s

f, N

f=F(t)

(15)

>> t=0:0.1:30;

>> a=2;

>> x=(f0./(m*(w0.^2+a.^2))).*(exp(-a.*t)-cos(w.*t)+a.*sin(w.*t)./w);

>> plot(t,x,'k-')

>> grid on

>> xlabel('t, s')

>> ylabel('X, m')

>> title('X=F(t)')

Fig. 3. Oscillation of the system under the influence of the force F = F0e.

с) F=F₀e^^atcosβt. The oscillation of the system under the action of such a force occurs according to the law

 

    

 

t e

  

t t

 

m t x F

at      







 









 







sin 2 cos sin

4 cos

2 2 2 2

2

2 2 2 2 2

2 2 2

0









 



 



>> f0=2; w0=1;

>>m=0.1;

>> t=0:0.1:4;

>> a=2;

>>b=0.1;

>> f=f0.*exp(-a.*t).*cos(b.*t);

>> plot(t,f,’k-‘)

>> grid

0 5 10 15 20 25 30

-10 -8 -6 -4 -2 0 2 4 6 8 10

t, s

X, m

X=F(t)

(16)

Fig.4. The force acting on the system is F = F0e cos βt.

>> t=0:0.1:30;

>> a=2;

>> b=0.1;

>> A=f0./(m.*((w.^2+a.^2-b.^2).^2+4.*a.*b));

>> B=(w.^2+a.^2-b.^2).^2;

>> C=a.*(w.^2+a.^2-+-b.^2).^2;

>> x=A.*(-B).*cos(w.*t)+C./w.*(sin(w.*t)+exp(-a.*t)).*(B.*cos(w.*t)-2.*b.*a.*sin(w.*t));

>> plot(t,x,’k-‘)

>> grid on

Fig.5. The oscillation of the system under the influence of force F = F0e cos βt.

d) Determine the oscillation of the system after the action of an external force varying according to the law F = 0 for t <0, F = F0 t / T for 0 <t <T, F = F0 for t> T (Fig. 6); Up to the instant t = 0 the system is at rest in the equilibrium position.

0 0.5 1 1.5 2 2.5 3 3.5 4

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

t, s

f, N

f=f0*exp(-at)cos(bt)

0 5 10 15 20 25 30

-1000 -500 0 500 1000 1500

t, s

X, m

x=F(t)

(17)

Soluti x m For t

c x For t

c1  The a

a Note Calcu

>> w=

>>T=

>>t=0

>>f0=

>> x1

>> x=

>> pl

>> gr

ion: for 0 <t



t mT

F 

³

0 

<T, we seek

1cos



t T

c 

= T, from th

3sin

0

mT F

 

amplitude of

2 2 2

1 c

c   that the smal ulation and v

=1;

=10;

0:0.1:30;

=2; m=0.1;

1=f0./(m.*T.

=x1.*(w.*t-si ot(t,x,'k-') rid on

F

<T



t

sin

the solution



c2sin T 

e continuity ,

n ₂

c m

T  the oscillatio

2 sin

3 0

mT

F 

 ller the slow isualization p

*w.^3);

in(w.*t));

ig.7. The oscill

00 10 20 30 40 50 60 70

X, m

Fig. 6. The

n in the form

 

 m

T F t  of x and x ̇ w



¹ ^co

3 0

mT F

 ^

ons 2 .

T

wer the "powe program

lation of the sys

5 10

e force acting on

2 0

 m

F

we find:



^.

osT

er" F0 is turn

stem under the i

0 15

t, s X=F(t)

n the system

ned on (ie, th

influence of for

20

he larger T).

rce at t> T, F =

25 30

F0

0

(18)

>> w=1;

>> T=10;

>> t=0:0.1:30;

>> f0=2; m=0.1;

>> x1=f0./(m.*T.*w.^3);

>> x=x1.*(w.*t-sin(w.*t));

>> plot(t,x,’k-‘)

>> grid on

>> c1=-x1.*sin(w.*T);

>> c2=x1.*(1-cos(w.*t));

>> X=c1.*cos(w.*(t-T))+c2.*sin(w.*(t-T))+f0./(m.*w.^2);

>> plot(t,X,’k-‘)

>> grid on

Fig.8. The oscillation of the system under the influence of force at 0 <t <T

e) Find the trajectory of the motion of the particle in the central field U = k r ^ 2/2, the so-called

spatial oscillator.

The calculated formulas w = √ (k / m) is the eigenvector frequency, x = acos (wt + α), y = bcos (wt + β) - where a is the amplitude of the oscillation.

>> a=1; b=2; w=1;

>> t=0:0.1:30;

>> alfa=pi./3; beta=pi./6;

>> fi=w.*t-alfa; delta=beta-alfa;

>> x=a.*cos(fi);

>> y=b.*cos(delta).*cos(fi)-b.*sin(delta).*sin(fi);

>> plot(x,y,'k-')

>> grid on

>> xlabel('X')

>> ylabel('Y')

>> title('Y=F(X)')

0 5 10 15 20 25 30

16 17 18 19 20 21 22 23

t, s

X, m

X=F(t)

(19)

Fig.9. Trajectory of the motion of a particle

Presented laboratory works were performed by third-year students of our university, who are studying in the specialty 5B060400-physics, while conducting laboratory classes on the discipline "Computer simulation of physical phenomena." Especially I want to note that visualization of calculations in the form of graphs allows you to better understand the essence of physical processes and students with great desire perform this part of the task.

REFERENCES

[1] V. P. Dyakonov. MATHLAB training course. - SPb.: Peter, 2001. –P533. (in Russ.).

[2] K.A. Kabylbekov, Bayzhanova A. Application of multimedia possibilities of computer systems for expansion of demonstration resources of some physical phenomena.Works All-Russia scientifically-practical conference with the international participation. Tomsk 2011.,P210-215. (in Russ.).

[3] K. A. Kabylbekov, P.A. Saidakhmetov, A.S. Arysbaeva. Мodel of the form of the organisation of self-maintained performance of computer laboratory operation. News of NAS of RК, series physical-math., Almaty, 2013, №6, P82-89. (in Russ.).

[4] K.A. Kabylbekov,P.A. Saidahmetov, L.E. Baydullaeva, R.A. Abduraimov . Procedure of use of computer models for photoeffect studying, Compton effect, models of forms of the organisation of performance of computer laboratory operations.

News NAS of RК, series physical-math., Almaty, 2013. №6, P114-121. (in Russ.).

[5] K.A. Kabylbekov, N.S. Saidullayeva, P.A. Saidakhmetov. Multimedia demonstration models of electromagnetic phenomena and their use in the educational process.Proceedings of the International Scientific and Practical Conference

"Chemistry in Building Materials and Materials Science in the XX Century" 2008,p. 139-144. (in Russ.).

[6] K.A.,Kabylbekov, N.S. Saidullayeva, R.S. Spabekova. Experience of work of the Department of Physics on the creation and expansion of information resources on the discipline "Physics" for tehn. special use and their use in the educational process to improve the quality of educational services. Collected works of сonference. Actual problems of education, science and production. 2 vol. SKSU named after M.Auezov , Shymkent.2008 (in Russ.).

[7] K.A. Kabylbekov, N.S. Saidullayeva., P.A. Saidakhmetov, T.A. Turmambekov, Omasheva G.Sh. Computer model of laboratory work "Verification of the Kirchhoff rules" program for computers. Certificate of the State registration of the intellectual property object, №. 319. 22. 04.2009 (in Russ.).

[8] Kabylbekov K.A., Ashirbaev H.A., Saikdahmetov P.А., Baigulova Z.A., Baidullaeva L.E. Model of the form of the organisations of computer laboratory operation on examination of Newton's fringes. News NAS of RК, series physical-math/, Almaty, №1 (299), 2015, Р14-20. (in Russ.).

-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1

-2 -1.5 -1 -0.5 0 0.5 1 1.5 2

X

Y

Y=F(X)

(20)

[9] Kabylbekov K.A., Ashirbaev H.A., Sabalakhova A.P., Dzhumagalieva A.I. Model of the form of the organisation of computer laboratory operation on exam