Various mathematics and mechanics questions are connected with the Riemann (Riemann-Hilbert-Privalov) problem from the theory of analytic functions. This justifies the relevance of the equation associated with the specified Riemann problem. Until now, the specificity of the location of the singular points of the rational coefficient of the equation has not been considered. The paper aims to obtain general formulas for solving abstract equations in the corresponding ring of rational functions, considering this specificity. The authors propose a way to solve an equation and a related problem. For the first time, the sought general formulas were established and proved to be simpler than those obtained earlier. The research indicates the signs when the use of these formulas is justified. The authors apply the general provisions of (1) the theory of rings and functional analysis, (2) the theory of linear operators, (3) equations in rings with factorization pairs, (4) the problem of factorization by subrings, and (5) the properties of the used ring of rational functions. The authors' approach is free from the theory of integrals (Cauchy and Fourier types), the requirements of the Hölder condition, and the index. In a similar situation, it applies to (1) integral equations of the Wiener-Hopf type and other equations of the convolution type, as well as to related functional equations; (2) to matrix equations that can be used in mechanics with projectors onto the corresponding subrings and unknown triangular matrices.
1.1
The equations and problems considered in the paper are related to the corresponding
famous Riemann problem (Riemann-Hilbert, Riemann-Hilbert-Privalov) from the
theory of analytic functions, but for a narrower class of functions. This problem is
important in the theory of differential and integro-differential equations, integral
equations of convolution type, the corresponding differential equations of
mathematical physics, in the theory of elasticity, torsion problems, and others
This equation for = ∈ {−∞, ∞} expresses the boundary conditions of the following, called related, problem:
For given rational coefficient functions ( ), ( ), −∞ < < ∞ find a pair of rational functions +( ) ∈ ℜ+, −( ) ∈ ℜ − , ∈ ℂ. All poles of the first one (if they exist) are located only inside the lower one; the second one is located only inside the upper one of the half-plane bounded by the closed real axis and satisfying the linear equation on this axis: ( ) +( ) + −( ) = ( ); ∈ {−∞; ∞}.” (2)
The closed real axis here is the contour [1]. The authors consider the case when all zeros and poles of the coefficient ( ) (if they exist) are located simultaneously in the lower or upper half-plane of the extended complex plane. 1.2
The study of the Riemann-Hilbert problem’s solvability by exact methods goes back
to the works of I. I. Privalov, F. D. Gahov, Yu. I. Cherskij, M. G. Krejn, and others
The research aims to establish the form of general formulas for solutions of abstract
equations (1), considering the specifics of assumptions in cases where the
coefficient ( ) is a function of the considered ring of rational ones. Moreover, a
function, all zeros, and poles of which (if they exist) are finite, immaterial, and are
concentrated simultaneously either inside the upper half-plane or the lower
halfplane of the extended complex plane. The aim can be achieved using the
corresponding statements and theorems, decision formulas established in the works
of the abovelisted authors
The current research uses designations and statements from the works of the abovementioned authors. The symbol denotes an arbitrary commutative and associative ring with unit . Let +, − - be the commuting projectors → . Therefore:
0: = + −(= − +), ∓: = ∓ − 0, ∗: = + + −.
For any subset ⊆ let us denote ∓, 0: = ∓, 0 ; ∓: = ∓ ; ∗ = + + −; ∗ = + + −. For any element ∈ , we assume:
∓, 0: = ∓, 0 , ∓: = ∓ , ∗: = ∗ .
Definition 1. A pair of subrings ( +, −) of a ring with unit will be called a
factorization pair [FP], if it is generated in force in commuting projectors
+, −: ∓ = ∓( ), and the following axioms
(**) (***) Definition 2. Any ring
with unit 1′, considered together with a fixed FP of the subrings 2′ , in other words, subrings with axiomatically specified properties (*), (**), (***), will be called a “ring with a factorization pair” or, in short - a ring with FP. 2.2
namely: Let us say that the element ∈ admits factorization in the commutative ring by the factorization pair 2′ = 2− 20 2+ (– by the FP 2′ = 2− 20 2+), if there are elements + ∈ +, 0 ∈ 0, − ∈ − such that: = + 0 −. This factorization is called, 1. Correct factorization факторизацией [CF], if
invertible in their subrings, respectively; 2. Normalized factorization [NF], if 0 = 0 = ;
only one normalized correct factorization.
3. Normalized regular factorization [NRF] if it is CF and 0 = 0 = . The correct
factorization of an element from − by the FP 1−′ 10′ 1+′ +
2+′ 20′ 2−′ = + +
− can be normalized
Let us establish formulas for solutions for (1) considering corresponding to the − +), ℜ0 = ℜ+ ∩ ℜ−,
( ): = 0 + +( ) + −( ). research objective. 3 3.1
Results of individual works by G. S. Poletaev (1973–2020) and in co-authorship
( ))−1[ −( ) +( )]− , ∈ ℂ ∪ {∞}, (3) where: 0:= 0( ) = ∈ ℂ.
Using theorem 1, let us establish a statement corresponding to the research objective.
Theorem 2. Let the function A(z)∈ℜ с 0:= 0 ( ) = ( 1, 2 ∈ ), for which: in the case 1):
≠ 0 has no real zeros and all its zeros and poles (for existence) are located: 1) only inside the lower, or 2) only inside the upper, from the half-planes −, + of the extended complex plane, bounded by a closed real axis. Then for it there exists in the inverse - function −1( ) ∈ ℜ and there is a normalized correct factorization of −1( ) = +( ) 0( ) −( ), ∈ ℂ ∪ {∞} by the factorization pair →∞ ( ) = = −( ) = 1, ∈ ℂ ∪ {∞}; +( ) = [ −1( ) 0]+ = −1( ) 0, 0:= [ ( )]0,
0( ) = [ 0]−1:= [ −1( )]0, and in case 2) it will be: +( ) = 1 , 0( ) = [ 0] −1, −( ) = [ 0 −1( )] − = 0 −1( ), ∈ ℂ ∪ {∞}.
For arbitrary ( ) ∈ ℜ
there is a unique solution to equation (1) and the problem. This solution in case 1) can be found according to following the formulas: +( ) = −1( ) +( ), −( ) = −( ), ∈ ℂ ∪ {∞}; and in case 2) - according to the formulas: +( ) = [ −1( ) +( )] +,
−( ) = −( ) + ( )[ −1( ) +( )] −, ∈ ℂ ∪ {∞}. and the assumptions, in
case
Proof. Under the conditions of theorem 2, for the coefficient ( ) ∈ ℜ there exists an inverse in ℜ , and by virtue of the definitions of the subrings ℜ +
, ℜ − 1) ( ), −1( ) ∈ ℜ+, and in case 2) ( ), −1( ) ∈ ℜ−. Therefore, there is a normalized correct factorization of −1( ) = +( ) 0( ) −( ), ∈ ℂ ∪ {∞} over the factorization pair ( 1, 2 ∈ ), which will be: in the case 1) +( ) = [ −1( ) 0)]+ == −1( ) 0, 0:= [ ( )] 0, 0( ) = [ 0]−1 = [ −1( )] 0, −( ) = 1, ∈ ℂ ∪ {∞}; and in case 2) +( ) = 1, 0( ) = [ 0]−1, −( ) = [ 0 −1( )]− = 0 −1( ), ∈ ℂ ∪ {∞}. Therefore, by theorem 1, for an arbitrary right-hand side ( ) ∈ ℜ there is a (4) (5)
In this case, unique solution in ℜ to equation (1) and the problem. It can be found by formulas (3), which, in case 1), give: +( ) = −1( ) 0 [ 0]−1 [1 ⋅ +( )]+ = −1( ) +( ),
−( ) = −( ) + (1)−1[1 ⋅ +( )]− = −( ).
The formulas (4) are valid. In case 2), from the same formulas (3), it follows that: +( ) = 1 ⋅ [ 0]−1[ 0 −1( ) +( )]+ = [ −1( ) +( )]+, −( ) = −( ) + [ 0]−1 ( )[ 0 −1( ) +( )]− =
= −( ) + ( )[ −1( ) +( )]− , that is, formulas (5) are valid. The theorem is proved.
Consequence. Under the conditions of theorem 2, equation (1) and the problem with the right-hand side ( ) = 1, ( ) ∈ ℜ , ∈ ℂ ∪ {∞} have in ℜ the only solution +( ), − ( ). In case 1) from theorem 2, it can be found by the formulas: +( ) = −1( ), − ( ) = 0, ∈ ℂ ∪ {∞}, (6) and in case 2) - according to the following formulas:
+( ) = [ −1( )]0 = [ 0]−1, − ( ) = ( )[ −1( )]−, ∈ ℂ ∪ {∞}. (7) The transformation of the second of equalities (7) can be represent in the as: +( ) = [ 0( )]−1, − ( ) = 1 − ( )[ 0]−1, ∈ ℂ ∪ {∞}. (8) Let us find a solution to an equation of the form (1) and problem in ℜ for: 11 2−33 −22 5 +5 .
( ) = 3 2−24 −45 ; ( ) = 2+22 −120 0 = 131; ( ) = 31(1(−−3 ))(( −−52 )) ; −1( ) = 13(1(−−3 ))(( −−52 ));
5( + ) ( ) = ( + 10 )( + 12 ) = +( ) = +( ),
−( ) = 0 = −( ).
In this example, all zeros and poles of the function ( ) are located in the upper half-plane. Thus, ( ), −1( ) ∈ ℜ− and, by virtue of theorem 2, formulas (5) are applicable. Using these formulas, let us find a unique solution to equation (1) and the problem in the form:
3( − 3 )( − 5 ) 5( + ) + + +( ) = 11( − )( − 2 ) ⋅ ( + 10 )( + 12 ) =
15(z − 3i)(z − 5i)(z + i) = 11(z − i)(z − 2i)(z + 10i)(z + 12i)
+ , −( ) =
5( + ) ( + 10 )( + 12 ) − + 11 ( − )( − 2 ) 3 ( − 3 )( − 5 )
× ⋅ ⋅ × 3( − 3 )( − 5 ) 11( − )( − 2 )
5( + ) ( + 10 )( + 12 ) = =
⋅ 5 (−467 + 1363 )
.
8008 2 − 8 − 15
By substitution in (1), one can make sure that the solution is actually found. The desired solution is as follows:
X+ (z) =
225
In the range of questions related to integral equations of the convolution type and
the Riemann problem, research methods based on the integral theory (Cauchy type)
are distinguished by a significant analytical barrier. It is quite challenging to bring
research to the level of observable theorems. In different classes of assumptions, it
is often necessary to rebuild the theory from scratch. Examples of new approaches
go back, particularly to the research of M. G.
The authors have reflected the research results in many publications. With the received materials (abstract and specific), they participated in scientific conferences of the following structures: 1. Kherson National Technical University (KNTU) in Kherson (Ukraine); 2. Taras Shevchenko National University of Kyiv (KNU); 3. National Technical University of Ukraine “Igor Sikorsky Kyiv Polytechnic
Institute” (NTUU KPI) in Kyiv (Ukraine); 4. Bauman Moscow State Technical University (BMSTU) in Moscow (Russia); 5. Odessa State Academy of Civil Engineering and Architecture in Odessa (Ukraine) (2015–2020), and others.
Experts showed a significant interest to the obtained results. 5
The paper considers the specifics of the assumptions about the location of the singular points of the rational coefficient of equation (1). The authors obtained final results that are easy to use, and the main results of the research are consistent with the objective. For the first time, under the conditions of theorem 2, signs are established that consider the specifics of assumptions about the coefficients of equation (1) and general formulas (4)-(8) of solutions in ℜ . They are more convenient than the ones obtained by the authors earlier. The results are applicable when studying the existence, uniqueness, or non-uniqueness of solutions, their construction for specific equations of the form (1), and corresponding related problems. Their practical significance lies in the possibility of facilitating their use for finding solutions of specific equations of the considered form and demonstrating a new general approach. The new research methodology for this problem is also of practical interest. Other researchers did not use the proposed method in the considered ring. The authors want to study the cases further when factorization concerning the factorization pair for −1( ) is not correct. The expansion of classes of equations, examples of rings with factorization pairs, the expansion of the scope of the approach based on the approximation of functions, and applied aspects are also promising. The obtained results indicate the fruitfulness of the approach from the theory of equations in rings with factorization pairs for the considered issues and problems.
According to assumptions, they deepen the fundamental understanding of issues related to the Riemann problem and simplify the process of solving specific examples. The formulation of the problem of solvability of the equation, considering the specifics of the location of the singular points of the rational coefficient, is given for the first time and the “circular” approach to the research. It demonstrates the possibility of a unified approach based on the provisions on equations in rings with FP and provides an opportunity to see the interpretation and results of research.