4D Separated Multi-point Boundary Problem for the ...

4D Separated Multi-point Boundary Problem for the General Case in the Non-Classical Treatment for a Higher-Order Hyperbolic Equation with Dominating Mixed Derivative

Ilgar Gurbat oglu Mamedov

https://scholar.google.com/citations?user=RPILyO8AAAAJ&hl=en

http://www.researcherid.com/rid/E-9473-2015

http://orcid.org/0000-0001-6354-1371

https://www.researchgate.net/profile/Ilgar_Mamedov3

http://www.mathnet.ru/rus/person27537

Institute of Control Systems Azerbaijan National Academy of Sciences http://isi.az/en/77-mamedov-ilgar.html

B. Vahabzade St.9, Baku city, AZ 1141, Azerbaijan Republic

Email address: [email protected] Abstract In this paper substantiated for a higher-order equation with non-smooth coefficients a four dimensional multi-point boundary problem -4 D separated multi-point boundary problem for the general case with non-classical boundary conditions is considered, which requires no matching conditions. Equivalence of these conditions four dimensional boundary condition is substantiated classical, in the

case if the solution of the problem in the anisotropic S. L. Sobolev's space is found. The considered equation as a hyperbolic equation even in particular e.g. in the two-dimensional case generalizes not only classic equations of mathematical physics (heat-conductivity equations, string vibration equation) and also many models differential equations (telegraph equation, Aller's equation , moisture transfer generalized equation, generalized Manjeron equation, Boussinesq - Love equation and etc.). It is grounded that the 4D multi-point boundary conditions for the general case in the classic and non-classic treatment are equivalent to each other. Thus, namely in this paper, the non-classic problem with 4D separated multi-point boundary conditions is grounded for a hyperbolic equation of higher- order. For simplicity, this was demonstrated for the general case in one of S.L. Sobolev anisotropic space Wpl1 ,l2 ,l3 ,l4  G . Key words : 4D separated multi-point boundary problem for the general case, higher-order hyperbolic equations, equations with non-smooth coefficients, equations with dominating mixed derivative.

1. Introduction Hyperbolic equations are attracted for sufficiently adequate description of a great deal of real processes occurring in the nature, engineering and etc. In particular, many processes arising in the theory of fluid filtration in cracked media are described by non-smooth coefficient hyperbolic equations. Urgency of investigations conducted in this field is explained by appearance of local and non-local problems for non-smooth coefficients equations connected with different applied problems. Such type problems arise for example, while studying the problems of moisture, transfer in soils, heat transfer in heterogeneous media, diffusion of thermal neutrons in inhibitors, simulation of different biological processes, phenomena and etc. In the present paper, for the general case here consider four dimensional separated multi-point boundary problem- 4D separated multi-point boundary problem for higher- order hyperbolic equation with dominating mixed derivative. The coefficients in this hyperbolic equation are not necessarily differentiable; therefore, there does not exist a formally adjoint differential equation making a certain sense. For this reason, this question cannot be investigated by the well-known methods using classical integration by parts and Riemann functions or classical-type fundamental solutions. The theme of the present paper, devoted to the investigation 4D separated multi-point boundary problem for higherorder differential equation with dominating mixed derivative of hyperbolic type, according to the above-stated is very actual for the solution of theoretical and practical problems. From this point of view, the paper even in particular, in the two-dimensional and three-dimensional case is devoted to the actual problems of applied mathematics and physics. 2. Problem Statement For the general case consider higher-order equation

V

l1 ,l 2 ,l3 ,l 4 l1

l2



u x   D1l1 D2l2 D3l3 D4l4 u x   l3

l4

     a j1 , j2 , j3 , j4 x D1j1 D2j2 D3j3 D4j4 u x   Z l1 ,l2 ,l3 ,l4 x  L p G  j1  0 j2  0 j3  0 j4  0

,

(1)

j1  j2  j3  j4 l1  l 2  l3  l 4

Here u x   u x1 , x2 , x3 , x4  is a desired function determined on G; a j1 , j2 , j3 , j4  x  are the given

measurable functions on G  G1  G2  G3  G4 , where Gk  0, hk , k  1,4; Zl1 ,l2 ,l3 ,l4 x is a given measurable function on G ; Dk    / x k is a generalized differentiation operator in S.L.Sobolev sense, Dk0 is an identity transformation operator. Equation (1) is a hyperbolic equation with four multiple real characteristics xk  const, k  1,4. Therefore, in some sense we can consider equation (1) as a pseudoparabolic equation [1]. This equation is a Boussinesq - Love generalized equation from the vibrations theory [2] and Aller's equation under mathematical modeling [3, p.261] of the moisture absorption process in biology. In the present paper equation (1) is considered in the general case when the coefficients a j1 , j2 , j3 , j4  x  are non-smooth functions satisfying only the following conditions: a j1 , j2 , j3 , j4  x   L p G , al1 , j2 , j3 , j4 x Lx1,, xp2, ,px,3p, x4 G, a j1 ,l2 , j3 , j4 x Lxp1,,x2, ,px,3p, x4 G, a j1 , j2 ,l3 , j4 x Lxp1,,px2,, x,3p, x4 G,

a j1 , j2 , j3 ,l4 x Lxp1,,px2, p, x,3, x4 G, al1 ,l2 , j3 , j4 x Lx1,,x2, ,px,3p, x4 G, al1 , j2 ,l3 , j4 x Lx1,, xp2,, x,3p, x4 G, al1 , j2 , j3 ,l4 x Lx1,, xp2, ,px,3, x4 G,

a j1 ,l2 ,l3 , j4  x   Lxp1,,x2,, x,3p, x4 G , a j1 ,l2 , j3 ,l4  x   Lxp1,,x2, ,px,3, x4 G , a j1 , j2 ,l3 ,l4  x   Lxp1,,px2,, x,3, x4 G , al1 ,l2 ,l3 , j4  x   Lx1,,x2,, x,3p, x4 G , al1 ,l2 , j3 ,l4  x   Lx1,,x2, ,px,3, x4 G , a j1 ,l2 ,l3 ,l4  x   Lxp1,,x2,, x,3, x4 G ,

al1 , j2 ,l3 ,l4 x Lx1,, xp2,, x,3, x4 G, jk  0, lk 1, k  1,4; Under these conditions, we'll look for the solution u  x  of equation (1) in S.L.Sobolev anisotropic space





Wpl1 ,l2 ,l3 ,l4  G  ux : D1j1 D2j2 D3j3 D4j4 ux Lp G, jk  0, lk , k  1,4 ,

where 1  p   . We'll define the norm in the space Wpl1 ,l2 ,l3 ,l4  G by the equality l1

l2

l3

l4

u W l1 ,l2 ,l3 ,l4  G       D1j1 D2j2 D3j3 D4j4 u p

j1 0 j2 0 j3 0 j4 0

L p G 

For hyperbolic equation (1) we can give the classic form 4D separated multi-point boundary conditions in the form :   D j1 u ( x)  1jk1  x2 , x3 , x4 ,  x2 , x3 , x4   G2  G3  G4 , j1  0, l1  1; 1 k x  x  1 1  j2 j2  D2 u ( x) x2  x2k   2 k  x1 , x3 , x4 , x1 , x3 , x4   G1  G3  G4 , j2  0, l2  1;  j3 j  D3 u ( x) x  x k  3k3  x1 , x2 , x4 ,  x1 , x2 , x4   G1  G2  G4 , j3  0, l3  1; 3 3  j4  D4 u ( x) k   4jk4  x1 , x2 , x3 ,  x1 , x2 , x3   G1  G2  G3 , j4  0, l4  1; x4  x4  

(2)

where 1jk1  x2 , x3 , x4 ,  2jk2  x1 , x3 , x4 , 3jk3  x1 , x2 , x4  and  4jk4 x1 , x2 , x3  are the given measurable

functions on G . Notice that here we assume x k  x1k , x2k , x3k , x4k , k  1,2,..., N are the fixed points from



G . It is obvious that in the case of conditions (2), in addition to the conditions

1jk x2 , x3 , x4 Wpl 1

2 , l3 , l 4



G2  G3  G4 , j1  0, l1 1;

2jk x1 , x3 , x4 Wpl ,l ,l  G1  G3  G4 , j2  0, l2 1; 1 3 4

2

3jk x1 , x2 , x4   Wpl1 ,l2 ,l4  G1  G2  G4 , j3  0, l3 1; 3

and

4jk x1, x2 , x3 Wpl ,l 4

1

2 ,l3



G1  G2  G3 , j4  0, l4  1;

the given functions should also satisfy the following agreement conditions:

 D2j 2 1jk1  x2 , x3 , x4  k  D1j12jk2  x1 , x3 , x4  k , x2  x2 x1  x1  j j  D 3 j1  x , x , x   D1j13k3 x1 , x2 , x4  k , x1  x1  3 1k 2 3 4 x3  x3k  j1 j 4 j 4 j1  D1  4 k  x1 , x2 , x3  x1  x1k  D4 1k  x2 , x3 , x4  x4  x4k ,  j j j j  D3 3 2 k2  x1 , x3 , x4  x  x k  D2 2 3k3  x1 , x2 , x4  x  x k , 3 3 2 2  j4 j2 j2 j4  D4  2 k  x1 , x3 , x4  k  D2  4 k x1 , x2 , x3  k , x4  x4 x2  x2   D j 4  j3  x , x , x   D3j34jk4  x1 , x2 , x3  k x3  x3  4 3k 1 2 4 x4  x4k   jk  0, lk  1, k  1,4. Consider the following non-classical boundary conditions :

V jk1 , j2 , j3 , j4 u  D1j1 D2j2 D3j3 D4j4 u( x1k , x2k , x3k , x4k ) = Z kj1 , j2 , j3 , j4 , jk  0,...,lk  1, k  1,...,4;

V

k l1 , j2 , j3 , j4



u x1   D1l1 D2j2 D3j3 D4j4 u( x1 , x2k , x3k , x4k )  Zlk1 , j2 , j3 , j4 ( x1 ),

j2  0,..., l2  1, j3  0,..., l3  1, j4  0,..., l4  1;

V

k j1 ,l2 , j3 , j4



u x2   D1j1 D2l2 D3j3 D4j4 u( x1k , x2 , x3k , x4k )  Z kj1 ,l2 , j3 , j4 ( x2 )

j1  0,..., l1  1, j3  0,..., l3  1, j4  0,..., l4  1;

V

k j1 , j2 ,l3 , j4



u x3   D1j1 D2j2 D3l3 D4j4 u( x1k , x2k , x3 , x4k )  Z kj1 , j2 ,l3 , j4 ( x3 ),

j1  0,..., l1  1, j2  0,..., l2  1, j4  0,..., l4  1;

V

k j1 , j2 , j3 ,l4



u x4   D1j1 D2j2 D3j3 D4l4 u( x1k , x2k , x3k , x4 )  Z kj1 , j2 , j3 ,l4 ( x4 ),

j1  0,..., l1  1, j2  0,..., l2  1, j3  0,..., l3  1;

V

k l1 ,l2 , j3 , j4



u x1, x2   D1l1 D2l2 D3j3 D4j4 u( x1, x2 , x3k , x4k )  Zlk1 ,l2 , j3 , j4 ( x1 , x2 ),

(3)

j3  0,..., l3  1, j4  0,..., l4  1;

V

k l1 , j2 ,l3 , j4



u x1, x3   D1l1 D2j2 D3l3 D4j4 u( x1, x2k , x3 , x4k )  Zlk1 , j2 ,l3 , j4 ( x1 , x3 ),

j2  0,..., l2  1, j4  0,..., l4  1;

V

k l1 , j2 , j3 ,l4



u x1, x4   D1l1 D2j2 D3j3 D4l4 u( x1, x2k , x3k , x4 )  Zlk1 , j2 , j3 ,l4 ( x1 , x4 ),

j2  0,..., l2  1, j3  0,..., l3  1;

V

k j1 ,l2 ,l3 , j4

u x2 , x3   D1j1 D2l2 D3l3 D4j4 u( x1k , x2 , x3 , x4k )  Z kj1 ,l2 ,l3 , j4 ( x2 , x3 ), j1  0,..., l1  1, j4  0,..., l4  1;



V

j1 ,l2 , j3 ,l4

u x2 , x4   D1j1 D2l2 D3j3 D4l4 u( x1k , x2 , x3k , x4 )  Z kj1 ,l2 , j3 ,l4 ( x2 , x4 ),



j1  0,..., l1  1, j3  0,..., l3  1;

V

k j1 , j2 ,l3 ,l4



u x3 , x4   D1j1 D2j2 D3l3 D4l4 u( x1k , x2k , x3 , x4 )  Z kj1 , j2 ,l3 ,l4 ( x3 , x4 ),

j1  0,..., l1  1, j2  0,..., l2  1;

V

k l1 ,l2 ,l3 , j4



u x1, x2 , x3   D1l1 D2l2 D3l3 D4j4 u( x1, x2 , x3 , x4k )  Zlk1 ,l2 ,l3 , j4 ( x1 , x2 , x3 ),

j4  0,..., l4  1;

V

k l1 ,l2 , j3 ,l4



u x1, x2 , x4   D1l1 D2l2 D3j3 D4l4 u( x1, x2 , x3k , x4 )  Zlk1 ,l2 , j3 ,l4 ( x1 , x2 , x4 ),

j3  0,..., l3  1;

V

k j1 ,l2 ,l3 ,l4



u x2 , x3 , x4   D1j1 D2l2 D3l3 D4l4 u( x1k , x2 , x3 , x4 )  Z kj1 ,l2 ,l3 ,l4 ( x2 , x3 , x4 ),

j1  0,..., l1  1;

V

k l1 , j2 ,l3 ,l4



u x1 , x3 , x4   D1l1 D2j2 D3l3 D4l4 u( x1, x2k , x3 , x4 )  Zlk1 , j2 ,l3 ,l4 ( x1 , x3 , x4 ),

j2  0,..., l2  1;

(4)

3. Methodology There with, the important principal moment is that the considered equation possesses nonsmooth coefficients satisfying only some p -integrability and boundedness conditions i.e. the considered hyperbolic operator Vl1 ,l2 ,l3 ,l4 has no traditional conjugated operator. In other words, the Riemann function for this equation can't be investigated by the classical method of characteristics. In the papers [4-6] the Riemann function is determined as the solution of an integral equation. This is more natural

than the classical way for deriving the Riemann function. The matter is that in the classic variant, for determining the Riemann function, the rigid smooth conditions on the coefficients of the equation are required. The Riemann’s method does not work for hyperbolic equations with non-smooth coefficients. In the present paper, a method that essentially uses modern methods of the theory of functions and functional analysis is worked out for investigations of such problems. In the main, this method it requested in conformity to hyperbolic equations of higher- order with three multiple characteristics. Notice that, in this paper the considered equation even in particular e.g. in the two-dimensional case is a generation of many model equations of some processes (for example, heat-conductivity equations , telegraph equation, Aller's equation , moisture transfer generalized equation, Manjeron equation, string vibrations equations and etc). If the function u Wpl1 ,l2 ,l3 ,l4  G  is a solution of the classical form 4D separated multi-point boundary problem (1), (2), then it is also a solution of problem (1), (4) for Z kj1 , j2 , j3 , j4 , defined by the following equalities: Z kj1 , j2 , j3 , j4 = D2j2 D3j3 D4j4 1jk1 x2 , x3 , x4  x2  x2kk  D1j1 D3j3 D4j4  2jk2 x1 , x3 , x4  x1  x1kk  x3  x 3 x 4  x 4k

x3  x 3 x 4  x 4k

= D1 1 D2 2 D4 4 3k3  x1 , x2 , x4  x1  x1k = D1 1 D2 2 D3 3  4 k4  x1 , x2 , x3  x1  x1k , jk  0,..., lk  1, k  1,..., 4; j

j

j

j

j

k

j

j

j

k

x2  x2 x 4  x 4k

x2  x2 x3  x 3k

Zlk1 , j2 , j3 , j4 ( x1 ) = D1l1 D3j3 D4j4 2jk2 x1 , x3 , x4  x3  x3k = D1l1 D2j2 D4j4 3jk3 x1 , x2 , x4  x2  x2k = x4  x4k

x4  x4k

=

D1l1 D2j2 D3j34jk4 x1 , x2 , x3  x2  x2k , j2  0,..., l2  1, j3  0,..., l3  1, j4  0,..., l4  1; x3  x3k

Z kj1 ,l2 , j3 , j4 ( x2 ) = D2l2 D3j3 D4j4 1jk1 x2 , x3 , x4  x3  x3k = D1j1 D2l2 D4j4 3jk3 x1 , x2 , x4  x1  x1k = = D1 1 D22 D3 34 k4 j

l

j

j

x1 , x2 , x3  x  x 1

k 1

x3  x3k

x4  x4k

x4  x4k

, j1  0,..., l1  1, j3  0,..., l3  1, j4  0,..., l4  1;

Z kj1 , j2 ,l3 , j4 ( x3 ) = D2j2 D3l3 D4j4 1jk1 x2 , x3 , x4  x2  x2k = D1j1 D3l3 D4j4 2jk2 x1 , x3 , x4  x1  x1k = = D1 1 D2 2 D334 k4 j

j

l

j

x1 , x2 , x3  x  x 1

k 1

x2  x2k

x4  x4k

x4  x4k

, j1  0,..., l1  1, j2  0,..., l2  1, j4  0,..., l4  1;

Z kj1 , j2 , j3 ,l4 ( x4 ) = D2j2 D3j3 D4l4 1jk1 x2 , x3 , x4  x2  x2k  D1j1 D3j3 D4l4 2jk2 x1 , x3 , x4  x1  x1k = = D1 1 D2 2 D44 3k3 j

j

l

j

x1 , x2 , x4  x  x 1

k 1

x2  x2k

x3  x3k

x3  x3k

, j1  0,..., l1  1, j2  0,..., l2  1, j3  0,..., l3  1;

Zlk1 ,l2 , j3 , j4 ( x1 , x2 ) = D1l1 D2l2 D4j4 3jk3 x1 , x2 , x4  j3  0,..., l3  1, j4  0,..., l4  1;

Zlk1 , j2 ,l3 , j4 ( x1 , x3 ) = D1l1 D3l3 D4j4  2jk2 x1 , x3 , x4 

j2  0,..., l2  1, j3  0,..., l3  1;

Z kj1 ,l2 ,l3 , j4 ( x2 , x3 ) = D2l2 D3l3 D4j4 1jk1 x2 , x3 , x4 

l

j

x3  x3k

j

x 2  x 2k

= D11 D2 2 D44 3k3  x1 , x2 , x4  j

j

l

= D1 1 D22 D33  4 k4  x1 , x2 , x3  j

x4  x4k

j

= D11 D2 2 D33 4 k4  x1 , x2 , x3 

l

x3  x3k

j

l

l

x4  x4k

j2  0,..., l2  1, j4  0,..., l4  1;

Zlk1 , j2 , j3 ,l4 ( x1 , x4 ) = D1l1 D3j3 D4l4  2jk2 x1 , x3 , x4 

= D11 D22 D3 3 4 k4  x1 , x2 , x3  l

x4  x4k

l

l

x2  x2k

j

x1  x1k

,

,

,

,

j1  0,..., l1  1, j4  0,..., l4  1;

Z kj1 ,l2 , j3 ,l4 ( x2 , x4 ) = D2l2 D3j3 D4l4 1jk1 x2 , x3 , x4  j1  0,..., l1  1, j3  0,..., l3  1;

Z kj1 , j2 ,l3 ,l4 ( x3 , x4 ) = D2j2 D3l3 D4l4 1jk1 x2 , x3 , x4 

= D1 1 D22 D44 3k3  x1 , x2 , x4  j

x3  x3k

l

j

= D1 1 D33 D44  2 k2  x1 , x3 , x4  j

x2  x2k

l

l

l

x1  x1k

j

x1  x1k

,

,

j1  0,..., l1  1, j2  0,..., l2  1;

Zlk1 ,l2 ,l3 , j4 ( x1, x2 , x3 ) = D1l D2l D3l  4jk  x1 , x2 , x3  , j4  0,..., l4  1; 1

3

2

4

Zlk1 ,l2 , j3 ,l4 ( x1, x2 , x4 ) = D1l D2l D4l  3jk  x1 , x2 , x4  , j3  0,..., l3  1; 1

2

3

4

Z kj1 ,l2 ,l3 ,l4 ( x2 , x3 , x4 ) = D2l D3l D4l 1jk  x2 , x3 , x4 , j1  0,..., l1  1; 2

3

4

1

Zlk1 , j2 ,l3 ,l4 ( x1, x3 , x4 ) = D1l D3l D4l  2jk  x1 , x3 , x4  , j2  0,..., l2  1; 1

3

4

2

The inverse one is easily proved. In other words, if the function u Wpl1 ,l2 ,l3 ,l4  G  is a solution of problem (1), (4), then it is also a solution of problem (1), (2) for the following functions: 1jk1  x2 , x3 , x4  = l2 1 l3 1 l4 1

( x2  x2k ) j2 ( x3  x3k ) j3 ( x4  x4k ) j4 k =  Z j1 , j2 , j3 , j4  j2 ! j3! j4 ! j2 0 j3 0 j4 0 l3 1 l4 1

+

( x3  x3k ) j3 ( x4  x4k ) j4   j! j4 ! j3  0 j4  0 3

( x2   2 )l2 1 k k (l2  1)! Z j1 ,l2 , j3 , j4 ( 2 )d 2  x x2

2

( x3   3 )l3 1 k k (l3  1)! Z j1 , j2 ,l3 , j4 ( 3 )d 3  x

l2 1 l4 1



k 2

( x2  x ) ( x 4  x ) j2 ! j4 ! j2  0 j4  0

j4 x3

l2 1 l3 1

k 2

j3 x4

j2

k 4

( x2  x ) ( x3  x ) j2 ! j3! j2  0 j3  0

  l 4 1

( x  x k ) j4  4 4 j4 ! j4  0

j2

k 3

( x4   4 )l4 1 k k (l4  1)! Z j1 , j2 , j3 ,l4 ( 4 )d 4  x 4

( x2   2 )l2 1 ( x3   3 )l3 1 k   (l2  1)! (l3  1)! Z j1 ,l2 ,l3 , j4 ( 2 , 3 )d 2 d 3  xk xk x2 x3

2

3

( x2   2 )l2 1 ( x4   4 )l4 1 k   (l2  1)! (l4  1)! Z j1 ,l2 , j3 ,l4 ( 2 , 4 )d 2 d 4  xk xk

l 3 1



k 3

( x3  x ) j3! j3  0

j3 x2 x4

l 2 1

k 2

j2 x3 x4

2

( x2  x ) j2 ! j2  0



3

4

( x3   3 )l3 1 ( x4   4 )l4 1 k   (l3  1)! (l4  1)! Z j1 , j2 ,l3 ,l4 ( 3 , 4 )d 3d 4  xk xk 3

4

( x2   2 ) ( x3   3 )l3 1 ( x4   4 )l4 1 k Z j1 ,l2 ,l3 ,l4 ( 2 , 3 , 4 )d 2 d 3d 4 , j1  0, l1  1; (l2  1)! (l3  1)! (l4  1)! xk xk xk l2 1

x2 x3 x4

 2

3

4

l1 1 l3 1 l4 1

( x1  x1k ) j1 ( x3  x3k ) j3 ( x4  x4k ) j4 k Z j1 , j2 , j3 , j4  j1! j3! j4 ! j1 0 j3 0 j4 0

 2jk  x1 , x3 , x4  =    2

( x3  x3k ) j3 ( x4  x4k ) j4   j! j4 ! j3  0 j4 0 3

( x1   1 )l1 1 k k (l1  1)! Zl1 , j2 , j3 , j4 ( 1 )d 1  x

l1 1 l4 1

( x3   3 )l3 1 k k (l3  1)! Z j1 , j2 ,l3 , j4 ( 3 )d 3  x

l3 1 l4 1

+

+

( x1  x ) ( x4  x )   j1! j4 ! j1  0 j4  0 k 1

j1

k 4

x1

1

j4 x3

3

(5)

l1 1 l3 1

+

+

+

( x1  x1k ) j1 ( x3  x3k ) j3   j! j3! j1  0 j3  0 1

( x4   4 )l4 1 k k (l4  1)! Z j1 , j2 , j3 ,l4 ( 4 )d 4  x x4

4

( x1   1 ) ( x3   3 )l3 1 k k k (l1  1)! (l3  1)! Z l1 , j2 ,l3 , j4 ( 1 , 3 )d 1d 3  x x

l 4 1

k 4

( x4  x )  j4 ! j4  0

j4 x1 x3

l 3 1

k 3

( x3  x )  j3! j3  0

j3 x1 x4

l 11

k 1

j1 x3 x4

1

l1 1

3

( x1   1 )l1 1 ( x4   4 )l4 1 k k k (l1  1)! (l4  1)! Zl1 , j2 , j3 ,l4 ( 1 , 4 )d 1d 4  x x 1

4

( x3   3 )l3 1 ( x4   4 )l4 1 k k k (l3  1)! (l4  1)! Z j1 , j2 ,l3 ,l4 ( 3 , 4 )d 3d 4  x x

+

( x1  x )  j1! j1  0

+

( x1   1 ) ( x3   3 )l3 1 ( x4   4 )l4 1 k k k k (l1  1)! (l3  1)! (l4  1)! Zl1 , j2 ,l3 ,l4 ( 1 , 3 , 4 )d 1d 3d 4 , j2  0, l2  1; x x x

3

1

3

4

l1 1

x1 x3 x4

(6)

4

l1 1 l2 1 l4 1

( x1  x1k ) j1 ( x2  x2k ) j2 ( x4  x4k ) j4 k Z j1 , j2 , j3 , j4  j1! j2 ! j4 ! j1 0 j2 0 j4 0

 3jk  x1 , x2 , x4  =    3

l2 1 l4 1

( x2  x2k ) j2 ( x4  x4k ) j4 + j2 ! j4 ! j2  0 j4  0 l1 1 l4 1

( x  x ) ( x4  x ) + 1 j1! j4 ! j1  0 j4  0 j1

k 1

l1 1 l2 1

k 4

( x  x ) ( x2  x ) + 1 j1! j2 ! j1  0 j2  0 j1

k 1

l 4 1

k 4

l 2 1

k 2

l 11

k 1

(x  x ) + 4 j4 ! j4  0 (x  x ) + 2 j2 ! j2  0 (x  x ) + 1 j1! j1  0

k 2

( x1   1 )l1 1 k  (l1  1)! Zl1 , j2 , j3 , j4 ( 1 )d 1  xk x1

1

( x2   2 )l2 1 k  (l2  1)! Z j1 ,l2 , j3 , j4 ( 2 )d 2  xk

j 4 x2

2

( x4   4 )l4 1 k  (l4  1)! Z j1 , j2 , j3 ,l4 ( 4 )d 4  xk

j 2 x4

4

( x1   1 ) ( x2   2 )l2 1 k   (l1  1)! (l2  1)! Zl1 ,l2 , j3 , j4 ( 1 , 2 )d 1d 2  xk xk l1 1

j4 x1 x2

1

2

( x1   1 )l1 1 ( x4   4 )l4 1 k   (l1  1)! (l4  1)! Zl1 , j2 , j3 ,l4 ( 1 , 4 )d 1d 4  xk xk

j2 x1 x4

1

4

( x2   2 )l2 1 ( x4   4 )l4 1 k   (l2  1)! (l4  1)! Z j1 ,l2 , j3 ,l4 ( 2 , 4 )d 2d 4  xk xk

j1 x2 x4

2

4

( x1   1 ) ( x2   2 )l2 1 ( x4   4 )l4 1 k Z l1 ,l2 , j3 ,l4 ( 1 , 2 , 4 )d 1d 2 d 4 , j3  0, l3  1; + (l1  1)! (l2  1)! (l4  1)! xk xk xk x1 x2 x4

1

2

l1 1

4

l1 1 l2 1 l3 1

( x1  x1k ) j1 ( x2  x2k ) j2 ( x3  x3k ) j3 k Z j1 , j2 , j3 , j4  j1! j2 ! j3! j1 0 j2 0 j3 0

 4jk x1 , x2 , x3  =    4

l2 1 l3 1

( x2  x2k ) j2 ( x3  x3k ) j3 + j2 ! j3! j2 0 j3  0 +

( x1   1 )l1 1 k  (l1  1)! Zl1 , j2 , j3 , j4 ( 1 )d 1  xk x1

1

( x2   2 )l2 1 k  (l2  1)! Z j1 ,l2 , j3 , j4 ( 2 )d 2  xk

l1 1 l3 1



k 1

( x1  x ) ( x3  x ) j1! j3! j1  0 j3  0

j3 x2

l1 1 l2 1

k 1

j2 x3

j1

k 3

( x  x ) ( x2  x ) + 1 j1! j2 ! j1  0 j2  0 j1

k 2

2

( x3   3 )l3 1 k  (l3  1)! Z j1 , j2 ,l3 , j4 ( 3 )d 3  xk 3

(7)

( x3  x3k ) j3  j3! j3  0

( x1   1 )l1 1 ( x2   2 )l2 1 k k k (l1  1)! (l2  1)! Zl1 ,l2 , j3 , j4 ( 1 , 2 )d 1d 2  x x

l 2 1

k 2

( x1   1 )l1 1 ( x3   3 )l3 1 k k k (l1  1)! (l3  1)! Z l1 , j2 ,l3 , j4 ( 1 , 3 )d 1d 3  x x

l 11

k 1

l 3 1

+

+

( x2  x )  j2 ! j2  0

x1 x2

1

2

j2 x1 x3

1

3

( x2   2 )l2 1 ( x3   3 )l3 1 k k k (l2  1)! (l3  1)! Z j1 ,l2 ,l3 , j4 ( 2 , 3 )d 2 d 3  x x

+

( x1  x )  j1! j1  0

+

( x1   1 ) ( x2   2 )l2 1 ( x3   3 )l3 1 k k k k (l1  1)! (l2  1)! (l3  1)! Zl1 ,l2 ,l3 , j4 ( 1 , 2 , 3 )d 1d 2 d 3 , j4  0, l4  1; x x x

j1 x2 x3

2

1

2

3

l1 1

x1 x2 x3

(8)

3

Note that the functions (5)-(8) possess one important property, more exactly, for all Z kj1 , j2 , j3 , j4 , the agreement conditions (3) possessing the above-mentioned properties are fulfilled for them automatically. Therefore, equalities (5)-(8) may be considered as a general kind of all the functions 1jk1  x 2 , x 3 , x 4 ,  2jk2 x1 , x3 , x 4 , 3jk3 x1 , x2 , x4  and  4jk4 x1 , x2 , x3  satisfying the agreement conditions (3). We have thereby proved the following assertion. Theorem. The 4D separated multi-point boundary problems of the form (1), (2) and the non-classical form (1), (4) are equivalent. Note that the 4D separated multi-point boundary problem in the non-classical treatment (1), (4) can be studied with the use of multi-point integral representations of special form for the functions u Wpl1 ,l2 ,l3 ,l4  G [7-14], u x  

l1 1 l 2 1 l3 1 l 4 1

( x1  x1k ) j1 ( x2  x2k ) j2 ( x3  x3k ) j3 ( x4  x4k ) j4 j1 j2 j3 j4 D1 D2 D3 D4 u ( x1k , x2k , x3k , x4k )      j1! j2 ! j3 ! j4 ! j1  0 j 2  0 j3  0 j 4  0

l 2 1 l3 1 l 4 1

( x  x2k ) j2 ( x3  x3k ) j3 ( x4  x4k ) j4   2 j2 ! j3 ! j4 ! j 2  0 j3  0 j 4  0 l1 1 l3 1 l 4 1

( x1  x ) ( x3  x ) j1! j3 ! j1  0 j3  0 j 4  0 k 1

 l1 1 l2 1 l4 1

k 1

l1 1 l2 1 l3 1

k 1

j1

( x  x ) ( x2  x )  1 j1! j2 ! j1  0 j2  0 j4  0 j1

k 2

( x  x ) ( x2  x )  1 j1! j2 ! j1  0 j2  0 j3  0 l3 1 l4 1

k 3

l2 1 l4 1

k 2

j1

k 2

( x  x ) ( x4  x )  3 j3! j4 ! j3  0 j 4  0 (x  x )  2 j2 ! j2  0 j4  0

j3

j2

k 4

( x4  x ) j4 ! k 4

( x4  x ) j4 !

( x2   2 ) l2 1 j1 l2 j3 i4 k k k k (l2  1)! D1 D2 D3 D4 u ( x1 , 2 , x3 , x4 )d 2  x

j3

j2

( x4  x ) j4 !

j4 x3

j2

( x3  x ) j3!

j3 x4

k 4

k 3

1

j4 x2

k 3

k 4

( x1   1 ) l1 1 l1 j2 j3 i4 k k k k (l1  1)! D1 D2 D3 D4 u ( 1 , x2 , x3 , x4 )d 1  x x1

2

( x3   3 )l3 1 j1 j2 l3 j4 k k k k (l3  1)! D1 D2 D3 D4 u( x1 , x2 , 3 , x4 )d 3  x 3

( x4   4 )l4 1 j1 j2 j3 l4 k k k k (l4  1)! D1 D2 D3 D4 u( x1 , x2 , x3 , 4 )d 4  x 4

( x1   1 ) ( x2   2 )l2 1 l1 l2 j3 j4 k k k k (l1  1)! (l2  1)! D1 D2 D3 D4 u( 1 , 2 , x3 , x4 )d 1d 2  x x

j4 x1 x2

1

l1 1

2

( x1   1 )l1 1 ( x3   3 )l3 1 l1 j2 l3 j4 k k k k (l1  1)! (l3  1)! D1 D2 D3 D4 u( 1 , x2 , 3 , x4 )d 1d 3  x x

j4 x1 x3

1

3

( x2  x2k ) j2 ( x3  x3k ) j3 j2 ! j3! j 2  0 j3  0

( x1   1 ) l1 1 ( x4   4 ) l4 1 l1 j2 j3 l4 k k k k (l1  1)! (l4  1)! D1 D2 D3 D4 u ( 1 , x2 , x3 , 4 )d 1d 4  x x

l1 1 l 4 1

( x2   2 ) l2 1 ( x3   3 ) l3 1 j1 l2 l3 j4 k k k k (l2  1)! (l3  1)! D1 D2 D3 D4 u ( x1 , 2 , 3 , x4 )d 2 d 3  x x

l 2 1 l3 1

 

( x  x k ) ( x4  x4k )  1 1 j1! j4 ! j1  0 j4  0 j1

x1 x4

1

4

j4 x2 x3

2

3

( x2   2 ) l2 1 ( x4   4 ) l4 1 j1 l2 j3 l4 k k k k (l2  1)! (l4  1)! D1 D2 D3 D4 u ( x1 , 2 , x3 , 4 )d 2 d 4  x x

l1 1 l3 1



k 1

( x1  x ) ( x3  x ) j1! j3! j1  0 j3  0

j3 x 2 x 4

l1 1 l 2 1

k 1

j2 x3 x4

j1

k 3

( x1  x ) ( x2  x ) j1! j2 ! j1  0 j2  0



j1

k 2

2

4

( x3   3 ) l3 1 ( x4   4 ) l4 1 j1 j2 l3 l4 k k k k (l3  1)! (l4  1)! D1 D2 D3 D4 u ( x1 , x2 , 3 , 4 )d 3d 4  x x 3

4

( x4  x4k ) j4 j4 ! j4  0

( x1   1 ) l1 1 ( x2   2 ) l2 1 ( x3   3 ) l3 1 l1 l2 l3 j4 k k k k (l1  1)! (l2  1)! (l3  1)! D1 D2 D3 D4 u ( 1 , 2 , 3 , x4 )d 1d 2 d 3  x x x

l3 1



k 3

( x3  x ) j3! j3  0

j3 x1 x2 x4

( x1   1 ) l1 1 ( x2   2 ) l2 1 ( x4   4 ) l4 1 l1 l2 j3 l4 k k k k (l1  1)! (l2  1)! (l4  1)! D1 D2 D3 D4 u ( 1 , 2 , x3 , 4 )d 1d 2 d 4  x x x

l1 1

k 1

j1 x2 x3 x4

l 4 1



( x1  x ) j1! j1  0



l 2 1

1

1

( x2  x2k ) j2 j2 ! j2  0



x1 x2 x3

2

2

3

4

( x2   2 ) l2 1 ( x3   3 ) l3 1 ( x4   4 ) l4 1 j1 l2 l3 l4 k k k k (l2  1)! (l3  1)! (l4  1)! D1 D2 D3 D4 u ( x1 , 2 , 3 , 4 )d 2 d 3d 4  x x x 2

3

4

( x1   1 ) l1 1 ( x3   3 ) l3 1 ( x4   4 ) l4 1 l1 j2 l3 l4 k k k k (l1  1)! (l3  1)! (l4  1)! D1 D2 D3 D4 u ( 1 , x2 , 3 , 4 )d 1d 3d 4  x x x x1 x3 x4

1

3

4

( x1   1 ) ( x2   2 ) l2 1 ( x3   3 ) l3 1 ( x4   4 ) l4 1 l1 l2 l3 l4 D1 D2 D3 D4 u ( 1 , 2 , 3 , 4 )d 1d 2 d 3 d 4 ( l  1 )! ( l  1 )! ( l  1 )! ( l  1 )! k k k k 1 2 3 4 x x x x x1 x2 x3 x4

 1

2

3

l1 1

4

4. Result So, the classical form 4D separated multi-point boundary problems (1), (2) and in non-classical treatment (1), (4) are equivalent in the general case. However, the 4D separated multi-point boundary problem in non-classical statement (1), (4) is more natural by statement than problem (1), (2). This is connected with the fact that in statement of problem (1), (4) the right sides of boundary conditions don't require additional conditions of agreement type. Note that some boundary -value problems in non-classical treatments for pseudo-parabolic and also hyperbolic equations were investigated in the author’s papers [15-28]. 5. Discussion and Conclusions In this paper a non-classical type 4D separated multi-point boundary problem for the general case is substantiated for a hyperbolic equation with non-smooth coefficients and with a higher-order dominating mixed derivative. Classic 4D separated multi-point boundary conditions for the general case are reduced to non-classic 4D separated multi-point boundary conditions by means of integral representations. Such statement of the problem has several advantages: 1) No additional agreement conditions are required in this statement; 2) One can consider this statement as a 4D separated multi-point boundary problem for the general case formulated in terms of traces in the S.L. Sobolev anisotropic space Wpl1 ,l2 ,l3 ,l4  G ;

3) In this statement the considered higher-order hyperbolic equation with dominating mixed derivative even in particular e.g. in the two-dimensional case is a generalization of many model equations of some processes (e.g. heat-conductivity equations , telegraph equation, Aller's equation , moisture transfer generalized equation, generalized Manjeron equation, Boussinesq - Love equation, string vibrations equations and etc.).

REFERENCES [1]

[2]

[3] [4]

[5]

[6]

[7]

[8]

[9]

[10]

[11]

[12]

[13]

[14]

[15]

D.Colton, “Pseudoparabolic equations in one space variable”, J. Different. equations, 1972, vol.12, No3, pp.559-565. A.P.Soldatov, M.Kh.Shkhanukov, “Boundary value problems with A.A.Samarsky general nonlocal condition for higher order pseudoparabolic equations”, Dokl. AN SSSR, 1987, vol.297, No 3. pp.547-552 . A.M. Nakhushev, Equations of mathematical biology. M.: Visshaya Shkola, 1995, 301p. S.S. Akhiev, “Fundamental solution to some local and non - local boundary value problems and their representations”, DAN USSR, 1983, vol.271, No 2, pp.265-269. S.S. Akhiev, “Riemann function equation with dominant mixed derivative of arbitrary order”, DAN USSR, 1985, vol. 283, No 4, pp.783-787. V.I.Zhegalov, E.A.Utkina, “On a third order pseudoparabolic equation”, Izv. Vuzov, Matem., 1999, No 10, pp.73-76. Т.И. Аманов, Пространства дифференцируемых функций с доминирующей смешанной производной, Наука, Алма-Ата, 1976, 171 с. C. M. Никольский, Приближение функций многих переменных и теоремы вложения, Наука, M., 1969, 455 с. П. И. Лизоркин, С. М. Никольский, “Классификация дифференцируемых функций на основе пространств с доминирующей производной”, Труды МИАН СССР, 77, 1965, 143–167. О.В. Бесов, В.П. Ильин, С. М. Никольский, Интегральные представления функций и теоремы вложения, Наука, M., 1975, 480 с. А.Дж. Джабраилов, “О некоторых функциональных пространствах. Прямые и обратные теоремы вложения”, Докл. АН СССР, 159 (1964), 254–257. С.С. Ахиев, “Об общем виде линейных ограниченных функционалов в одном функциональном пространстве типа С. Л. Соболева”, Докл. АН. Азерб. ССР, 35:6 (1976), 3–7. А.М. Наджафов, “Об интегральных представлениях функций из пространств с доминирующей смешанной производной”, Вестник БГУ. Cер. физ.-мат. наук, 3 (2005), 31–39. И.Г. Мамедов, “Об одном разложении для непрерывной функции многих переменных”, Вестник БГУ. Cер. физ.-мат. наук, №3-4, с.144-152 ,1999. I.G. Mamedov, “On the well-posed solvability of the Dirichlet problem for a generalized Mangeron equation with nonsmooth coefficients”, Differential Equations, 51 (6), 745-754, 2015.

[16]

[17]

[18]

[19]

[20]

[21]

[22]

[23]

[24]

[25]

[26]

[27]

[28]

I.G. Mamedov, “Nonclassical analog of the Goursat problem for a three-dimensional equation with highest derivative “, Mathematical Notes, 96 (1-2), 239-247, 2014. И.Г. Мамедов “Формула интегрирования по частям неклассического типа при исследовании задачи Гурса для одного псевдопараболического уравнения“, Владикавк. матем. журн. 13 (4), 40-51, 2011. I.G. Mamedov (Co-authors: R.A. Bandaliyev, V.S. Guliyev, A.B. Sadigov ), “The optimal control problem in the processes described by the Goursat problem for a hyperbolic equation in variable exponent Sobolev spaces with dominating mixed derivatives”, Elsevier: Journal of Computational and Applied Mathematics 305, 11-17, 2016. I.G. Mamedov, “Fundamental solution of initial-boundary value problem for fourth-order pseudoparabolic equations with nonsmooth coefficients”, .Vladikavkazskiĭ Matematicheskiĭ Zhurnal 12 (1), 17-32, 2010. I.G. Mamedov, “3D Goursat Problem for the General Case in the Non-classical Treatment for a Higher-order Hyperbolic Equation with Dominating Mixed Derivative and Their Application to the Means of 3D Technology in Biology”, Caspian Journal of Applied Mathematics, Ecology and Economics 2 (2), 93-101 , 2014. I.G. Mamedov, “On a nonclassical interpretation of the four-dimensional Goursat problem for one hyberbolic equation”, Vladikavkazskii Matematicheskii Zhurnal, 17 (4), 59-66, 2015. I.G. Mamedov, “3D Goursat problem in the non-classical treatment for Manjeron generalized equation with non-smooth coefficients”, Applied and Computational Mathematics, 4 (1-1), 1-5, 2015. И.Г. Мамедов, “Локальные и нелокальные краевые задачи для гиперболических уравнений с чисто смешанными производными третьего порядка с негладкими коэффициентами”, Деп. в АЗНИИНТИ, 65с. ,2000. I.G. Mamedov, “Investigation of a problem with integro-multipoint boundary conditions for generalized moisture transfer equation”, Izv. NAS of Azerb. Ser. Fiz-tekhn. i matem. nauk, v. 27, № 2-3, 121-126, 2007. И.Г. Мамедов, “Смешанная задача с нелокальными краевыми условиями типа Бицадзе–Самарского и Самарского–Ионкина, возникающая при моделировании фильтрации жидкости в трещиноватых средах” , Известия НАН Азербайджана (сер. физ.-техн. и матем. наук), 26 (3), 32-37, 2006. .I.G. Mamedov ,”The optimal control problem in the processes, described by the non-local problem with loadings for hyperbolic integro-differential equation” , Izv. NASA, series FTMN, v.24, № 2, 74-79, 2004. I.G. Mamedov, “Nonlocal problem for one hyperbolic integro-differential equation with loadings in boundary conditions “, Trans. Acad. Sci. Azerb. Ser. Phys.-Tech. Math. Sci, v. 23, №4, 69-78, 2003. I.G. Mamedov, “Three-dimensional nonlocal boundary-value problem with integral conditions for loaded Volterra-hyperbolic integro-differential equations”, Proc. Inst. Math. Mech. Natl. Acad. Sci. Azerb , vol. 24, 153-162, 2006.