MATHEMATICAL MODELING. MATHEMATICS МАТЕМАТИЧНЕ МОДЕЛЮВАННЯ. МАТЕМАТИКА

ISSN 1727-7108. Web: visnyk.tntu.edu.ua

MATHEMATICAL MODELING.

MATHEMATICS

МАТЕМАТИЧНЕ МОДЕЛЮВАННЯ.

МАТЕМАТИКА

UDC 517.52/524

COMPUTATION OF IMPROPER INTEGRAL ACCORDING TO

EIGEN-ELEMENTS OF 1

ST

_{-GENUS HANKEL –}

(KONTOROVYCH-LEBEDEV) HYBRID DIFFERENTIAL OPERATOR – 2

ND

GENUS

LEGENDRE 2

ND

GENUS – FOURIER ON POLAR AXIS

Iryna Hotynchan

Chernivtsi Institute of Trade and Economics of Kyiv National University of

Trade and Economics, Chernivtsi, Ukraine

Summary. There were calculated using comparison method of solving the boundary problem on the polar

axis segment with three junction points for the separate system consisting of Hankel, Kontorovich-Lebedev, Legendre and Fourier differential equations for the modified functions, which was built, on one side, by Cauchy function method, and on the other side, by the definite hybrid integral transformation, poly-parametric family of the improper integrals according to the eigen-elements of 1st_{-genus Hankel differential operator –} (Kontorovich-Lebedev) 2nd_{–genus – Legendre 2}nd_{–genus – Fourier.}

Key words: improper integrals, Eigen-elements, hybrid differential operator, integral transformation,

main solutions.

Received 07.04.2017

Problem setting. Thin-walled construction composite elements usually are in steady

mode after series of dramatic alterations of temperature or power stress. The investigation of their physical and technical characteristics leads to thermo-mechanic (mechanic) problems of lump-heterogeneous ambience [1]. The practice shows that even in the simplest cases the values, which characterize the steady conditions of composite material, are marked as multi-parametric integrals that can be conventionally convergent even when they describe analytic function. Hence, it is necessary to convert the improper integral due to its convergence (function) that is especially important during engineering computing. The issues [2, 3] are dedicated to calculation of improper integrals.

Research objective is to elaborate methods of calculation of multi-parametrical

medium of improper integrals according to Eigen-elements of 1st-genus Hankel – (Kontorovych-Lebedev) Hybrid Differential Operator – 2nd genus Legendre 2nd genus – Fourier.

Task setting. Let us construct the limited on multiplication



 

 

 







_

_

_





 : 0; _{1 1}; _{2 2}; _{3 3};

3 r r R R R R R R

I solution of separated system of ordinary



_{, 1}2



₁( ) ₁( ),



0; ₁

,

1 q u r g r r R B_   





2( ) 2( ),



1; 2



, 2 2 2 q u r g r r R R B_   





3( ) 3( ),



2; 3

,

2 3 ) ( q u r g r r R R  _ (1)







          ₄2 ₄( ) ₄( ), ₃; 2 2 R r r g r u q dr d

under conjugation conditions

. 3 , 2 , 1 ; 2 , 1 , ) ( ) ( _{2 2 1} 1 1                _        _   r j k u dr d r u dr d jk R r k k j k j k k j k j    _k   (2)

The differential operators of Fourier – ₂ 2 dr d , Hankel – ; 2 1 , 1 2 1 2 1 2 1 2 , 1               r dr d r dr d B Kontorovych-Lebedev –





,



0;



, 2 1 , 1 2 ₂ 2 2 2 ₂ 2 2 2 2            r dr d r dr d r B Legendre generalized operator – _{ } 2 1 , 1 1 2 1 4 1 2 1 2 1 2 2                 _     chr chr dr d cthr dr d participate in the system (1) [4].

Let us assume that conditions on coefficients are accomplished q_s 0, m_jk 0,

0  m jk  , kj k j k j k j jk c ₂ ₁ ₁ ₂ , c_1kc_2k 0, j 1,2, k,m1,2,3, s1,4.

Research results. І. Method of Cauchy functions. The fundamental solution system for

Hankel differential equation



_{, 1}2



0

1 q v

B_ are created by 1st-genus Bessel imaginary argument functions (modified Bessel functions) першого роду v₁ I _,

 

q₁r

1    та другого роду

 

qr K v_{2 , 1} 1  

 [4]. The fundamental solution system for Kontorovych-Lebedev differential equation



₂2



0

2 q v

B_ are created by 1st _{genus Bessel modified cylindrical functions}

 

r I v _{q }  2 2, 1 and 2 nd_{-genus v K}

 

_r q2,2 

2  [4]. The fundamental solution system for

Lagrange differential equation



 q32



v0 are created by 1st-genus Legendre generalized

functions v P_ 

 

chr 3 1 and 2nd-genus  

_{ }

chr L v _ 3 2  , where 3 3 2 1 q     [3]. The fundamental solution system for Fourier differential equation 2 42 0

2        q v dr d

are created with

functions v eq4r 1 та r q e v 4 2   [4].

 

_

_

   

  1 1 1 0 1 2 1 1 1 , 1 1( ) , R d g r E r q I A r u      ,

 

_

 

_

_

   

  2 1 2 2 2 2 2 1 2 2 2 , 2 , 2 2( ) , R R q q r B K r E r g d I A r u _  _      ,  

_{ }

 

_{ }

_{   }



   3 2 3 3 3 3 3 3 3( ) , R R d sh g r E chr L B chr P A r u _ _    , (3)

   



  _  3 4 4 4 4 4( ) , R r q d g r E e B r u   ,

where Cauchy’s functions are represented as:

 





 





 





             . 0 , , , 0 , , , 1 1 1 1 1 11 ; , 1 , 1 1 1 1 1 11 ; , 1 , 1 1 11 11 ; , 2 1 1 1 1 1 1 1 1 R r r q R q q I R r q R q r q I R q U q r E                 (4)

 





















                . , , , , , , , , , 2 1 1 2 11 ; , 1 1 12 ; , 2 1 1 2 11 ; , 1 1 12 ; , 2 1 11 ; , 2 2 2 2 2 2 2 2 2 2 2 2 2 R r R r R R R r R R r R R R r E q q q q q                     (5)

 

 

 

 

_

_

 

_

_

 

_

_

 

_

_

 

_

_

            . , , , , , , , , , 3 2 1 3 , 11 ; 1 2 , 12 ; 3 2 1 3 , 11 ; 1 2 , 12 ; 3 2 11 ; 3 3 3 3 3 3 3 F chR ch F chR chr R r R R r R ch chR F chr chR F chR chR q B r E      _           (6)

 

_

_

_{ }









                    . , , , , , 1 , 3 4 3 4 3 12 3 4 3 4 3 12 3 12 4 3 12 4 4 3 4 3 4 r R q R q e r R r q R q e q q r E R r q R q        (7)

The functions participating in equations (4) – (7), are universally accepted [2, 3, 4]. The conjugation conditions (2) to determine the values A_i



i1,2,3



and B_j ( j 2,3,4)

produce the algebraic system of six equations

 

_

_

 

_

_

_

_

13 4 3 4 32 12 3 3 32 , 11 ; 3 3 31 , 11 ; 3 3      chR A Z chR B V q R B  Z ,  

_

_

 

_

_

_

_

34 13 4 3 4 32 22 3 3 32 , 21 ; 3 3 31 , 21 ; ₃ 3 chR A Z chR B V q R B G Z_  _    .

The following functions participate in the system (8)

 





 









 





    _    2 1 2 2 2 2 2 2 1 1 1 1 1 1 2 2 2 1 11 11 ; , 2 2 11 ; , 1 2 1 21 0 1 2 1 1 1 11 11 ; , 1 , 1 2 1 11 12 , , R R q q R d g R R R R c d g R q U q I R c G            _          ,









 

 

_

_

 

_

_

 





 _    _  3 2 3 3 3 1 2 2 2 2 , 2 2 3 3 2 11 ; 3 3 , 11 ; 2 22 1 2 2 2 1 11 11 ; , 1 1 12 ; , 1 2 2 12 23 , , , , R R R R q q d sh g chR chR ch chR F shR c d g R R R R c G                    ,  

_

_

 

_

_

 

 

 





   _   3 3 4 3 2 3 3 4 3 12 4 3 12 23 3 3 2 11 ; 2 2 , 12 ; 3 13 34 , , R R q R R d g q e c d sh g chR chR ch chR F shR c G              .

Let us introduce the functions:

 

 











1 1



, ;2



1 2



11 11 ; , 2 1 1 ; , 1 1 11 21 ; , ; , j q U ₁ qR q₂₂ j R ,R U ₁ q R q₂₂ j R,R       ,  

_{ }

_

_

 

_

_

_

_

 

_

_

3 2 2 ; 2 1 1 ; , 3 2 1 ; 2 1 2 ; , ; , 2 2 2 , 3 , 2 2 , 3 , 2 jk q q j R R k chR chR q j R R k chR chR q                  ,    

_{ }

 ;2

 

 ;1



2 3



, ;1

 

 ;2



2 3



, ; , q q 3 chR ,chR q 3 chR ,chR A__{ j} _{ } _ _j _{ } _ _j ,    

_{ }

_

_

 

_{ }

_

_

 

_{ }

q R q V q R q V q B__{, ;j 22}32 _{4 3 q}_{, ;j1 12}32 _{4 3 q}_{, ;j2} 2 2 2 2     ,  

_{ }

_

_

 

_

_

_

_

 

_

_

3 2 2 ; 3 4 32 12 3 2 1 ; 3 4 32 22 ; 3 , 3 , 3 j q V q R j chR chR V q R j chR chR            ,  

_{ }

 

_{ }

_

_

 

_{ }

_

_

r R q r R q q r _{q j q j} j q               , , 2, 2 1 ; , 2 ; 2 2 2 ; , 1 ; ; , 2 3 2, 2 3 2 2 2      ,  

_{ }

_{r,q V}

_

_{q R}

_

_F 

_

_{chR ,chr}

_

_V

_

_{q R}

_

_F 

_

_{chR ,chr}

_

3 3 , 21 ; 3 4 32 12 3 3 , 11 ; 3 4 32 22 1 ; 3 3 3          ,  _{ j}

 

r q U_{ }



q R



_{q  j}



R r



U_{ }



qR



_{q  j}



R r



 , , 1, 1 2 ; , 1 1 11 11 ; , 1 1 1 ; , 1 1 11 21 ; , ; ,  1  2 2  1 1 2 2  ,  

_{ }

 

 

q F 



chR chr



 

 

q F 



chR chr



q r, , 2, 2 , 22 ; 1 ; , 2 2 , 12 ; 2 ; , 2 ; 3 3 3              , j,k 1,2.

Let us suppose the condition of univalent solution for boundary problem (1), (2) has been accomplished: for any non-zero vector q 



q1,q2,q3,q4



the identification item of

   

_{ }

_

_

   

_{ }

_

_

   

_{ }





 _{ }

_{ }

_

_

   

_{ }

. 0 2 ; , 1 1 11 11 ; , 1 ; , 1 1 11 21 ; , 2 ; , 3 4 32 21 1 ; , 3 4 32 22 , 1 1 1 1        q B R q U q B R q U q A R q V q A R q V q                    (9)

Let us determine the main solutions for the problem (1), (2):

1) originated by the inhomogeneity of conjugation conditions for Green’s function

   

_{ }

  

 

   

_{ }

4 3 , 2 ; , 4 , 13 ; , , R r q e q q A q r R     _         ,   

 

   

_{ }

   

_{ }

4 3 , 1 ; , 4 , 23 , , R r q e q q A q r R     _         ,    

_{ }

 

 





   1

_{ }

4 3 2 , 1 1 11 21 ; , 3 2 2 13 1 2 2 12 4 , 11 ; , , R r q e q R q U q B R sh c R c q r R _     _          ,    

_{ }

 

 





   1

_{ }

4 3 2 , 1 1 11 11 ; , 3 2 2 13 1 2 2 12 4 , 21 , , R r q e q R q U q B R sh c R c q r R _     _          ,    

_{ }

 

 

 

 

   

_{ }

4 3 , 2 ; , 3 2 2 13 4 , 12 ; , , R r q e q q q B R sh c q r R      _         ,    

_{ }

 

 

 

 

   

_{ }

4 3 , 1 ; , 3 2 2 13 4 , 22 ; , , R r q e q q q B R sh c q r R      _         ;

2) originated by the inhomogeneity of the system of influence function

   

_

_

   

_{ }

 

 

 

_

_

 





 

 

                ; , , , , , , , , , 2 1 1 ; , 2 ; , 2 1 1 ; , 2 ; , , 2 22 ; , 2 2 2 2 2 R r R q r q R r R q q r q q r H q q       _                  

_

_

 

 

   

_{ }

 

  _{  }          , 1 , 1 ; 1 2 2 12 1 2 1 11 31 ; , 1 3 12 1 , , , I q q q r R c R c q r H    _{ } ,    

_

_

 

 

   

_{ }

 



q



q q r R c q r H , , q , _;1 , , 2 ; , 1 2 2 12 32 ; , 3 2 2 12              _    _ ,    

_

_

   

_{ }

 

 

4 3 3 , , , _;2 , 23 34 ; , R q e q r q c q r H      _{ }        ,    

_

_

 

 

   

_{ }

 

_{ }

 

_

_

 

_

_

 

_{ }

                ; , , , , , , , , , 3 2 1 ; 2 ; 3 2 1 ; 2 ; , 3 33 ; , 3 3 3 3 R r R q r q R r R q q r q q B q r H      _                  

_

_

 

 

     

_{ }

 

  _{  }         , 1 , 3 3 2 13 1 2 2 12 1 2 1 11 41 ; , 1 3 4 2 1 , , I q q e q B R sh c R c R c q r H R r q   _{ }   ,    

_

_

 

 

 

_

_

   2

_{ }

4 3 2 2 , 2 ; , 3 3 2 13 1 2 2 12 42 ; , , , , q e q r R q q q B R sh c R c q r H _      _            ,    

_

_

 

 

 

_

_

   3

_{ }

4 3 , 2 ; 3 3 2 13 43 ; , , , , e q r R q q q B R sh c q r H      _           ,    

_

_

   

_{ }

   

  



 

  







   

  



 

  







                               . , , , , , , , 1 , , 3 4 3 4 3 12 2 ; , 4 3 4 3 22 1 ; , 3 4 3 4 3 12 2 ; , 4 3 4 3 22 1 ; , , 4 44 ; , 3 4 3 4                     r R q R q q q R q q e r R r q R q q r q R q q e q q q r H R r q R q

   

_

_{  }

   

_

_{  }

_  

_



_

3 2 2 1 2 3 3 ; , 1 2 2 2 ; , , , , , R R j R R j r q g d H r q g sh d H__      __        

_

_{, ,}

_{  }

_, 3 4 4 ; ,



  R j r q g d H__    j1,4. ₍₁₀₎

II. Method of integral transformations. The hybrid differential operator M_{ ,} _{ } was built in [5] ([5], c. 83), the direct H_ _{,  ;3} and the reverse H__, _{ } _;3 1st_{-genus Hankel}

(Kontorovych-Lebedev) 2nd genus Legendre 2nd genus Fourier hybrid integral transformation was determined ([5], c. 86), which was originated on multiplication I₃ with hybrid differential operator (HDO)

   

  ,

M , and the main identity of integral transformation of hybrid differential operator (HDO)

   

  ,

M was proved ([5], c. 87).

The direct H_ _{,  ;3}, and reverse H__, _{ }_{ ;3} hybrid integral transformation and the main identity of integral transformation of hybrid differential operator (HDO) M_{ ,} _{ } make up the mathematical instrument to solve the boundary problem (1), (2).

Let us record the system (1) in matrix form













 

 

 

 

                                           r g r g r g r g r u q dr d r u q r u q B r u q B 4 3 2 1 4 2 4 2 2 3 2 3 ) ( 2 2 2 1 2 1 , ) ( ) ( ) ( ) ( 2 1     . (11)

Integral operator H_ _{,  ;3} will be presented in the form of operator matrix-row

   

_{ }

   

_{ }

   

_{ }

   

_{ }

   

_{ }

_{, .} , , , 3 3 2 2 1 2 1 1 4 4 ; , 3 3 ; , 1 2 2 2 ; , 0 1 2 1 1 ; , 3 ; ,         









    R R R R R R dr r V shrdr r V dr r r V dr r r V H                               (12)

In the equation (12) V_ _,_{  ;j}



r,



( j 1,4) the components of spectral vector-function

   

_

_

_

 , r,

V ([5], c. 86), а _j ( j 1,4) – weight coefficients of weight function 

 

r ([5], p. 84). Let us apply the operator matrix-row (12) to the system (11) according to the rule of matrix multiplication. As the result of main identity, we will obtain the algebraic equation

where q max



q₁;q₂;q₃;q₄



,

 





 

4 ~ ~ j j u u   ,

 





 

4 ~ ~ j j g g   , 3 2 1 2 1 23 22 21 1 2 1 13 12 1 1 1 shR shR R c c c R c c d _   _ , 3 2 23 22 13 2 shR shR c c c d  , 23 3 1 c d  ,  _{ }

 

 _{ }

 

k R r k k j k j k j V r dr d Z    _      _            2 2 , ; 1 , 1 , 2 ; , (j 1,2,k 1,3).

Hence, the function is as follows

   

_

  

 

  

 

_{ }

               3 1 1 2 2 , 22 ; , 2 2 2 , 12 ; , 2 2 ~ ~ ) ( ~ k k k k k k q Z q Z d g q g u                  . (13)

Integral operator H__, _{ }_{ ;3} as the one that is reverse to H_ _{,  ;3} will be represented in the form of operator matrix-column

   

_{ }

   

_{   }

   

_{   }

   

_{   }

   

_{   }

                                









        0 4 ; , 0 3 ; , 0 2 ; , 0 1 ; , 3 ; , , 2 , 2 , 2 , 2                                d r V d r V d r V d r V H      . (14)

   

_{ }

   

_

_{  }

_{ }





_           1  0 3 3 0 3 ; , 2 2 ; , , , 2 R j d sh g d V q r V                     

_{ }

   

_

_,

_{  }

_{ }

_, , 2 1 0 4 4 0 4 ; , 2 2 ; ,





            R j d g d V q r V                 . 4 , 1  j (15)

Having compared the solutions (10) and (15) as the result of unification, we will get the following formulae to compute the improper integrals.

   

_{ }

   

_

_

   

_

_

, 4 , 1 , , , , , , 2 ; , 1 0 ; , ; ,    



S j r  V k   d k H jk r  q j k           (16)    

_{ }

   

_{ }

   

_

_

2 , 1 , 4 , 1 , , , , 2 , 2 ; , 1 0 , 12 ; , ; ,     



S _j r  Z k  d d_k R j _k r  q j k           , (17)    

_{ }

   

_{ }

   

_

_

2 , 1 , 4 , 1 , , , , 2 , 1 ; , 1 0 , 22 ; , ; ,     



S _j r  Z k  d d_k R j _k r  q j k           , (18)    

_{ }





   

_{ }

   

_{ }

. 4 , 1 , , , 2 ₁2 1 _{, ; ,} ; ,      j r V q r S__{ j}   __{ j}  __ 

The conclusion of accomplished investigation is the statement.

Main theorem. Let the vector-function f(r)



B,₁[g1],B₂[g2],_{ } [g3],g4''



is

continuous on the set I₃, and the functions g_j(r) satisfy the limitation conditions

   

_{ }

  

 

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

lim

_          dr r dV r g dr r dg r V r r             

_{ }

  

 

, 0 , ) ( ) ( , 4 ₄ , ;4 4 ; , 1 2 1

lim

_           dr r dV r g dr r dg r V r r       

and conjugation conditions (2). If the condition of univalent solution for the boundary (1), (2) problem (9) is adhered, then there appear the formulae (16) – (18) to calculate the multi-parameter improper integrals due to Eigen-elements of hybrid differential operator M_ _{, } , indicated in the research project ([5], с. 83).

Conclusions. The obtained formulae (16) – (18) contribute to reference literature in the

sphere of computation of improper integrals from superposition of special functions of mathematical physics.

References

1. Lenyuk M.P., Petryk M.R. Intehral'ni peretvorennya Fur’ye, Besselya iz spektral'nym parametrom v zadachakh matematychnoho modelyuvannya masoperenosu v neodnoridnykh seredovyshchakh, Kyyiv, Naukova dumka, 2000, 372 p. [In Ukraine].

3. Lenyuk M., Shelestovs'kyy B. Obchyslennya nevlasnykh intehraliv za vlasnymy hibrydnoho dyferentsial'noho operatora Eylera-Fur’ye na sehmenti polyarnoyi osi, Visnyk Ternopil's'koho derzhavnoho tekhnichnoho universytetu, vol. 16, no. 4, 2011, pp. 193 – 201. [In Ukraine].

4. Lenyuk M.P., Shynkaryk M.I. Hibrydni intehral'ni peretvorennya (Fur’ye, Besselya, Lezhandra), ch. 1, Ternopil, Ekonomichna dumka, 2004, 368 p. [In Ukraine].

5. Hotynchan H.I. Modelyuvannya tekhnolohichnykh protsesiv v bahatosharovykh navipobmezhenykh tilakh metodom intehral'noho peretvorennya Hankelya 1-ho rodu – (Kontorovycha-Lyebyedyeva) 2-ho rodu – Lezhandra 2-ho rodu Fur’ye, Visnyk Khmel'nyts'koho natsional'noho universytetu, Tekhnichni nauky, vol. 1, Khmel'nyts'kyy, KhNU, 2005, pp. 82 – 89. [In Ukraine].