Hyperbolic Boundary Value Problem for Unlimited Piecewise-Homogeneous Hollow Cylinder

Hyperbolic Boundary Value Problem for Unlimited Piecewise-Homogeneous Hollow Cylinder

By means of the method of integral and hybrid integral transforms, in combination with the method of main solutions (influence functions and Green functions) the integral image of exact analytical solution of hyperbolic boundary value problem of mathematical physics for unlimited piecewise-homogeneo...

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Datum:2016
Hauptverfasser: Konet, I.M., Pylypiuk, T.M.
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Veröffentlicht: Інститут кібернетики ім. В.М. Глушкова НАН України 2016
Schriftenreihe:Математичне та комп'ютерне моделювання. Серія: Фізико-математичні науки
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Zitieren:Hyperbolic Boundary Value Problem for Unlimited Piecewise-Homogeneous Hollow Cylinder / I.M. Konet, T.M. Pylypiuk // Математичне та комп'ютерне моделювання. Серія: Фізико-математичні науки: зб. наук. пр. — Кам’янець-Подільський: Кам'янець-Подільськ. нац. ун-т, 2016. — Вип. 14. — С. 91-101. — Бібліогр.: 18 назв. — англ.

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spelling nasplib_isofts_kiev_ua-123456789-1339142025-02-09T17:38:00Z Hyperbolic Boundary Value Problem for Unlimited Piecewise-Homogeneous Hollow Cylinder Konet, I.M. Pylypiuk, T.M. By means of the method of integral and hybrid integral transforms, in combination with the method of main solutions (influence functions and Green functions) the integral image of exact analytical solution of hyperbolic boundary value problem of mathematical physics for unlimited piecewise-homogeneous hollow cylinder is obtained for the first time. Методом інтегральних і гібридних інтегральних перетворень у поєднанні з методом головних розв’язків (функцій впливу та функцій Гріна) вперше побудовано інтегральне зображення точного аналітичного розв'язку гіперболічної крайової задачі математичної фізики для необмеженого кусково-однорідного порожнистого циліндра. 2016 Article Hyperbolic Boundary Value Problem for Unlimited Piecewise-Homogeneous Hollow Cylinder / I.M. Konet, T.M. Pylypiuk // Математичне та комп'ютерне моделювання. Серія: Фізико-математичні науки: зб. наук. пр. — Кам’янець-Подільський: Кам'янець-Подільськ. нац. ун-т, 2016. — Вип. 14. — С. 91-101. — Бібліогр.: 18 назв. — англ. 2308-5878 https://nasplib.isofts.kiev.ua/handle/123456789/133914 517.946 en Математичне та комп'ютерне моделювання. Серія: Фізико-математичні науки application/pdf Інститут кібернетики ім. В.М. Глушкова НАН України
institution Digital Library of Periodicals of National Academy of Sciences of Ukraine
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language English
description By means of the method of integral and hybrid integral transforms, in combination with the method of main solutions (influence functions and Green functions) the integral image of exact analytical solution of hyperbolic boundary value problem of mathematical physics for unlimited piecewise-homogeneous hollow cylinder is obtained for the first time.
format Article
author Konet, I.M.
Pylypiuk, T.M.
spellingShingle Konet, I.M.
Pylypiuk, T.M.
Hyperbolic Boundary Value Problem for Unlimited Piecewise-Homogeneous Hollow Cylinder
Математичне та комп'ютерне моделювання. Серія: Фізико-математичні науки
author_facet Konet, I.M.
Pylypiuk, T.M.
author_sort Konet, I.M.
title Hyperbolic Boundary Value Problem for Unlimited Piecewise-Homogeneous Hollow Cylinder
title_short Hyperbolic Boundary Value Problem for Unlimited Piecewise-Homogeneous Hollow Cylinder
title_full Hyperbolic Boundary Value Problem for Unlimited Piecewise-Homogeneous Hollow Cylinder
title_fullStr Hyperbolic Boundary Value Problem for Unlimited Piecewise-Homogeneous Hollow Cylinder
title_full_unstemmed Hyperbolic Boundary Value Problem for Unlimited Piecewise-Homogeneous Hollow Cylinder
title_sort hyperbolic boundary value problem for unlimited piecewise-homogeneous hollow cylinder
publisher Інститут кібернетики ім. В.М. Глушкова НАН України
publishDate 2016
url https://nasplib.isofts.kiev.ua/handle/123456789/133914
citation_txt Hyperbolic Boundary Value Problem for Unlimited Piecewise-Homogeneous Hollow Cylinder / I.M. Konet, T.M. Pylypiuk // Математичне та комп'ютерне моделювання. Серія: Фізико-математичні науки: зб. наук. пр. — Кам’янець-Подільський: Кам'янець-Подільськ. нац. ун-т, 2016. — Вип. 14. — С. 91-101. — Бібліогр.: 18 назв. — англ.
series Математичне та комп'ютерне моделювання. Серія: Фізико-математичні науки
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fulltext Серія: Фізико-математичні науки. Випуск 14 91 UDC 517.946 I. M. Konet, Doctor of Physics and Mathematics Sciences, Professor, T. M. Pylypiuk, Ph. D. of Physics and Mathematics Sciences Kamianets-Podilsky Ivan Ohienko National University, Kamianets-Podilsky HYPERBOLIC BOUNDARY VALUE PROBLEM FOR UNLIMITED PIECEWISE-HOMOGENEOUS HOLLOW CYLINDER By means of the method of integral and hybrid integral trans- forms, in combination with the method of main solutions (influ- ence functions and Green functions) the integral image of exact analytical solution of hyperbolic boundary value problem of ma- thematical physics for unlimited piecewise-homogeneous hollow cylinder is obtained for the first time. Keywords: hyperbolic equation, initial and boundary conditions, conjugate conditions, integral transforms, the main solutions. Introduction. The theory of hyperbolic boundary value problems for partial differential equations is an important section of the modern theory of differential equations which is intensively developing in the present time. The popularity of the problem is the consequence of the significance of its results in the development of many mathematical problems, as well as of its numer- ous applications in mathematical modeling of different processes and pheno- menon of mechanics, physics, engineering, new technologies. Significant results from the theory of Cauchy and boundary value prob- lems for hyperbolic equations were obtained in the works of J. Hadamard [1], L. Gording [2], Yu. Mitropolsky, G. Khoma, M. Hromyak [3], A. Samoilen- ko, B. Tkach [4], M. Smirnov [5], V. Chernyatyn [6] and others. It is well known that the complexity of a boundary-value problem significantly depends on the coefficients of equations (different types of degeneracy and features) and the geometry of domain (smoothness of the boundary, the presence of corner points, etc.) in which the problem is con- sidered. The dependence of the properties of solutions of boundary value problems for linear, quasi-linear, and certain classes of nonlinear equations (hyperbolic, parabolic, elliptic) in homogeneous domains on the above- mentioned properties of the coefficients of equations and geometry of do- main are studied in detail, and functional spaces of correctness of prob- lems in the sense of Hadamard are constructed. However, many important applied problems of thermophysics, ther- modynamics, theory of elasticity, theory of electrical circuits, theory of vibrations lead to boundary value problems for partial differential equa- tions not only in homogeneous domains (when the coefficients of the equ- © I. M. Konet, T. M. Pylypiuk, 2016 Математичне та комп’ютерне моделювання 92 ations are continuous), but also in inhomogeneous and piecewise homoge- neous domains if the coefficients of the equations are piecewise conti- nuous or piecewise constant [7, 8]. The method of hybrid integral transforms generated by hybrid differen- tial operators when in each component of connectivity of piecewise homo- geneous domain are treated different differential operators or differential operators look the same, but with different sets of coefficients is an effective method of constructing exact solutions for a fairly broad class of linear boundary value problems in piecewise homogeneous domains [9–12]. By means of the method of hybrid integral transforms the exact solution of hyperbolic boundary value problem of mathematical physics for unlimited piecewise homogeneous hollow cylinder is obtained in this article. Formulation of the problem. Let’s consider the problem of struc- ture of 2 -periodic for angular variable  solution of partial differential equations of hyperbolic type of 2nd order [13] 2 22 2 2 2 2 2 2 2 2 2 2 1 ( , , , ); ; 1, 1 j j rj zj j j j j j u a a a u r rt r r z u f t r z r I j n                              (1) which is bounded in the set 1 1 1 0 1 1 1 {( , , , ) : 0; ( ; ), 0, ; n n n j j j n j j D t r z t r I I R R R R R                  [0;2 ); ( ; )}z     with initial conditions 1 2 0 0( , , ); ( , , ); j j t j t j u u g r z g r z t        ;jr I 1, 1j n  , (2) boundary conditions 0 0 0 11 11 1 0( , , ); r R u g t z r          1 1 22 22 1 ( , , );n n n r R u g t z r            (3) 0; s j s z u z     0; s j s z u z     0,1;s  (4) and conjugate conditions 1 1 2 2 1 0; 1, 2; 1, . k k k k k j j k j j k r R u u r r j k n                           (5) Серія: Фізико-математичні науки. Випуск 14 93 Here ,, , , ,k k rj j zj j js jsa a a    — some not negative constants; 0 0 0 0 11 11 11 110; 0; 0;       1 1 1 1 22 22 22 220; 0; 0;n n n n          2 1 1 2 0;k k k k jk j j j jс       1 2 0;k kc c   1 2 1( , , , ) ( , , , ), ( , , , ), , ( , , , ) ;nf t r z f t r z f t r z f t r z x      1 1 1 1 1 2 1( , , ) ( , , ), ( , , ), , ( , , )ng r z g r z g r z g r z     ;  2 2 2 2 1 2 1( , , ) ( , , ), ( , , ), , ( , , ) ;ng r z g r z g r z g r z     0 ( , , ); ( , , )g t z g t z  are known bounded continuous functions;  1 2 1( , , , ) ( , , , ), ( , , , ), , ( , , , )nu t r z u t r z u t r z u t r z     is the desired function. The main part. Let’s assume that the solution of the problem (1)–(5) exists and defined and the unknown functions satisfy the conditions of applicability of integral transformations (6)–(8) [14–16]. Let’s apply the integral Fourier transform on Cartesian axis ( ; )  relative to variable z to the problem (1)–(5) [14]:  ( ) ( ) exp( ) ( ),F g z g z i z dz g        1i   , (6)  1 1 ( ) ( ) exp( ) ( ), 2 F g g i z d g z           (7) 2 2 2 2 [ ( )] ( ). d g F F g z g dz               (8) The integral operator F due to the formula (6) as a result of identi- ty (8) three-dimensional initial boundary value problem of conjugation (1)– (5) puts in accordance the task of constructing solution which is limited in the set  ( , , ); 0; ; [0;2 )nD t r t r I       and is 2 -periodical of an- gular variable  of differential equations   2 22 2 2 2 2 2 2 2 2 2 1 ( , , , ); ; 1, 1 j j rj j zj j j j j u a a u a u r rt r r f t r r I j n                                  (9) with initial conditions 1 0 ( , , );j jt u g r      2 0 ( , , );j j t u g r t         ;jr I 1, 1;j n  (10) Математичне та комп’ютерне моделювання 94 boundary conditions 0 0 0 11 11 1 0( , , ); r R u g t r             1 1 22 22 1 ( , , );n n n r R u g t r               (11) and conjugate conditions 1 1 2 2 1 0; 1, 2; 1, . k k k k k j j k j j k r R u u r r j k n                             (12) Let’s apply finite integral Fourier transform relative to the variable  to the problem (9)–(12) [15]:   2 0 ( ) ( ) exp( ) ,m mF g g im d g        (13) 1 0 Re [ ] exp( ) ( ), 2m m m m m F g g im g         (14) 2 2 2 ,2 [ ( )]m m m d g F m F g m g d             (15) here Re( ) — the real part of the expression ( ) relative to the variable ; 0 1,  2;k  1,2,3k   The integral operator mF due to the formula (13) as a result of identi- ty (15) two-dimensional initial boundary value problem of conjugation (9)–(12) puts in accordance the task of constructing solution which is li- mited in the set  ( , ); 0; nD t r t r I     of differential equations   2 22 2 2 2 2 2 2 2 1 ( , , ); ; 1, 1; jm jm rj jm zj j jm jm j u a u a u r rt r r f t r r I j n                           (16) with initial conditions 1 0 ( , );jm jmt u g r     2 0 ( , );jm jm t u g r t        ;jr I 1, 1,j n  (17) boundary conditions 0 0 0 11 11 1 0 ( , );m m r R u g t r            1 1 22 22 1, ( , );n n n m m r R u g t r              (18) and conjugate conditions Серія: Фізико-математичні науки. Випуск 14 95 1 1 2 2 1, 0; 1, 2; 1, . k k k k k j j km j j k m r R u u r r j k n                             (19) Let’s apply finite hybrid integral Hankel transform of 2nd kind rela- tive to the variable r in piecewise homogeneous segment nI  of n conju- gation points to the problem (16)–(19) [16]:   0 ( ) ( ) ( , ) ( ) ( ), R sn s s R H f r f r V r r rdr f     (20) 1 2 1 ( , ) ( ) ( ) ( ), ( , ) s sn s s s s V r H f f f r V r u               (21) 1 1 2 2 ( ) 1 [ ( )] ( ) ( ) ( , ) k k Rn sn m s s k k s k k R H B f r f f r V r rdr                   0 12 0 0 0 1 0 1 11 1 0 11 11( , )s r R df a R V R f dr               (22)   12 1 1 1 1 1 22 22 22 .n n n n n r R df a R f dr                 Spectral function ( , )sV r  , weight function ( )r and hybrid Bessel differential operator 1 2 ( ) 1 1 ( ) ( ) , km n m rk k k k B a r R R r B        written in [16], take part in formulas (20)–(22). Here 22 2 2 1 km kmB r rr r        is Bessel differential operator, ( )x is the Heaviside step function. Let’s write the system (16) and the initial conditions (17) in matrix form 1 2 1, 2 2 2 1 1 12 2 2 2 2 2 22 2 2 2 , 1 1 1,2 ( ) ( ) ( ) m m n m r m r m r n n n m a B q u t a B q u t a B q u t                                                  = 1 2 1, ( , , ) ( , , ) ( , , ) m m n m f t r f t r f t r                            , (23) Математичне та комп’ютерне моделювання 96 1 11 1 2 2 11, 1,0 ( , )( , , ) ( , , ) ( , ) ; ( , , ) ( , ) mm m m n m n mt g ru t r u t r g r u t r g r                                    2 11 2 2 2 21, 1,0 ( , )( , , ) ( , , ) ( , ) , ( , , ) ( , ) mm m m n m n mt g ru t r u t r g r t u t r g r                                     (24) here 2 2 2 2( ) ;j zj jq a    1, 1.j n  The integral operator snH is represented as an operator matrix-row due to the rule (20):   1 2 0 1 1 1 1 2 2 1 1 ( , ) ( , ) ( , ) ( , ) . n n n R R sn s s R R R R n s n n s n R R H V r rdr V r rdr V r rdr V r rdr                           . (25) Let’s apply the operator matrix-row (25) to the problem (23), (24) according to the matrix multiplication rule. As a result of the identity (22), we get a Cauchy problem     21 1 2 2 2 2 1 0 12 1 1 1 10 2 1 11 1 0 0 1 1 22 1 ( ) ( , , ) ( , , ) ( , ) ( , ) ( , ) ( , ), n n j j jm jm j j n s m n n n s m d q u t f t a R dt V R g t a R V R g t                                            (26) 1 1 1 1 10 1 1 2 1 10 ( , , ) ( , ); ( , , ) ( , ), n n jm s jm s j jt n n jm s jm s j jt u t g d u t g dt                             (27) here 1 ( , , ) ( , , ) ( , ) ; j j R jm s jm j s j R u t u t r V r rdr         1, 1,j n  1 ( , , ) ( , , ) ( , ) , j j R jm s jm j s j R f t f t r V r rdr          1, 1,j n  Серія: Фізико-математичні науки. Випуск 14 97 1 ( , ) ( , ) ( , ) ; j j R k k jm s jm j s j R g g r V r rdr         1, 2;k  1, 1.j n  Let’s suppose that  2 2 2 2 1 2 1 1max , , ..., nq q q q  and put everywhere 2 2 2 1 ;j jq q   1, 1j n  . Cauchy problem (26), (27) takes the form     2 12 2 0 1 0 1 11 1 0, 02 12 1 1 1 22 1 ( , ) ( , , ) ( ) ( , ) ( , ) ( , ), m s m m s s m n n n n s m d u u f t a R V R g t dt a R V R g t                             (28) 1 0 ( , , ) ( , );m s m s t u t g        2 0 ( , ),m m s t du g dt       (29) where 1 1 ( , , ) ( , , ); n m s jm s j u t u t          1 1 ( , , ) ( , , ), n m s jm s j f t f t          1 1 1 1 ( , ) ( , ), n m s jm s j g g          1 2 2 1 ( , ) ( , ), n m s jm s j g g          2 2 2 2 2 1 1( , ) .s s za        It is directly verify that the only solution of the inhomogeneous Cauchy problem (28), (29) is a function 2 1sin( ( , ) ) sin( ( , ) ) ( , , ) ( , ) ( , ) ( , ) ( , ) s s m s m s m s s s t td u t g g dt                             1 2 0 1 0 1 11 1 0, 0 sin( ( , )( )) ( , , ) ( ) ( , ) t s m s s s t f a R V R                  (30)   12 1 0 1 1 22 1( , ) ( , ) ( , ) .n m n n n s mg t a R V R g t d               Integral operator 1 snH  , as inverse to snH , we represent as the opera- tor matrix-column:   1 2 1 2 21 1 1 2 1 ( , ) ( , ) ( , ) ( , ) ( , ) ( , ) s s s s ssn s n s s s V r V r V r H V r V r V r                                                (31) Математичне та комп’ютерне моделювання 98 Let’s apply operator matrix-column (31) to the matrix-element ( , , )m su t       , where the function ( , , )m su t   is defined by formula (30) due to matrices multiplication rule. As a result we get the only solution of one- dimensional hyperbolic initial boundary problem of conjugation (16)–(19):                1 12 ),( ~~ ),( )),(sin( ),( ~~ ),( )),(sin( ),,(~ s sm s s sm s s jm g t t g t rtu          1 2 0 1 0 1 112 1 0 ( , ) sin( ( , )( )) ( , , ) ( , )( , ) t j s s m s sss V r t f a R V r                      (32)     12 1 1 0 0 1 1 22 1 2 ( , ) ( , ) ( , ) ( , ) ( , ) . , j sn s m n n n s m s V r V R g a R V R g d V r                     If to apply consistently inverse operators 1F  and 1 mF  to functions ( , , ),jmu t r  which are defined by formulas (32) and perform the some simple transformation, we get functions           ),,,(),,,,(),,,( 1 1 0 2 01   k n k t R R jkj fzrtEzrtu k k             ),,(),,,,( 1 1 1 2 01   k n k R R jkk gzrtE t dddd k k 1 21 2 1 0 ( , , , , ) ( , , ) k k Rn k jk k k R d d d E t r z g                          (33)   2 1 , 0 0 0 , , , ( , , ) t k j rd d d W t r z g                         2 , , , , ( , , ) ;j rW t r z g d d d             1, 1,j n  Functions (33) define the only solution of hyperbolic initial boundary problem of conjugation (1)–(5). In formulas (33) there are components 2 2 0 1 0 ( , , , , ) ( , ) ( , )sin( ( , ) )1 cos( ) cos( ) ( , )2 ( , ) jk j s k ss m sm s s E t r z V r Vt z d m V r                            of matrix of influence (function of influence), components 1 2 0 1 , 1 0 1 11 1 0( , , , ) ( ) ( , , , , )j r jW t r z a R E t r R z     of left radial Green’s Серія: Фізико-математичні науки. Випуск 14 99 matrix (left Green’s function) and components 2 , ( , , , )j rW t r z    12 1 1 1 22 , 1( , , , , )n n n j na R E t r R z       of right radial Green’s matrix (right Green’s function) of considered problem. Using a properties of functions of influence ( , , , , )jkE t r z  and ra- dial Green’s functions , ( , , , ), ( 1, 2)k j rW t r z k  we can verify that func- tions ( , , , )ju t r z which are defined by formulas (33) satisfy the equation (1), the initial conditions (2), the boundary conditions (3), (4) and conju- gate conditions (5) in the sense of theory of generalized functions [17]. The uniqueness of the solution (33) follows from its structure (inte- grated image) and from uniqueness of the main solutions (functions of influence and Green’s functions) of problem (1)–(5). By methods from [17, 18] can be proved that under appropriate con- ditions on the initial data, formulas (33) define a limited classical solution of the hyperbolic initial boundary problem of conjugation (1)–(5). We get the following theorem as the summary of the above results. Theorem. If functions ( , , , ),jf t r z 1 ( , , ),jg r z 2 ( , , )jg r z satisfy conditions: 1) are continuously differentiated twice for each variable; 2) have a limited variation for the geometric variables; 3) are absolutely summable with the variable z in ( ; )  ; 4) conjugate conditions are true and functions 0 ( , , ),g t z ( , , )Rg t z are continuously differentiated twice for each variable, have a limited variation for the geometric variables, are absolutely summable with the variable z in ( ; )  , then hyperbolic initial boundary value problem (1)-(5) has the only limited classical solution, which is determined by formula (33). Remark 1. In the case of 0rj j zj ja a a a    formulas (33) define the structure of the solution of hyperbolic initial boundary value problem (1)– (5) in an infinite isotropic piecewise homogeneous hollow cylinder. Remark 2. Parameters 0 0 11 11, ;  1 1 22 22,n n   allow to allocate the solutions of initial boundary value problems from formulas (33) in the case of boundary conditions of the 1st, 2nd and 3rd kind and their possible combinations on the radial surface 0 ,r R r R  . Remark 3. Analysis of the solution (33) is done directly from the general structure according to the analytical expression of functions ( , , , ),jf t r z ( , , ), ( 1, 2),k jg r z k  0 ( , , ),g t z ( , , )g t z . Математичне та комп’ютерне моделювання 100 Remark 4. In the case of 2 0j  equation (1) is a classic three- dimensional inhomogeneous wave equation (the equation of fluctuations) for an orthotropic environment in cylindrical coordinates. Remark 5. In the case of 11 110, 1;k k   12 0,k  12 1;k  21 1 ,k kE  21 0;k  22 2 ,k kE  22 0,k  here 1 ,kE 2 kE — Young's mod- ulus ( 1,k n ), the conjugate conditions (5) coincide with conditions of ideal mechanical contact. Thus, in these cases 4, 5 (at ( , , , ) 0jf t r z  ) considered hyperbolic boundary value problem (1)–(5) is a mathematical model of free oscillat- ing processes in unlimited piecewise homogeneous hollow cylinder. Conclusions. By means of method of integral and hybrid integral trans- forms with the method of principal solutions (influence functions and Green’s functions) integral image of exact analytical solution of hyperbolic boundary- value problem of mathematical physics in unlimited piecewise homogeneous hollow cylinder is obtained. The obtained solution is of algorithmic character, continuously depend on the parameters and data of problem and can be used in further theoretical research and in practical engineering calculations of real processes which are modeled by hyperbolic boundary-value problems of ma- thematical physics in piecewise homogeneous domains. References: 1. Hadamard J. The Cauchy problem for linear partial differential equations of parabolic type / J. Hadamard. — Moscow : Nauka, 1978. 2. Gording L. Cauchy's problem for hyperbolic equations / L. Gording. — Мoscow : IL, 1961. 3. Mytropol’skiy Yu. Asymptotic methods of investigation of quasi-wave equa- tions of hyperbolic type / Yu. Mytropol’skiy, G. Khoma, M. Gromiak. — Кyiv : Naukova Dumka, 1991. 4. Samoilenko A. Numerical-analytic methods in the theory of periodic solutions of partial differential equations / A. Samoilenko, B. Tkach. — Kyiv: Naukova Dumka, 1992. 5. Smirnov M. Degenerating elliptic and hyperbolic equations / M. Smirnov. — Мoscow : Nauka, 1962. 6. Cherniatyn V. Fourier method in mixed problem for partial differential equa- tions / V. Cherniatyn. — Мoscow : Izd. MGU, 1991. 7. Sergienko I. Mathematic modeling and the study of processes in heterogeneous environments / I. Sergienko, V. Skopetsky, V. Deineka. — Kyiv : Naukova Dumka, 1991. 8. Deineka V. Models and methods of solving of problems with conjugate conditions / V. Deineka, I. Sergienko, V. Skopetsky. — Kyiv : Naukova Dumka, 1998. 9. Konet I. The temperature fields in the piece-homogeneous cylindrical domains / I. Konet, M. Leniuk. — Chernivtsi : Prut, 2004. Серія: Фізико-математичні науки. Випуск 14 101 10. Gromyk A. The temperature fields in the piece-homogeneous spatial environments / A. Gromyk, I. Konet, M. Leniuk. — Kamenets-Podilsky : Abetka-Svit, 2011. 11. Konet I. Hyperbolic boundary-value problems of mathematical physics in piecewise homogeneous spacial environments / I. Konet. — Kamenets- Podilsky : Abetka-Svit, 2013. 12. Konet I. Parabolic boundary value problems in piecewise homogeneous envi- ronments / I. Konet, T. Pylypiuk. — Kamenets-Podilsky : Abetka-Svit, 2016. 13. Perestiuk M. The theory of equations of mathematical physics / M. Perestiuk, V. Marynets’. — Kyiv : Lybid’, 2006. 14. Sneddon I. Fourier transforms / I. Sneddon. — Мoscow : IL, 1955. 15. Тranter К. Integral transformations in mathematical physics / К. Тranter. — Мoscow : Gostehteorizdat, 1956. 16. Bybliv O. Integral Hankel transform of the 2nd kind for piecewise- homogeneous segments / O. Bybliv, M. Lenyuk // Izv. vuzov. Маthematics. — 1987. — № 5. — P. 82–85. 17. Shilov G. Mathematical analysis. Second special course / G. Shilov. — Mos- cow : Nauka, 1965. 18. Gelfand I. Some questions in the theory of differential equations / I. Gelfand, G. Shilov. — Мoscow : Fizmatgiz, 1958. Методом інтегральних і гібридних інтегральних перетворень у по- єднанні з методом головних розв’язків (функцій впливу та функцій Гріна) вперше побудовано інтегральне зображення точного аналітич- ного розв'язку гіперболічної крайової задачі математичної фізики для необмеженого кусково-однорідного порожнистого циліндра. Ключові слова: гіперболічне рівняння, початкові та крайові умо- ви, умови спряження, інтегральні перетворення, головні розв’язки. 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