International Journal of Mathematics and Mathematical Sciences

International Journal of Mathematics and Mathematical Sciences / 2010 / Article

Research Article | Open Access

Volume 2010 |Article ID 714534 | 38 pages | https://doi.org/10.1155/2010/714534

Benjamin-Ono Equation on a Half-Line

Academic Editor: Michael Tom
Received18 Sep 2009
Accepted28 Jan 2010
Published11 Mar 2010

Abstract

We consider the initial-boundary value problem for Benjamin-Ono equation on a half-line. We study traditionally important problems of the theory of nonlinear partial differential equations, such as global in time existence of solutions to the initial-boundary value problem and the asymptotic behavior of solutions for large time.

1. Introduction

In this paper we study the large time asymptotic behavior of solutions to the initial-boundary value problem for the Benjamin-Ono equation on a half-line:

๐‘ข๐‘ก+๐‘ข๐‘ข๐‘ฅ+โ„‹๐‘ข๐‘ฅ๐‘ฅ๐‘ข=0,๐‘ฅ>0,๐‘ก>0,(๐‘ฅ,0)=๐‘ข0(๐‘ฅ),๐‘ฅ>0,๐‘ข(0,๐‘ก)=0,๐‘ก>0,(1.1) where โˆซโ„‹๐‘ข=PV0+โˆž๐‘ข((๐‘ฆ,๐‘ก)/(๐‘ฆโˆ’๐‘ฅ))๐‘‘๐‘ฆ is the Hilbert transformation, and PV means the principal value of the singular integral. We note that in the case of the whole line we have the relations โ„‹๐œ•2๐‘ฅ=๐œ•๐‘ฅ(โˆ’๐œ•2๐‘ฅ)1/2 since the operator โ„‹ can be written as follows: โ„‹=โˆ’โ„ฑโˆ’1(๐‘–๐œ‰/|๐œ‰|)โ„ฑ=(โˆ’๐œ•2๐‘ฅ)โˆ’1/2๐œ•๐‘ฅ, where โˆš(โ„ฑ๐œ‘)(๐œ‰)=(1/โˆซ2๐œ‹)๐œ‘(๐‘ฅ)eโˆ’๐‘–๐‘ฅ๐œ‰๐‘‘๐‘ฅ is the usual Fourier transform, and โ„ฑโˆ’1 denotes the inverse Fourier transform. This equation is of great interest in many areas of Physics (see [1, 2]). The Cauchy problem (1.1) was studied by many authors. The existence of solutions in the usual Sobolev spaces ๐‡๐‘ ,0 was proved in [3โ€“9] and the smoothing properties of solutions were studied in [10โ€“14]. In paper [15] it was proved that for small initial data in ๐‡2,0โˆฉ๐‡1,1 solutions decay as ๐‘กโ†’โˆž in ๐‹โˆž norm at the same rate โˆš1/๐‘ก as for the case of the linear Benjamin-Ono equation, where

๐‡๐‘š,๐‘ =๎‚†๐œ™โˆˆ๐‹2โˆถโ€–๐œ™โ€–๐‘š,๐‘ =โ€–โ€–๎€ท1+๐‘ฅ2๎€ธ๐‘ /2๎€ท1โˆ’๐œ•2๐‘ฅ๎€ธ๐‘š/2๐œ™โ€–โ€–๐‹2๎‚‡<โˆž.(1.2)

The initial-boundary value problem (1.1) plays an important role in the contemporary mathematical physics. For the general theory of nonlinear equations on a half-line we refer to the book [16], where it was developed systematically a general theory of the initial-boundary value problems for nonlinear evolution equations with pseudodifferential operators on a half-line, where pseudodifferential operator ๐•‚ on a half-line was introduced by virtue of the inverse Laplace transformation of the product of the symbol ๐พ(๐‘)=๐‘‚(๐‘๐›ฝ) which is analytic in the right complex half-plane, and the Laplace transform of the derivative ๐œ•๐‘ฅ[๐›ฝ]๐‘ข. Thus, for example, in the case of ๐พ(๐‘)=๐‘3/2 we get the following definition of the fractional derivative ๐œ•๐‘ฅ3/2:

๐œ•๐‘ฅ3/2๐œ™=โ„’โˆ’1๎‚ป๐‘3/2๎‚ตโ„’๐œ™โˆ’๐œ™(0)๐‘๎‚ถ๎‚ผ.(1.3) Here and below ๐‘๐›ฝ is the main branch of the complex analytic function in the complex half-plane Re๐‘โ‰ฅ0, so that 1๐›ฝ=1 (we make a cut along the negative real axis (โˆ’โˆž,0)). Note that due to the analyticity of ๐‘๐›ฝ for all Re๐‘>0 the inverse Laplace transform gives us the function which is equal to 0 for all ๐‘ฅ<0. In spite of the importance and actuality there are few results about the initial-boundary value problem for pseudodifferential equations with nonanalytic symbols. For example, in paper [17] there was considered the case of rational symbol ๐พ(๐‘) which have some poles in the right complex half-plane. There was proposed a new method for constructing the Green operator based on the introduction of some necessary condition at the singularity points of the symbol ๐พ(๐‘). In the paper [18] one of the authors considered the initial-boundary value problem for a pseudodifferential equation with symbol ๐พ(๐‘)=|๐‘|1/2 and nonlinearity |๐‘ข|๐œŽ๐‘ข.

As far as we know the case of nonanalytic conservative symbols ๐พ(๐‘) was not studied previously. In the present paper we fill this gap, considering as example the Benjamin-Ono equation (1.1) with a symbol ๐พ(๐‘)=โˆ’๐‘|๐‘|. There are many natural open questions which we need to study. First we consider the following question: how many boundary data we should pose on problem (1.1) for its correct solvability? Also we study traditionally important problems of a theory of nonlinear partial differential equations, such as global in time existence of solutions to the initial-boundary value problem and the asymptotic behavior of solutions for large time. We adopt here the approach of book [16] based on the estimates of the Green function. The main difficulty for nonlocal equation (1.1) on a half-line is that the symbol ๐พ(๐‘)=โˆ’๐‘|๐‘| is non analytic in the complex plane. Therefore we cannot apply the Laplace theory directly. To construct Green operator we proposed a new method based on the integral representation for sectionally analytic function and theory of singular integrodifferential equations with Hilbert kernel and the discontinues coefficients (see [18, 19]).

To state precisely the results of the present paper we give some notations. We denote โˆšโŸจ๐‘กโŸฉ=1+๐‘ก2,{๐‘ก}=๐‘ก/โŸจ๐‘กโŸฉ. Direct Laplace transformation โ„’๐‘ฅโ†’๐œ‰ is

ฬ‚๐‘ข(๐œ‰)โ‰กโ„’๐‘ฅโ†’๐œ‰๎€œ๐‘ข=0+โˆž๐‘’โˆ’๐œ‰๐‘ฅ๐‘ข(๐‘ฅ)๐‘‘๐‘ฅ,(1.4) and the inverse Laplace transformation โ„’โˆ’1๐œ‰โ†’๐‘ฅ is defined by

๐‘ข(๐‘ฅ)โ‰กโ„’โˆ’1๐œ‰โ†’๐‘ฅฬ‚๐‘ข=(2๐œ‹๐‘–)โˆ’1๎€œ๐‘–โˆžโˆ’๐‘–โˆž๐‘’๐œ‰๐‘ฅฬ‚๐‘ข(๐œ‰)๐‘‘๐œ‰.(1.5) Weighted Lebesgue space is ๐‹๐‘ž,๐‘Ž(๐‘+)={๐œ‘โˆˆ๐’ฎ๎…ž;โ€–๐œ‘โ€–๐‹๐‘ž,๐‘Ž<โˆž}, where

โ€–๐œ‘โ€–๐‹๐‘ž,๐‘Ž=๎‚ต๎€œ0+โˆž๐‘ฅ๐‘Ž๐‘ž||||๐œ‘(๐‘ฅ)๐‘ž๎‚ถ๐‘‘๐‘ฅ1/๐‘ž(1.6) for ๐‘Ž>0, 1โ‰ค๐‘ž<โˆž and

โ€–๐œ‘โ€–๐‹โˆž=esssup๐‘ฅโˆˆ๐‘+||||.๐œ‘(๐‘ฅ)(1.7) Sobolev space is

๐‡1๎€ท๐‘+๎€ธ=๎€ฝ๐œ‘โˆˆ๐’ฎ๎…ž;โ€–โ€–โŸจ๐œ•๐‘ฅโ€–โ€–โŸฉ๐œ‘๐‹2๎€พ.<โˆž(1.8) We define a linear functional ๐‘“:

๐‘“๎€œ(๐œ™)=0+โˆž๐‘ฆ๐œ™(๐‘ฆ)๐‘‘๐‘ฆ.(1.9) Now we state the main results.

Theorem 1.1. Suppose that the initial data ๐‘ข0โˆˆ๐™โ‰ก๐‡1(๐‘+)โˆฉ๐‹1,๐‘Ž+1(๐‘+) with ๐‘Žโˆˆ(0,1) are such that the norm โ€–โ€–๐‘ข0โ€–โ€–๐™โ‰ค๐œ€(1.10) is sufficiently small. Then there exists a unique global solution ๎€ท[๐‘ขโˆˆ๐‚0,โˆž);๐‡1๎€ท๐‘+๎€ธ๎€ธ(1.11) to the initial-boundary value problem (1.1). Moreover the following asymptotic is valid in ๐‹โˆž(๐‘+)โˆถ1๐‘ข=๐‘ก๎€ท๐ดฮ›๐‘ฅ๐‘กโˆ’1/2๎€ธ๎ƒฉ๐‘ฅ+min1,โˆš๐‘ก๎ƒช๐‘‚๎€ท๐‘กโˆ’1โˆ’(๐‘Ž/2)๎€ธ(1.12) for ๐‘กโ†’โˆž, where ฮ›(๐‘ฅ๐‘กโˆ’1/2)โˆˆ๐‹โˆž(๐‘+),ฮ›(0)=0 is defined below by the formula (2.191), and the constant ๎€ท๐‘ข๐ด=๐‘“0๎€ธโˆ’๎€œ0+โˆž๐‘“(๐’ฉ(๐‘ข))๐‘‘๐œ,๐’ฉ(๐‘ข)=๐‘ข๐‘ฅ๐‘ข.(1.13)

Remark 1.2. Note that the time decay rate of the solution is faster comparing with the case of the corresponding Cauchy problem. So the nonlinearity ๐‘ข๐‘ข๐‘ฅ in (1.1) is not the super critical case for our problem.

Remark 1.3. In the case of the negative half line ๐‘ฅ<0 we expect that the solutions have an oscillation character, and the time decay rate of the solution is the same as the case of the corresponding Cauchy problem. so the nonlinearity ๐‘ข๐‘ข๐‘ฅ in (1.1) will be the super critical case.

2. Preliminaries

In subsequent consideration we will have frequently to use certain theorems of the theory of functions of complex variable, the statements of which we now quote. The proofs can be found in [19].

Theorem 2.1. Let ๐œ™(๐‘ž) be a complex function, which obeys the Hรถlder condition for all finite ๐‘ž and tends to a definite limit ๐œ™โˆž as |๐‘ž|โ†’โˆž, such that for large ๐‘ž the following inequality holds: ||๐œ™(๐‘ž)โˆ’๐œ™โˆž||||๐‘ž||โ‰ค๐ถโˆ’๐œ‡,๐œ‡>0.(2.1) Then Cauchy type integral 1๐น(๐‘ง)=๎€œ2๐œ‹๐‘–๐‘–โˆžโˆ’๐‘–โˆž๐œ™(๐‘ž)๐‘žโˆ’๐‘ง๐‘‘๐‘ž(2.2) constitutes a function analytic in the left and right semiplanes. Here and below these functions will be denoted ๐น+(๐‘ง) and ๐นโˆ’(๐‘ง), respectively. These functions have the limiting values ๐น+(๐‘) and ๐นโˆ’(๐‘) at all points of imaginary axis Re๐‘=0, on approaching the contour from the left and from the right, respectively. These limiting values are expressed by Sokhotzki-Plemelj formula: ๐น+(๐‘)=lim๐‘งโ†’๐‘,Re๐‘ง<01๎€œ2๐œ‹๐‘–๐‘–โˆžโˆ’๐‘–โˆž๐œ™(๐‘ž)1๐‘žโˆ’๐‘ง๐‘‘๐‘ž=๎€œ2๐œ‹๐‘–๐‘ƒ๐‘‰๐‘–โˆžโˆ’๐‘–โˆž๐œ™(๐‘ž)1๐‘žโˆ’๐‘๐‘‘๐‘ž+2๐น๐œ™(๐‘),โˆ’(๐‘)=lim๐‘งโ†’๐‘,Re๐‘ง>01๎€œ2๐œ‹๐‘–๐‘–โˆžโˆ’๐‘–โˆž๐œ™(๐‘ž)1๐‘žโˆ’๐‘ง๐‘‘๐‘ž=๎€œ2๐œ‹๐‘–๐‘ƒ๐‘‰๐‘–โˆžโˆ’๐‘–โˆž๐œ™(๐‘ž)1๐‘žโˆ’๐‘๐‘‘๐‘žโˆ’2๐œ™(๐‘).(2.3)

Subtracting and adding the formula (2.3) we obtain the following two equivalent formulas:

๐น+(๐‘)โˆ’๐นโˆ’๐น(๐‘)=๐œ™(๐‘),+(๐‘)+๐นโˆ’1(๐‘)=๎€œ๐œ‹๐‘–๐‘ƒ๐‘‰๐‘–โˆžโˆ’๐‘–โˆž๐œ™(๐‘ž)๐‘žโˆ’๐‘๐‘‘๐‘ž,(2.4) which will be frequently employed hereafter.

Theorem 2.2. An arbitrary function ๐œ™(๐‘) given on the contour Re๐‘=0, satisfying the Hรถlder condition, can be uniquely represented in the form ๐œ™(๐‘)=๐‘ˆ+(๐‘)โˆ’๐‘ˆโˆ’(๐‘),(2.5) where ๐‘ˆยฑ(๐‘) are the boundary values of the analytic functions ๐‘ˆยฑ(๐‘ง) and the condition ๐‘ˆยฑโˆž=0 holds. These functions are determined by formula 1๐‘ˆ(๐‘ง)=๎€œ2๐œ‹๐‘–๐‘–โˆžโˆ’๐‘–โˆž๐œ™(๐‘ž)๐‘žโˆ’๐‘ง๐‘‘๐‘ž.(2.6)

Theorem 2.3. An arbitrary function ๐œ‘(๐‘) given on the contour Re๐‘=0, satisfying the Hรถlder condition, and having zero index, 1ind๐œ‘(๐‘ก)โˆถ=๎€œ2๐œ‹๐‘–๐‘–โˆžโˆ’๐‘–โˆž๐‘‘ln๐œ‘(๐‘)=0,(2.7) is uniquely representable as the ratio of the functions ๐‘‹+(๐‘) and ๐‘‹โˆ’(๐‘), constituting the boundary values of functions, ๐‘‹+(๐‘ง) and ๐‘‹โˆ’(๐‘ง), analytic in the left and right complex semiplane and having in these domains no zero. These functions are determined to within an arbitrary constant factor and given by formula ๐‘‹ยฑ(๐‘ง)=๐‘’ฮ“ยฑ(๐‘ง)1,ฮ“(๐‘ง)=๎€œ2๐œ‹๐‘–๐‘–โˆžโˆ’๐‘–โˆž1๐‘žโˆ’๐‘งln๐œ‘(๐‘ž)๐‘‘๐‘ž.(2.8)

We consider the following linear initial-boundary value problem on half-line ๐‘ข๐‘ก๎€œโˆ’๐‘ƒ๐‘‰0+โˆž๐‘ข๐‘ฆ๐‘ฆ(๐‘ฆ,๐‘ก)๐‘ฅโˆ’๐‘ฆ๐‘‘๐‘ฆ=0,๐‘ก>0,๐‘ฅ>0,๐‘ข(๐‘ฅ,0)=๐‘ข0(๐‘ฅ),๐‘ฅ>0,๐‘ข(0,๐‘ก)=0,๐‘ก>0.(2.9)

Setting

||๐‘ž||๐พ(๐‘ž)=โˆ’๐‘ž,๐พ1(๐‘ž)=โˆ’๐‘ž2||๐œ‰||,๐‘˜(๐œ‰)=1/2๐‘’(1/2)๐‘–arg๐œ‰,(2.10)

where Re๐‘˜(๐œ‰)>0 for Re๐œ‰>0, we define

๐’ข๎€œ(๐‘ก)๐œ™=0+โˆž๐บ(๐‘ฅ,๐‘ฆ,๐‘ก)๐œ™(๐‘ฆ)๐‘‘๐‘ฆ,(2.11)

where the function ๐บ(๐‘ฅ,๐‘ฆ,๐‘ก) is given by formula

1๐บ(๐‘ฅ,๐‘ฆ,๐‘ก)=12๐œ‹๐‘–๎€œ2๐œ‹๐‘–๐œ€+๐‘–โˆž๐œ€โˆ’๐‘–โˆž๐‘‘๐œ‰๐‘’๐œ‰๐‘ก๎€œ๐‘–โˆžโˆ’๐‘–โˆž๐‘‘๐‘๐‘’๐‘๐‘ฅ1๐พ1(๐‘)+๐œ‰(๐‘’โˆ’๐‘๐‘ฆโˆ’1+ฮจ(๐œ‰,๐‘ฆ))12๐œ‹๐‘–๎€œ2๐œ‹๐‘–๐œ€+๐‘–โˆž๐œ€โˆ’๐‘–โˆž๐‘‘๐œ‰๐‘’๐œ‰๐‘ก๎€œ๐‘–โˆžโˆ’๐‘–โˆž๐‘‘๐‘๐‘’๐‘๐‘ฅ1๐พ1๐‘Œ(๐‘)+๐œ‰โˆ’(๐‘,๐œ‰)ร—lim๐‘งโ†’๐‘,Re๐‘ง>01๎€œ2๐œ‹๐‘–๐‘–โˆžโˆ’๐‘–โˆž11๐‘žโˆ’๐‘ง๐‘Œ+(๐พ๐‘ž,๐œ‰)1(๐‘ž)โˆ’๐พ(๐‘ž)๐พ1(๐‘ž)+๐œ‰(๐‘’โˆ’๐‘ž๐‘ฆ+ฮจ(๐œ‰,๐‘ฆ))๐‘‘๐‘ž(2.12) for ๐œ€>0,๐‘ฅ>0,๐‘ฆ>0,๐‘ก>0. Here and below

๐‘Œยฑ=๐‘’ฮ“ยฑ๐‘คยฑ.(2.13)ฮ“+(๐‘,๐œ‰) and ฮ“โˆ’(๐‘,๐œ‰) are a left and right limiting values of sectionally analytic function ฮ“(๐‘ง,๐œ‰) given by

1ฮ“(๐‘ง,๐œ‰)=๎€œ2๐œ‹๐‘–๐‘–โˆžโˆ’๐‘–โˆž1๐‘žโˆ’๐‘งln๎‚ป๎‚ต๐พ(๐‘ž)+๐œ‰๐พ1๎‚ถ๐‘ค(๐‘ž)+๐œ‰โˆ’(๐‘ž)๐‘ค+๎‚ผ(๐‘ž)๐‘‘๐‘ž,(2.14) where

๐‘คโˆ’๎‚ต๐‘ง(๐‘ง)=๎‚ถ๐‘ง+๐‘˜(๐œ‰)1/2,๐‘ค+๎‚ต๐‘ง(๐‘ง)=๎‚ถ๐‘งโˆ’๐‘˜(๐œ‰)1/2,๐‘’ฮจ(๐œ‰,๐‘ฆ)=โˆ’โˆ’๐‘˜(๐œ‰)๐‘ฆ๐‘Œโˆ’(+1๐‘˜(๐œ‰),๐œ‰)๎€œ2๐œ‹๐‘–๐‘–โˆžโˆ’๐‘–โˆž11๐‘žโˆ’๐‘˜(๐œ‰)๐‘Œ+(๐พ๐‘ž,๐œ‰)1(๐‘ž)โˆ’๐พ(๐‘ž)๐พ1(๐‘’๐‘ž)+๐œ‰โˆ’๐‘ž๐‘ฆ๐‘‘๐‘ž.(2.15)

All the integrals are understood in the sense of the principal values.

Proposition 2.4. Let the initial data be ๐‘ข0โˆˆ๐‹1(๐‘+). Then there exists a unique solution ๐‘ข(๐‘ฅ,๐‘ก) of the initial-boundary value problem (2.9), which has integral representation ๐‘ข(๐‘ฅ,๐‘ก)=๐’ข(๐‘ก)๐‘ข0.(2.16)

Proof. To derive an integral representation for the solutions of the problem (2.9) we suppose that there exists a solution ๐‘ข(๐‘ฅ,๐‘ก) of problem (2.9), which is continued by zero outside of ๐‘ฅ>0:
๐‘ข(๐‘ฅ,๐‘ก)=0,โˆ€๐‘ฅ<0.(2.17)
Let ๐œ™(๐‘) be a function of the complex variable ๐‘, which obeys the Hรถlder condition for all finite ๐‘ and tends to 0 as ๐‘โ†’ยฑ๐‘–โˆž. We define the operator 1โ„™๐œ™(๐‘ง)=โˆ’๎€œ2๐œ‹๐‘–๐‘–โˆžโˆ’๐‘–โˆž1๐‘žโˆ’๐‘ง๐œ™(๐‘ž)๐‘‘๐‘ž.(2.18)
Since the operator โ„™ is defined by a Cauchy type integral, it is readily observed that โ„™๐œ™(๐‘ง) constitutes a function analytic in the entire complex plane, except for points of the contour of integration Re๐‘ง=0. Also by Sokhotzki-Plemelj formula we have for Re๐‘=0โ„™+1๐œ™=๎€œ2๐œ‹๐‘–๐‘ƒ๐‘‰๐‘–โˆžโˆ’๐‘–โˆž11๐‘žโˆ’๐‘๐œ™(๐‘ž)๐‘‘๐‘ž+2โ„™๐œ™(๐‘),โˆ’1๐œ™=๎€œ2๐œ‹๐‘–๐‘ƒ๐‘‰๐‘–โˆžโˆ’๐‘–โˆž11๐‘žโˆ’๐‘๐œ™(๐‘ž)๐‘‘๐‘žโˆ’2๐œ™(๐‘).(2.19) Here โ„™+๐œ™ and โ„™โˆ’๐œ™ are limits of โ„™๐œ™ as ๐‘ง tends to ๐‘ from the left and right semi-plane, respectively.
We have for the Laplace transform โ„’๎€ฝโ„‹๐‘ข๐‘ฅ๐‘ฅ๎€พ๎‚ปโˆ’||๐‘||๐‘๎‚ต=โ„™โ„’{๐‘ข}โˆ’๐‘ข(0,๐‘ก)๐‘โˆ’๐‘ข๐‘ฅ(0,๐‘ก)๐‘2๎‚ถ๎‚ผ.(2.20) Since โ„’{๐‘ข} is analytic for all Re๐‘ž>0, we have ฬ‚๐‘ข(๐‘ž,๐‘ก)=โ„’{๐‘ข}=โ„™ฬ‚๐‘ข(๐‘,๐‘ก).(2.21) Therefore applying the Laplace transform with respect to ๐‘ฅ to problem (2.9) we obtain for ๐‘ก>0โ„™๎‚ปฬ‚๐‘ข๐‘ก+๐พ(๐‘)ฬ‚๐‘ข(๐‘,๐‘ก)โˆ’๐พ(๐‘)๐‘๐‘ข(0,๐‘ก)โˆ’๐พ(๐‘)๐‘2๐‘ข๐‘ฅ๎‚ผ(0,๐‘ก)=0,ฬ‚๐‘ข(๐‘,0)=ฬ‚๐‘ข0(๐‘),(2.22) where ||๐‘||๐พ(๐‘)=โˆ’๐‘.(2.23) We rewrite (2.22) in the form ฬ‚๐‘ข๐‘ก+๐พ(๐‘)ฬ‚๐‘ข(๐‘,๐‘ก)โˆ’๐พ(๐‘)๐‘๐‘ข(0,๐‘ก)โˆ’๐พ(๐‘)๐‘2๐‘ข๐‘ฅ(0,๐‘ก)=ฮฆ(๐‘,๐‘ก),ฬ‚๐‘ข(๐‘,0)=ฬ‚๐‘ข0(๐‘),(2.24) with some function ฮฆ(๐‘,๐‘ก) such that for all Re๐‘>0โ„™{ฮฆ(๐‘,๐‘ก)}=0(2.25) and for |๐‘|>1||||1ฮฆ(๐‘,๐‘ก)โ‰ค๐ถ||๐‘||.(2.26) Applying the Laplace transformation with respect to time variable to problem (2.24) we find for Re๐‘>0ฬ‚1ฬ‚๐‘ข(๐‘,๐œ‰)=๎‚ต๐พ(๐‘)+๐œ‰ฬ‚๐‘ข0(๐‘)+๐พ(๐‘)๐‘ฬ‚๐‘ข(0,๐œ‰)+๐พ(๐‘)๐‘2ฬ‚๐‘ข๐‘ฅ๎๎‚ถ(0,๐œ‰)+ฮฆ(๐‘,๐œ‰).(2.27) Here the functions ฬ‚๎ฬ‚๐‘ข(๐‘,๐œ‰),ฮฆ(๐‘,๐œ‰),ฬ‚๐‘ข(0,๐œ‰), and ฬ‚๐‘ข๐‘ฅ(0,๐œ‰) are the Laplace transforms for ฬ‚๐‘ข(๐‘,๐‘ก),ฮฆ(๐‘,๐‘ก),๐‘ข(0,๐‘ก), and ๐‘ข๐‘ฅ(0,๐‘ก) with respect to time, respectively. We will find the function ๎ฮฆ(๐‘,๐œ‰) using the analytic properties of function ฬ‚ฬ‚๐‘ข in the right-half complex planes Re๐‘>0 and Re๐œ‰>0. We have for Re๐‘=0ฬ‚1ฬ‚๐‘ข(๐‘,๐œ‰)=๎€œ๐œ‹๐‘–๐‘ƒ๐‘‰๐‘–โˆžโˆ’๐‘–โˆž1ฬ‚๐‘žโˆ’๐‘ฬ‚๐‘ข(๐‘ž,๐œ‰)๐‘‘๐‘ž.(2.28) In view of Sokhotzki-Plemelj formula via (2.27) the condition (2.28) can be written as ฮ˜+(๐‘,๐œ‰)=โˆ’ฮ›+(๐‘,๐œ‰),(2.29) where the sectionally analytic functions ฮ˜(๐‘ง,๐œ‰) and ฮ›(๐‘ง,๐œ‰) are given by Cauchy type integrals: 1ฮ˜(๐‘ง,๐œ‰)=๎€œ2๐œ‹๐‘–๐‘–โˆžโˆ’๐‘–โˆž11๐‘žโˆ’๐‘ง๎1๐พ(๐‘ž)+๐œ‰ฮฆ(๐‘ž,๐œ‰)๐‘‘๐‘ž,(2.30)ฮ›(๐‘ง,๐œ‰)=๎€œ2๐œ‹๐‘–๐‘–โˆžโˆ’๐‘–โˆž11๐‘žโˆ’๐‘ง๎‚ต๐พ(๐‘ž)+๐œ‰ฬ‚๐‘ข0(๐‘ž)+๐พ(๐‘ž)๐‘žฬ‚๐‘ข(0,๐œ‰)+๐พ(๐‘ž)๐‘ž2ฬ‚๐‘ข๐‘ฅ๎‚ถ(0,๐œ‰)๐‘‘๐‘ž.(2.31) To perform the condition (2.29) in the form of nonhomogeneous Riemann problem we introduce the sectionally analytic function: 1ฮฉ(๐‘ง,๐œ‰)=๎€œ2๐œ‹๐‘–๐‘–โˆžโˆ’๐‘–โˆž1๐‘žโˆ’๐‘งฮจ(๐‘ž,๐œ‰)๐‘‘๐‘ž,(2.32) where ฮจ(๐‘,๐œ‰)=๐พ(๐‘)๎๐พ(๐‘)+๐œ‰ฮฆ(๐‘,๐œ‰).(2.33) Taking into account the assumed condition (2.25) and making use of Sokhotzki-Plemelj formula (2.3) we get for limiting values of the functions ฮฉ(๐‘ง,๐œ‰) and ฮ˜(๐‘ง,๐œ‰)ฮฉโˆ’(๐‘,๐œ‰)=โˆ’๐œ‰ฮ˜โˆ’(๐‘,๐œ‰).(2.34) Also observe that from (2.30) and (2.32) by formula (2.4) ๐พ๎€ทฮ˜(๐‘)+(๐‘,๐œ‰)โˆ’ฮ˜โˆ’๎€ธ(๐‘,๐œ‰)=ฮจ(๐‘,๐œ‰)=ฮฉ+(๐‘,๐œ‰)โˆ’ฮฉโˆ’(๐‘,๐œ‰).(2.35) Substituting (2.29) and (2.34) into this equation we obtain nonhomogeneous Riemann problem ฮฉ+(๐‘,๐œ‰)=๐พ(๐‘)+๐œ‰๐œ‰ฮฉโˆ’(๐‘,๐œ‰)โˆ’๐พ(๐‘)ฮ›+(๐‘,๐œ‰).(2.36)
It is required to find two functions for some fixed point ๐œ‰, Re๐œ‰>0: ฮฉ+(๐‘ง,๐œ‰), analytic in Re๐‘ง<0 and ฮฉโˆ’(๐‘ง,๐œ‰), analytic in Re๐‘ง>0, which satisfy on the contour Re๐‘=0 the relation (2.36). Here, for some fixed point ๐œ‰, Re๐œ‰>0, the functions ๐‘Š(๐‘,๐œ‰)=๐พ(๐‘)+๐œ‰๐œ‰,๐‘”(๐‘,๐œ‰)=โˆ’๐พ(๐‘)ฮ›+(๐‘,๐œ‰)(2.37) are called the coefficient and the free term of the Riemann problem, respectively.
Note that bearing in mind formula (2.33) we can find unknown function ๎ฮฆ(๐‘,๐œ‰) which involved in the formula (2.27) by the relation ๎ฮฆ(๐‘,๐œ‰)=๐พ(๐‘)+๐œ‰๎€ทฮฉ๐พ(๐‘)+(๐‘,๐œ‰)โˆ’ฮฉโˆ’(๎€ธ.๐‘,๐œ‰)(2.38) The method for solving the Riemann problem ๐ด+(๐‘)=๐œ‘(๐‘)๐ดโˆ’(๐‘)+๐œ™(๐‘) is based on the Theorems 2.2 and 2.3.
In the formulations of Theorems 2.2 and 2.3 the coefficient ๐œ‘(๐‘) and the free term ๐œ™(๐‘) of the Riemann problem are required to satisfy the Hรถlder condition on the contour Re๐‘=0. This restriction is essential. On the other hand, it is easy to observe that both functions ๐‘Š(๐‘,๐œ‰) and ๐‘”(๐‘,๐œ‰) do not have limiting value as ๐‘โ†’ยฑ๐‘–โˆž. The principal task now is to get an expression equivalent to the boundary value problem (2.36), such that the conditions of theorems are satisfied. First, let us introduce some notation and let us establish certain auxiliary relationships. Setting ๐พ1(๐‘)=โˆ’๐‘2,(2.39) we introduce the function ๎‚‹๎‚ต๐‘Š(๐‘,๐œ‰)=๐พ(๐‘)+๐œ‰๐พ1๎‚ถ๐‘ค(๐‘)+๐œ‰โˆ’(๐‘)๐‘ค+(๐‘),(2.40) where for some fixed point ๐‘˜(๐œ‰) (Re๐‘˜(๐œ‰)>0) ๐‘คโˆ’๎‚ต๐‘ง(๐‘ง)=๎‚ถ๐‘ง+๐‘˜(๐œ‰)1/2,๐‘ค+๎‚ต๐‘ง(๐‘ง)=๎‚ถ๐‘งโˆ’๐‘˜(๐œ‰)1/2.(2.41) We make a cut in the plane ๐‘ง from point ๐‘˜(๐œ‰) to point โˆ’โˆž through 0. Owing to the manner of performing the cut the functions ๐‘คโˆ’(๐‘ง), ๐พ1(๐‘ง) are analytic for Re๐‘ง>0 and the function ๐‘ค+(๐‘ง) is analytic for Re๐‘ง<0.
We observe that the function ๎‚‹๐‘Š(๐‘,๐œ‰) given on the contour Re๐‘=0 satisfies the Hรถlder condition and under the assumption Re๐พ1(๐‘)>0 does not vanish for any Re๐œ‰>0. Also we have ๎‚‹1Ind.๐‘Š(๐‘,๐œ‰)=๎€œ2๐œ‹๐‘–๐‘–โˆžโˆ’๐‘–โˆž๎‚‹๐‘‘ln๐‘Š(๐‘,๐œ‰)=0.(2.42) Therefore in accordance with Theorem 2.3 the function ๎‚‹๐‘Š(๐‘,๐œ‰) can be represented in the form of the ratio ๎‚‹๐‘‹๐‘Š(๐‘,๐œ‰)=+(๐‘,๐œ‰)๐‘‹โˆ’,(๐‘,๐œ‰)(2.43) where ๐‘‹ยฑ(๐‘,๐œ‰)=๐‘’ฮ“ยฑ(๐‘,๐œ‰)1,ฮ“(๐‘ง,๐œ‰)=๎€œ2๐œ‹๐‘–๐‘–โˆžโˆ’๐‘–โˆž1๎‚‹๐‘žโˆ’๐‘งln๐‘Š(๐‘ž,๐œ‰)๐‘‘๐‘ž.(2.44) Now we return to the nonhomogeneous Riemann problem (2.36). Multiplying and dividing the expression (๐พ(๐‘)+๐œ‰)/๐œ‰ by (1/(๐พ1(๐‘)+๐œ‰))(๐‘คโˆ’(๐‘)/๐‘ค+(๐‘)) and making use of the formula (2.43) we get ๐‘Š(๐‘,๐œ‰)=๐พ(๐‘)+๐œ‰๐œ‰=๐‘Œ+(๐‘,๐œ‰)๐‘Œโˆ’๎‚ต๐พ(๐‘,๐œ‰)1(๐‘)+๐œ‰๐œ‰๎‚ถ,(2.45) where ๐‘Œยฑ(๐‘,๐œ‰)=๐‘‹ยฑ(๐‘,๐œ‰)๐‘คยฑ(๐‘).(2.46) Replacing in (2.36) the coefficient of the Riemann problem ๐‘Š(๐‘,๐œ‰) by (2.45) we reduce the nonhomogeneous Riemann problem (2.36) to the form ฮฉ+(๐‘,๐œ‰)๐‘Œ+=๎‚ต๐พ(๐‘,๐œ‰)1(๐‘)+๐œ‰๐œ‰๎‚ถฮฉโˆ’(๐‘,๐œ‰)๐‘Œโˆ’โˆ’1(๐‘,๐œ‰)๐‘Œ+(๐‘,๐œ‰)๐พ(๐‘)ฮ›+(๐‘,๐œ‰).(2.47) Now we perform the function ฮ›(๐‘ง,๐œ‰) given by formula (2.31) as ฮ›(๐‘ง,๐œ‰)=ฮ›1(๐‘ง,๐œ‰)+ฮ›2(๐‘ง,๐œ‰),(2.48) where ฮ›11(๐‘ง,๐œ‰)=๎€œ2๐œ‹๐‘–๐‘–โˆžโˆ’๐‘–โˆž11๐‘žโˆ’๐‘ง๐พ1๎‚ต(๐‘ž)+๐œ‰ฬ‚๐‘ข0๐พ(๐‘ž)+1(๐‘ž)๐‘ž๐พฬ‚๐‘ข(0,๐œ‰)+1(๐‘ž)๐‘ž2ฬ‚๐‘ข๐‘ฅ๎‚ถฮ›(0,๐œ‰)๐‘‘๐‘ž,(2.49)21(๐‘ง,๐œ‰)=๎€œ2๐œ‹๐‘–๐‘–โˆžโˆ’๐‘–โˆž1๐พ๐‘žโˆ’๐‘ง1(๐‘ž)โˆ’๐พ(๐‘ž)๎€ท๐พ(๐พ(๐‘ž)+๐œ‰)1๎€ธ๎‚ต(๐‘ž)+๐œ‰ฬ‚๐‘ข0๐œ‰(๐‘ž)โˆ’๐‘žฬ‚๐‘ข(0,๐œ‰)โˆ’ฬ‚๐‘ข๐‘ฅ๎‚ถ(0,๐œ‰)๐‘‘๐‘ž.(2.50) Firstly we calculate the left limiting value ฮ›+1(๐‘,๐œ‰). Since there exists only one root ๐‘˜(๐œ‰) of equation ๐พ1(๐‘ง)=โˆ’๐œ‰ such that Re๐‘˜(๐œ‰)>0 for all Re๐œ‰>0, therefore, taking limit ๐‘งโ†’๐‘ from the left-hand side of complex plane, by Cauchy theorem we get ฮ›+1๐‘˜(๐‘,๐œ‰)=โˆ’๎…ž(๐œ‰)๎‚ต๐‘โˆ’๐‘˜(๐œ‰)ฬ‚๐‘ข0๐œ‰(๐‘˜(๐œ‰))โˆ’๐‘ฬ‚๐‘ข(0,๐œ‰)โˆ’ฬ‚๐‘ข๐‘ฅ๎‚ถ(0,๐œ‰).(2.51) The last relation implies that (๐พ1(๐‘)+๐œ‰)ฮ›+1(๐‘,๐œ‰) can be expressed by the function ฮ›3(๐‘ง,๐œ‰) which is analytic in Re๐‘ง>0โˆถ๎€ท๐พ1๎€ธฮ›(๐‘)+๐œ‰+1(๐‘,๐œ‰)=ฮ›โˆ’3(๐‘,๐œ‰),(2.52) where ฮ›3(๐‘ง,๐œ‰)=โˆ’๐‘˜๎…ž๎‚ต๐พ(๐œ‰)1(๐‘ง)+๐œ‰๐‘งโˆ’๐‘˜(๐œ‰)๎‚ถ๎‚ตฬ‚๐‘ข0๐œ‰(๐‘˜(๐œ‰))โˆ’๐‘ฬ‚๐‘ข(0,๐œ‰)โˆ’ฬ‚๐‘ข๐‘ฅ๎‚ถ(0,๐œ‰).(2.53) By Sokhotzki-Plemelj formula (2.4) we express the left limiting value ฮ›+2(๐‘,๐œ‰) in the term of the right limiting value ฮ›โˆ’2(๐‘,๐œ‰) as ฮ›+2(๐‘,๐œ‰)=ฮ›โˆ’2(๐‘,๐œ‰)+ฬƒ๐‘”1(๐‘,๐œ‰),(2.54) where ฬƒ๐‘”1๐พ(๐‘,๐œ‰)=1(๐‘)โˆ’๐พ(๐‘)๎€ท๐พ(๐พ(๐‘)+๐œ‰)1๎€ธ๎‚ต(๐‘)+๐œ‰ฬ‚๐‘ข0๐œ‰(๐‘)โˆ’๐‘ฬ‚๐‘ข(0,๐œ‰)โˆ’ฬ‚๐‘ข๐‘ฅ๎‚ถ.(0,๐œ‰)(2.55) Bearing in mind the representation (2.48) and making use of (2.52), (2.55), and (2.45) after simple transformations we get โˆ’๐พ(๐‘)ฮ›+๐‘Œ=โˆ’+๐‘Œโˆ’๎€บฮ›โˆ’3+๎€ท๐พ1(๎€ธฮ›๐‘)+๐œ‰โˆ’2๎€ป+๐œ‰ฮ›+โˆ’๐‘”1(๐‘,๐œ‰),(2.56) where ๐‘”1(๐‘,๐œ‰)=(๐พ(๐‘)+๐œ‰)ฬƒ๐‘”1๐พ(๐‘,๐œ‰)=1(๐‘)โˆ’๐พ(๐‘)๐พ1๎‚ต(๐‘)+๐œ‰ฬ‚๐‘ข0๐œ‰(๐‘)โˆ’๐‘ฬ‚๐‘ข(0,๐œ‰)โˆ’ฬ‚๐‘ข๐‘ฅ๎‚ถ(0,๐œ‰).(2.57) Replacing in (2.47) โˆ’๐พ(๐‘)ฮ›+(๐‘,๐œ‰) by (2.56), we reduce the nonhomogeneous Riemann problem (2.47) in the form ฮฉ+1(๐‘,๐œ‰)๐‘Œ+=ฮฉ(๐‘,๐œ‰)โˆ’1(๐‘,๐œ‰)๐‘Œโˆ’โˆ’1(๐‘,๐œ‰)๐‘Œ+๐‘”(๐‘,๐œ‰)1(๐‘,๐œ‰),(2.58) where ฮฉ+1(๐‘,๐œ‰)=ฮฉ+(๐‘,๐œ‰)โˆ’๐œ‰ฮ›+ฮฉ(๐‘,๐œ‰),โˆ’1๎€ท๐พ(๐‘,๐œ‰)=1๐œ‰(๐‘)+๐œ‰๎€ธ๎€ทโˆ’1ฮฉโˆ’(๐‘,๐œ‰)โˆ’ฮ›โˆ’2๎€ธ(๐‘,๐œ‰)โˆ’ฮ›โˆ’3(๐‘,๐œ‰).(2.59) In subsequent consideration we will have to use the following property of the limiting values of a Cauchy type integral, the statement of which we now quote. The proofs may be found in [19].Lemma 2.5. If ๐ฟ is a smooth closed contour and ๐œ™(๐‘ž) a function that satisfies the Hรถlder condition on ๐ฟ, then the limiting values of the Cauchy type integral 1ฮฆ(๐‘ง)=๎€œ2๐œ‹๐‘–๐ฟ1๐‘žโˆ’๐‘ง๐œ™(๐‘ž)๐‘‘๐‘ž(2.60) also satisfy this condition.
Since ๐‘”1(๐‘,๐œ‰) satisfies on Re๐‘=0 the Hรถlder condition, on basis of this Lemma the function (1/๐‘Œ+(๐‘,๐œ‰))๐‘”1(๐‘,๐œ‰) also satisfies this condition. Therefore in accordance with Theorem 2.2 it can be uniquely represented in the form of the difference of the functions ๐‘ˆ+(๐‘,๐œ‰) and ๐‘ˆโˆ’(๐‘,๐œ‰), constituting the boundary values of the analytic function ๐‘ˆ(๐‘ง,๐œ‰), given by formula 1๐‘ˆ(๐‘ง,๐œ‰)=๎€œ2๐œ‹๐‘–๐‘–โˆžโˆ’๐‘–โˆž11๐‘žโˆ’๐‘ง๐‘Œ+๐‘”(๐‘ž,๐œ‰)1(๐‘ž,๐œ‰)๐‘‘๐‘ž.(2.61) Therefore the problem (2.58) takes the form ฮฉ+1(๐‘,๐œ‰)๐‘Œ+(๐‘,๐œ‰)+๐‘ˆ+ฮฉ(๐‘,๐œ‰)=โˆ’1(๐‘,๐œ‰)๐‘Œโˆ’(๐‘,๐œ‰)+๐‘ˆโˆ’(๐‘,๐œ‰).(2.62) The last relation indicates that the function (ฮฉ+1/๐‘Œ+)+๐‘ˆ+, analytic in Re๐‘ง<0, and the function (ฮฉโˆ’1/๐‘Œโˆ’)+๐‘ˆโˆ’, analytic in Re๐‘ง>0, constitute the analytic continuation of each other through the contour Re๐‘ง=0. Consequently, they are branches of unique analytic function in the entire plane. According to generalize Liouville theorem this function is some arbitrary constant ๐ด. Thus, bearing in mind the representations (2.59) and (2.52) we get ฮฉ+(๐‘,๐œ‰)=๐‘Œ+๎€ท๐ดโˆ’๐‘ˆ+๎€ธ+๐œ‰ฮ›+,ฮฉโˆ’๐œ‰(๐‘,๐œ‰)=๐พ1๐‘Œ(๐‘)+๐œ‰โˆ’(๐ดโˆ’๐‘ˆโˆ’๎€ทฮ›)+๐œ‰+1+ฮ›โˆ’2๎€ธ.(2.63) Since there exists only one root ๐‘˜(๐œ‰) of equation ๐พ1(๐‘ง)=โˆ’๐œ‰ such that Re๐‘˜(๐œ‰)>0 for all Re๐œ‰>0, therefore, in the expression for the function ฮฉโˆ’(๐‘ง,๐œ‰) the factor ๐œ‰/(๐พ1(๐‘ง)+๐œ‰) has a pole in the point ๐‘ง=๐‘˜(๐œ‰). Also the function ๐œ‰ฮ›+1 has a pole in the point ๐‘ง=๐‘˜(๐œ‰). Thus in general case the problem (2.36) is insolvable. It is soluble only when the functions ๐‘ˆโˆ’(๐‘ง,๐œ‰) and ๐œ‰ฮ›+1 satisfy additional conditions. For analyticity of ฮฉโˆ’(๐‘ง,๐œ‰) in points ๐‘ง=๐‘˜(๐œ‰) it is necessary that Res๐‘=๐‘˜(๐œ‰)๎‚ป1๐พ1๐‘Œ(๐‘)+๐œ‰โˆ’(๐ดโˆ’๐‘ˆโˆ’)+ฮ›+1๎‚ผ=0.(2.64) We reduce (2.64) to the form ๐ด๐‘Œโˆ’(๐‘˜(๐œ‰),๐œ‰)โˆ’๐‘Œโˆ’(๐‘˜(๐œ‰),๐œ‰)๐‘ˆโˆ’๎‚ตโˆ’๐œ‰(๐‘˜(๐œ‰),๐œ‰)+๐‘˜(๐œ‰)ฬ‚๐‘ข(0,๐œ‰)+ฬ‚๐‘ข0(๐‘˜(๐œ‰))โˆ’ฬ‚๐‘ข๐‘ฅ๎‚ถ(0,๐œ‰)=0.(2.65) Multiplying the last relation by 1/๐‘Œโˆ’(๐‘˜(๐œ‰),๐œ‰) and taking limit ๐œ‰โ†’โˆž we get that ๐ด=0. This implies that for solubility of the nonhomogeneous problem (2.36) it is necessary and sufficient that the following condition is satisfied: ๐‘Œโˆ’(๐‘˜(๐œ‰),๐œ‰)๐‘ˆโˆ’๐œ‰(๐‘˜(๐œ‰),๐œ‰)+๐‘˜(๐œ‰)ฬ‚๐‘ข(0,๐œ‰)โˆ’ฬ‚๐‘ข0(๐‘˜(๐œ‰))+ฬ‚๐‘ข๐‘ฅ(0,๐œ‰)=0.(2.66)
Therefore, we need to put in the problem (2.9) one boundary data and the rest of boundary data can be found from (2.66). Thus, for example, if we put ๐‘ข(0,๐‘ก)=0 from (2.66) we obtain for the Laplace transform of ๐‘ข๐‘ฅ(0,๐‘ก),โˆ’[๐‘Œโˆ’](๐‘˜(๐œ‰),๐œ‰)๐ผ(๐‘˜(๐œ‰),๐œ‰)โˆ’1ฬ‚๐‘ข๐‘ฅ(0,๐œ‰)=ฬ‚๐‘ข0(๐‘˜(๐œ‰))โˆ’๐‘Œโˆ’1(๐‘˜(๐œ‰),๐œ‰)๎€œ2๐œ‹๐‘–๐‘–โˆžโˆ’๐‘–โˆž1๐‘žโˆ’๐‘˜(๐œ‰)ฬ‚๐‘ข0(๐‘ž)๐‘Œ+๐พ(๐‘ž,๐œ‰)1(๐‘ž)โˆ’๐พ(๐‘ž)๐พ1(๐‘ž)+๐œ‰๐‘‘๐‘ž,(2.67) where 1๐ผ(๐‘˜(๐œ‰),๐œ‰)=๎€œ2๐œ‹๐‘–๐‘–โˆžโˆ’๐‘–โˆž11๐‘žโˆ’๐‘˜(๐œ‰)๐‘Œ+๐พ(๐‘ž,๐œ‰)1(๐‘ž)โˆ’๐พ(๐‘ž)๐พ1(๐‘ž)+๐œ‰๐‘‘๐‘ž.(2.68) Now we prove that the coefficient of ฬ‚๐‘ข๐‘ฅ(0,๐œ‰) does not vanish for all Re๐œ‰>0. We represent the function ๐ผ(๐‘˜(๐œ‰),๐œ‰) in the form ๐ผ=๐ผ1+๐ผ2,(2.69) where ๐ผ1=1๎€œ2๐œ‹๐‘–๐‘–โˆžโˆ’๐‘–โˆž11๐‘žโˆ’๐‘˜(๐œ‰)๐‘Œ+๐ผ(๐‘ž,๐œ‰)๐‘‘๐‘ž,21=โˆ’๎€œ2๐œ‹๐‘–๐‘–โˆžโˆ’๐‘–โˆž11๐‘žโˆ’๐‘˜(๐œ‰)๐‘Œ+(๐‘ž,๐œ‰)๐พ(๐‘ž)+๐œ‰๐พ1(๐‘ž)+๐œ‰๐‘‘๐‘ž.(2.70) Since, for Re๐‘งโ‰ 0,๎€œ๐‘–โˆžโˆ’๐‘–โˆž1๐‘žโˆ’๐‘ง๐‘‘๐‘ž=โˆ’๐œ‹๐‘–sgn(Re๐‘ง),(2.71) making use of analytic properties of the function ((1/๐‘Œ+(๐‘ž,๐œ‰))โˆ’1) by Cauchy Theorem we have ๐ผ1=1๎€œ2๐œ‹๐‘–๐‘–โˆžโˆ’๐‘–โˆž11๐‘žโˆ’๐‘˜(๐œ‰)๐‘‘๐‘ž+๎€œ2๐œ‹๐‘–๐‘–โˆžโˆ’๐‘–โˆž1๎‚ต1๐‘žโˆ’๐‘˜(๐œ‰)๐‘Œ+๎‚ถ1(๐‘ž,๐œ‰)โˆ’1๐‘‘๐‘ž=โˆ’2,(2.72) where ๐œ‰ is some fixed point, Re๐œ‰>0. To calculate the function ๐ผ2 we will use the identity (2.43). Observe that the function 1/๐‘Œโˆ’(๐‘ž,๐œ‰) is analytic for all Re๐‘ž>0. Therefore, setting the relation (2.43) into definition of ๐ผ2 and making use of Cauchy Theorem we find ๐ผ21=โˆ’๎€œ2๐œ‹๐‘–๐‘–โˆžโˆ’๐‘–โˆž11๐‘žโˆ’๐‘˜(๐œ‰)๐‘Œโˆ’๎‚ต1(๐‘ž,๐œ‰)๐‘‘๐‘ž=๐‘Œโˆ’๎‚ถ+1(๐‘˜(๐œ‰),๐œ‰)โˆ’12.(2.73) Thus, from (2.72) and (2.73) we obtain the following relation for the function ๐ผโˆถ1๐ผ(๐‘˜(๐œ‰),๐œ‰)=๐‘Œโˆ’(๐‘˜(๐œ‰),๐œ‰)โˆ’1.(2.74) Substituting this formula into (2.67) we get ฬ‚๐‘ข๐‘ฅ(0,๐œ‰)=ฬ‚๐‘ข0(๐‘˜(๐œ‰))๐‘Œโˆ’โˆ’1(๐‘˜(๐œ‰),๐œ‰)๎€œ2๐œ‹๐‘–๐‘–โˆžโˆ’๐‘–โˆž1๐‘žโˆ’๐‘˜(๐œ‰)ฬ‚๐‘ข0(๐‘ž)๐‘Œ+๐พ(๐‘ž,๐œ‰)1(๐‘ž)โˆ’๐พ(๐‘ž)๐พ1(๐‘ž)+๐œ‰๐‘‘๐‘ž.(2.75) Now we return to problem (2.36). From (2.63) under the conditions ๐‘ข(0,๐‘ก)=0 and (2.75) the limiting values of solution of (2.36) are given by ฮฉ+(๐‘,๐œ‰)=โˆ’๐‘Œ+๐‘ˆ++๐œ‰ฮ›+2,ฮฉโˆ’๐œ‰(๐‘,๐œ‰)=โˆ’๐พ1(๐‘Œ๐‘)+๐œ‰โˆ’๐‘ˆโˆ’+๐œ‰ฮ›โˆ’2.(2.76) From (2.76) with the help of the integral representations (2.61) and (2.50), for sectionally analytic functions ๐‘ˆ(๐‘ง,๐œ‰) and ฮ›2(๐‘ง,๐œ‰), making use of Sokhotzki-Plemelj formula (2.3) and relation (2.45) we can express the difference limiting values of the function ฮฉ(๐‘ง,๐œ‰) in the form ฮฉ+(๐‘,๐œ‰)โˆ’ฮฉโˆ’(๐‘,๐œ‰)=โˆ’๐‘Œ+๐‘ˆ++๐œ‰๐พ1(๐‘Œ๐‘)+๐œ‰โˆ’๐‘ˆโˆ’๎€ทฮ›+๐œ‰+2โˆ’ฮ›โˆ’2๎€ธ=โˆ’๐‘Œ+๎‚ต๐‘ˆ+โˆ’๐œ‰๐‘ˆ๐พ(๐‘)+๐œ‰โˆ’๎‚ถ๎€ทฮ›+๐œ‰+2โˆ’ฮ›โˆ’2๎€ธ1=โˆ’2๐พ(๐‘)๐‘”๐พ(๐‘)+๐œ‰1โˆ’(๐‘,๐œ‰)๐พ(๐‘)๐‘Œ๐พ(๐‘)+๐œ‰+1(๐‘,๐œ‰)๎€œ2๐œ‹๐‘–๐‘ƒ๐‘‰๐‘–โˆžโˆ’๐‘–โˆž11๐‘žโˆ’๐‘๐‘Œ+๐‘”(๐‘ž,๐œ‰)1(๐‘ž,๐œ‰)๐‘‘๐‘ž.(2.77) We now proceed to find the unknown function ๎ฮฆ(๐‘,๐œ‰) involved in the formula (2.27) for the solution ฬ‚ฬ‚๐‘ข(๐‘,๐œ‰) of the problem (2.9). Replacing the difference ฮฉ+(๐‘,๐œ‰)โˆ’ฮฉโˆ’(๐‘,๐œ‰) in the relation (2.38) by formula (2.77) we get ๎ฮฆ(๐‘,๐œ‰)=๐พ(๐‘)+๐œ‰๎€ทฮฉK(๐‘)+(๐‘,๐œ‰)โˆ’ฮฉโˆ’(๎€ธ1๐‘,๐œ‰)=โˆ’2๐‘”1(๐‘,๐œ‰)โˆ’๐‘Œ+1(๐‘,๐œ‰)๎€œ2๐œ‹๐‘–๐‘ƒ๐‘‰๐‘–โˆžโˆ’๐‘–โˆž11๐‘žโˆ’๐‘๐‘Œ+๐‘”(๐‘ž,๐œ‰)1(๐‘ž,๐œ‰)๐‘‘๐‘ž.(2.78) It is easy to observe that ๎ฮฆ(๐‘,๐œ‰) is boundary value of the function analytic in the left complex semi-plane and therefore satisfies our basic assumption for all Re๐‘ง>0โ„™{ฮฆ}=0.(2.79) Having determined the function ๎ฮฆ(๐‘,๐œ‰) bearing in mind formula (2.27) and conditions ๐‘ข(0,๐‘ก)=0 we determine required function ฬ‚ฬ‚๐‘ขโˆถฬ‚1ฬ‚๐‘ข(๐‘,๐œ‰)=๐พ๎€ท(๐‘)+๐œ‰ฬ‚๐‘ข0(๐‘)โˆ’ฬ‚๐‘ข๐‘ฅ๎€ธโˆ’1(0,๐œ‰)21๐พ๐‘”(๐‘)+๐œ‰1โˆ’1(๐‘,๐œ‰)๐‘Œ๐พ(๐‘)+๐œ‰+(1๐‘,๐œ‰)๎€œ2๐œ‹๐‘–๐‘ƒ๐‘‰๐‘–โˆžโˆ’๐‘–โˆž11๐‘žโˆ’๐‘๐‘Œ+๐‘”(๐‘ž,๐œ‰)1(๐‘ž,๐œ‰)๐‘‘๐‘ž,(2.80) where the function ๐‘”1(๐‘,๐œ‰) is given by formula (2.55): ๐‘”1๐พ(๐‘,๐œ‰)=1(๐‘ž)โˆ’๐พ(๐‘ž)๐พ1(๎€ท๐‘ž)+๐œ‰ฬ‚๐‘ข0(๐‘ž)โˆ’ฬ‚๐‘ข๐‘ฅ๎€ธ.(0,๐œ‰)(2.81) Now we prove that, in accordance with last relation, the function ฬ‚ฬ‚๐‘ข(๐‘,๐œ‰) constitutes the limiting value of an analytic function in Re๐‘ง>0.
With the help of the integral representations (2.61), (2.31), and (2.50) for sectionally analytic functions ๐‘ˆ(๐‘ง,๐œ‰),ฮ›(๐‘ง,๐œ‰), and ฮ›2(๐‘ง,๐œ‰), and making use of Sokhotzki-Plemelj formula (2.3) we have 1๐พ๎€ท(๐‘)+๐œ‰ฬ‚๐‘ข0(๐‘)โˆ’ฬ‚๐‘ข๐‘ฅ๎€ธ(0,๐œ‰)=ฮ›+โˆ’ฮ›โˆ’,1๐พ๐‘”(๐‘)+๐œ‰1(๐‘,๐œ‰)=ฮ›+2โˆ’ฮ›โˆ’2,1๎€œ2๐œ‹๐‘–PV๐‘–โˆžโˆ’๐‘–โˆž11๐‘žโˆ’๐‘๐‘Œ+๐‘”(๐‘ž,๐œ‰)1(1๐‘ž,๐œ‰)๐‘‘๐‘ž=2๎€ท๐‘ˆ++๐‘ˆโˆ’๎€ธ.(2.82) Substituting these relations into (2.80) we express the function ฬ‚ฬ‚๐‘ข in the following form: ฬ‚๎€ทฮ›ฬ‚๐‘ข=+โˆ’ฮ›โˆ’๎€ธโˆ’12๎€ทฮ›+2โˆ’ฮ›โˆ’2๎€ธโˆ’121๐พ๐‘Œ(๐‘)+๐œ‰+๎€ท๐‘ˆ++๐‘ˆโˆ’๎€ธ.(2.83) If it is taken into account that ฮ›(๐‘ง,๐œ‰)=ฮ›1(๐‘ง,๐œ‰)+ฮ›2(๐‘ง,๐œ‰) by virtue of the relation (2.45), the last expression agrees with formula ฬ‚ฬ‚๐‘ข=ฮ›+1โˆ’ฮ›โˆ’1+12๎€ทฮ›+2โˆ’ฮ›โˆ’2๎€ธโˆ’121๐พ๐‘Œ(๐‘)+๐œ‰+๐‘ˆ+โˆ’121๐พ1๐‘Œ(๐‘)+๐œ‰โˆ’๐‘ˆโˆ’.(2.84) Expressing the function ๐‘ˆ+ in the last equation in terms of ๐‘ˆโˆ’๐‘ˆ+=๐‘ˆโˆ’+1๐‘Œ+๎€ทฮ›(๐พ(๐‘)+๐œ‰)+2โˆ’ฮ›โˆ’2๎€ธ,(2.85) we arrive at the following relation: ฬ‚ฬ‚๐‘ข=ฮ›+1โˆ’ฮ›โˆ’1โˆ’1๐พ1๐‘Œ(๐‘)+๐œ‰โˆ’๐‘ˆโˆ’,(2.86) where by virtue of (2.49) and (2.66), ฮ›+1โˆ’ฮ›โˆ’1=1๐พ1๎€ท(๐‘)+๐œ‰ฬ‚๐‘ข0(๐‘)โˆ’ฬ‚๐‘ข๐‘ฅ๎€ธ.(0,๐œ‰)(2.87) Thus the function ฬ‚ฬ‚๐‘ข is the limiting value of an analytic function in Re๐‘ง>0. Note the fundamental importance of the proven fact that the solution ฬ‚ฬ‚๐‘ข constitutes an analytic function in Re๐‘ง>0 and, as a consequence, its inverse Laplace transform vanishes for all ๐‘ฅ<0. We now return to solution ๐‘ข(๐‘ฅ,๐‘ก) of the problem (2.9).
Under assumption ๐‘ข(0,๐‘ก)=0 the integral representation (2.61) takes form 1๐‘ˆ(๐‘ง,๐œ‰)=๎€œ2๐œ‹๐‘–๐‘–โˆžโˆ’๐‘–โˆž11๐‘žโˆ’๐‘ง๐‘Œ+๐พ(๐‘ž,๐œ‰)1(๐‘ž)โˆ’๐พ(๐‘ž)๐พ1๎€ท(๐‘ž)+๐œ‰ฬ‚๐‘ข0(๐‘ž)โˆ’ฬ‚๐‘ข๐‘ฅ๎€ธ(0,๐œ‰)๐‘‘๐‘ž,(2.88) where ฬ‚๐‘ข๐‘ฅ(0,๐œ‰) is defined by (2.75). Substituting this relation into (2.86) and taking inverse Laplace transform with respect to time and inverse Fourier transform with respect to space variables we obtain ๐‘ข(๐‘ฅ,๐‘ก)=๐’ข(๐‘ก)๐‘ข0=๎€œโˆž0๐บ(๐‘ฅ,๐‘ฆ,๐‘ก)๐‘ข0(๐‘ฆ)๐‘‘๐‘ฆ,(2.89) where the function ๐บ(๐‘ฅ,๐‘ฆ,๐‘ก) was defined by formula (2.12). Proposition 2.4 is proved.

Now we collect some preliminary estimates of the Green operator ๐’ข(๐‘ก). Let the contours ๐’ž๐‘– be defined as

๐’ž1=๎‚†๎€ท๐‘โˆˆโˆž๐‘’โˆ’๐‘–(๐œ‹/2+๐œ€)๎€ธ๎š๎€ท,00,โˆž๐‘’๐‘–(๐œ‹/2+๐œ€)๎€ธ๎‚‡๐’ž,(2.90)2=๎‚†๎€ท๐‘žโˆˆโˆž๐‘’โˆ’๐‘–((๐œ‹/2)+2๐œ€)๎€ธ๎š๎€ท,00,โˆž๐‘’๐‘–((๐œ‹/2)+2๐œ€)๎€ธ๎‚‡๐’ž,(2.91)3=๎‚†๎€ท๐‘žโˆˆโˆž๐‘’โˆ’๐‘–((๐œ‹+๐œ€)/2)๎€ธ๎š๎€ท,00,โˆž๐‘’๐‘–((๐œ‹+๐œ€)/2)๎€ธ๎‚‡,(2.92) where ๐œ€>0 can be chosen such that all functions under integration are analytic and Re๐‘˜(๐œ‰)>0 for ๐œ‰โˆˆ๐’ž1.

Lemma 2.6. The function ๐บ(๐‘ฅ,๐‘ฆ,๐‘ก) given by formula (2.12) has the following representation: 1๐บ(๐‘ฅ,๐‘ฆ,๐‘ก)=๎€œ2๐œ‹๐‘–๐‘–โˆžโˆ’๐‘–โˆž๐‘’๐‘๐‘ฅโˆ’๐พ(๐‘)๐‘ก๐‘’โˆ’๐‘๐‘ฆโˆ’1๐‘‘๐‘12๐œ‹๐‘–๎€œ2๐œ‹๐‘–๐’ž1๐‘‘๐œ‰๐‘’๐œ‰๐‘ก๎€œ๐’ž2๐‘’๐‘๐‘ฅ๐‘’ฮ“(๐‘,๐œ‰)๐‘ค+(๐‘,๐œ‰)(๐‘โˆ’๐‘˜(๐œ‰))๐พ(๐‘)+๐œ‰๐ผ(๐‘,๐œ‰,๐‘ฆ)๐‘‘๐‘,(2.93) where 1๐ผ(๐‘,๐œ‰,๐‘ฆ)=๎€œ2๐œ‹๐‘–๐‘–โˆžโˆ’๐‘–โˆž11(๐‘žโˆ’๐‘)(๐‘žโˆ’๐‘˜(๐œ‰))๐‘’ฮ“+(๐‘ž,๐œ‰)๐‘ค+๐‘’(๐‘ž,๐œ‰)โˆ’๐‘ž๐‘ฆ1๐‘‘๐‘ž,ฮ“(๐‘ง,๐œ‰)=๎€œ2๐œ‹๐‘–๐‘–โˆžโˆ’๐‘–โˆž1๐‘žโˆ’๐‘งln๎‚ป๎‚ต๐พ(๐‘ž)+๐œ‰๐พ1๎‚ถ๐‘ค(๐‘ž)+๐œ‰โˆ’(๐‘ž)๐‘ค+๎‚ผ(๐‘ž)๐‘‘๐‘ž.(2.94) The functions ๐‘คยฑ(๐‘ž,๐œ‰),๐‘˜(๐œ‰) were defined in formulas (2.13) and (2.10).

Proof. We rewrite formula (2.12) in the form ๐บ(๐‘ฅ,๐‘ฆ,๐‘ก)=๐ฝ1(๐‘ฅโˆ’๐‘ฆ,๐‘ก)+๐ฝ2(๐‘ฅ,๐‘ฆ,๐‘ก)+๐ฝ3(๐‘ฅ,๐‘ฆ,๐‘ก)+๐ฝ4(๐‘ฅ,๐‘ฆ,๐‘ก),(2.95) where ๐ฝ11(๐‘ฅโˆ’๐‘ฆ,๐‘ก)=๎€œ2๐œ‹๐‘–๐‘–โˆžโˆ’๐‘–โˆž๐‘’๐‘(๐‘ฅโˆ’๐‘ฆ)โˆ’๐พ1(๐‘)๐‘ก๐ฝ๐‘‘๐‘,21(๐‘ฅ,๐‘ฆ,๐‘ก)=12๐œ‹๐‘–๎€œ2๐œ‹๐‘–๐‘–โˆž+๐œ€โˆ’๐‘–โˆž+๐œ€๐‘‘๐œ‰๐‘’๐œ‰๐‘ก๎€œฮจ(๐œ‰,๐‘ฆ)๐‘–โˆžโˆ’๐‘–โˆž๐‘’๐‘๐‘ฅ1๐พ1(๐ฝ๐‘)+๐œ‰๐‘‘๐‘,31(๐‘ฅ,๐‘ฆ,๐‘ก)=โˆ’12๐œ‹๐‘–๎€œ2๐œ‹๐‘–๐‘–โˆž+๐œ€โˆ’๐‘–โˆž+๐œ€๐‘‘๐œ‰๐‘’๐œ‰๐‘ก๎€œ๐‘–โˆžโˆ’๐‘–โˆž๐‘’๐‘๐‘ฅ๐‘Œโˆ’(๐‘,๐œ‰)๐พ1ฮฅ(๐‘)+๐œ‰โˆ’1๐ฝ(๐‘,๐œ‰,๐‘ฆ)๐‘‘๐‘,(2.96)41(๐‘ฅ,๐‘ฆ,๐‘ก)=โˆ’12๐œ‹๐‘–๎€œ2๐œ‹๐‘–๐‘–โˆž+๐œ€โˆ’๐‘–โˆž+๐œ€๐‘‘๐œ‰๐‘’๐œ‰๐‘ก๎€œฮจ(๐œ‰,๐‘ฆ)๐‘–โˆžโˆ’๐‘–โˆž๐‘’๐‘๐‘ฅ๐‘Œโˆ’(๐‘,๐œ‰)๐พ1ฮฅ(๐‘)+๐œ‰โˆ’1(๐‘,๐œ‰,0)๐‘‘๐‘.(2.97) Here ฮฅ11(๐‘ง,๐œ‰,๐‘ฆ)=๎€œ2๐œ‹๐‘–๐‘–โˆžโˆ’๐‘–โˆž11๐‘žโˆ’๐‘ง๐‘Œ+๐พ(๐‘ž,๐œ‰)1(๐‘ž)โˆ’๐พ(๐‘ž)๐พ1๐‘’(๐‘ž)+๐œ‰โˆ’๐‘ž๐‘ฆ๐‘’๐‘‘๐‘ž,ฮจ(๐œ‰,๐‘ฆ)=โˆ’โˆ’๐‘˜(๐œ‰)๐‘ฆ๐‘Œโˆ’(๐‘˜(๐œ‰),๐œ‰)+ฮฅ1๐‘Œ(๐‘˜(๐œ‰),๐œ‰,๐‘ฆ),ยฑ=๐‘’ฮ“ยฑ๐‘คยฑ,1(2.98)ฮ“(๐‘ง,๐œ‰)=๎€œ2๐œ‹๐‘–๐‘–โˆžโˆ’๐‘–โˆž1๐พ๐‘žโˆ’๐‘งln๎‚ป๎‚ต(๐‘ž)+๐œ‰๐พ1(๎‚ถ๐‘ค๐‘ž)+๐œ‰โˆ’(๐‘ž)๐‘ค+(๎‚ผ๐‘ž)๐‘‘๐‘ž.(2.99) Firstly we consider the sectionally analytic function ฮฅ1(๐‘ง,๐œ‰,๐‘ฆ) given by Cauchy type integral (2.98).
On basis of the definition (2.98) its limiting value can be represent in the form ฮฅโˆ’1(๐‘,๐œ‰,๐‘ฆ)=๐ผ1(๐‘,๐œ‰,๐‘ฆ)+๐ผ2(๐‘,๐œ‰,๐‘ฆ),(2.100) where ๐ผ1(๐‘,๐œ‰,๐‘ฆ)=lim๐‘งโ†’๐‘,Re๐‘ง>01๎€œ2๐œ‹๐‘–๐‘–โˆžโˆ’๐‘–โˆž11๐‘žโˆ’๐‘ง๐‘Œ+๐‘’(๐‘ž,๐œ‰)โˆ’๐‘ž๐‘ฆ๐ผ๐‘‘๐‘ž,2(๐‘,๐œ‰,๐‘ฆ)=โˆ’lim๐‘งโ†’๐‘,Re๐‘ง>01๎€œ2๐œ‹๐‘–๐‘–โˆžโˆ’๐‘–โˆž11๐‘žโˆ’๐‘ง๐‘Œ+(๐‘ž,๐œ‰)๐พ(๐‘ž)+๐œ‰๐พ1๐‘’(๐‘ž)+๐œ‰โˆ’๐‘ž๐‘ฆ๐‘‘๐‘ž.(2.101) Making use of analytic properties of the functions (1/๐‘Œ+(๐‘ž,๐œ‰)โˆ’1), for Re๐‘ž<0, and ๐‘’โˆ’๐‘ž๐‘ฆ, for Re๐‘ž>0, by Cauchy theorem we have ๐ผ1(๐‘,๐œ‰,๐‘ฆ)=lim๐‘งโ†’๐‘,Re๐‘ง>01๎€œ2๐œ‹๐‘–๐‘–โˆžโˆ’๐‘–โˆž1๐‘’๐‘žโˆ’๐‘งโˆ’๐‘ž๐‘ฆ๐‘‘๐‘ž+lim๐‘งโ†’๐‘,Re๐‘ง>01๎€œ2๐œ‹๐‘–๐‘–โˆžโˆ’๐‘–โˆž1๎‚ต1๐‘žโˆ’๐‘ง๐‘Œ+๎‚ถ(๐‘ž,๐œ‰)โˆ’1(๐‘’โˆ’๐‘ž๐‘ฆโˆ’1)๐‘‘๐‘ž+lim๐‘งโ†’๐‘,Re๐‘ง>01๎€œ2๐œ‹๐‘–๐‘–โˆžโˆ’๐‘–โˆž1๎‚ต1๐‘žโˆ’๐‘ง๐‘Œ+๎‚ถ(๐‘ž,๐œ‰)โˆ’1๐‘‘๐‘ž=โˆ’๐‘’โˆ’๐‘๐‘ฆ+lim๐‘งโ†’๐‘,Re๐‘ง>01๎€œ2๐œ‹๐‘–๐‘–โˆžโˆ’๐‘–โˆž1๎‚ต1๐‘žโˆ’๐‘ง๐‘Œ+๎‚ถ(๐‘ž,๐œ‰)โˆ’1(๐‘’โˆ’๐‘ž๐‘ฆโˆ’1)๐‘‘๐‘ž,(2.102) where ๐œ‰ is some fixed point, Re๐œ‰>0.
To calculate the function ๐ผ2(๐‘,๐œ‰,๐‘ฆ) we will use the following identity: 1๐‘Œ+(๐‘ž,๐œ‰)๐พ(q)+๐œ‰๐พ1=1(๐‘ž)+๐œ‰๐‘Œโˆ’.(๐‘ž,๐œ‰)(2.103) Observe that the function 1/๐‘Œโˆ’(๐‘ž,๐œ‰) is analytic for all Re๐‘ž>0. Therefore, setting the relation (2.103) into definition of ๐ผ2(๐‘,๐œ‰,๐‘ฆ) and making use of Cauchy theorem we find ๐ผ2(๐‘,๐œ‰,๐‘ฆ)=โˆ’lim๐‘งโ†’๐‘,Re๐‘ง>01๎€œ2๐œ‹๐‘–๐‘–โˆžโˆ’๐‘–โˆž11๐‘žโˆ’๐‘ง๐‘Œโˆ’๐‘’(๐‘ž,๐œ‰)โˆ’๐‘ž๐‘ฆ1๐‘‘๐‘ž=๐‘Œโˆ’๐‘’(๐‘,๐œ‰)โˆ’๐‘๐‘ฆ.(2.104) Thus from (2.102) and (2.104) we obtain the following relation: ฮฅโˆ’1(๐‘,๐œ‰,๐‘ฆ)=โˆ’๐‘’โˆ’๐‘๐‘ฆ+ฮฅโˆ’1(๐‘,๐œ‰,๐‘ฆ)+๐‘Œโˆ’๐‘’(๐‘,๐œ‰)โˆ’๐‘๐‘ฆ,(2.105) where 1ฮฅ(๐‘ง,๐œ‰,๐‘ฆ)=๎€œ2๐œ‹๐‘–๐‘–โˆžโˆ’๐‘–โˆž1๎‚ต1๐‘žโˆ’๐‘ง๐‘Œ+๎‚ถ๐‘’(๐‘ž,๐œ‰)โˆ’1โˆ’๐‘ž๐‘ฆ๐‘‘๐‘ž.(2.106) In the same way (see also proof of relation (2.74)) we can prove that ฮฅโˆ’11(๐‘,๐œ‰,0)=โˆ’1+๐‘Œโˆ’.(๐‘,๐œ‰)(2.107) Also we observe that ฮจ(๐œ‰,๐‘ฆ)=โˆ’๐‘’โˆ’๐‘˜(๐œ‰)๐‘ฆ+ฮฅ(๐‘˜(๐œ‰),๐œ‰,๐‘ฆ).(2.108) Inserting into definition (2.96) the expression (2.105) for ฮฅโˆ’1(๐‘,๐œ‰,๐‘ฆ) we obtain the function ๐ฝ3(๐‘ฅ,๐‘ฆ,๐‘ก) in the form ๐ฝ31(๐‘ฅ,๐‘ฆ,๐‘ก)=โˆ’12๐œ‹๐‘–๎€œ2๐œ‹๐‘–๐‘–โˆžโˆ’๐‘–โˆž๐‘‘๐œ‰๐‘’๐œ‰๐‘ก๎€œ๐‘–โˆžโˆ’๐‘–โˆž๐‘’๐‘๐‘ฅ๐‘Œโˆ’(๐‘,๐œ‰)๐พ1(๐‘)+๐œ‰(โˆ’๐‘’โˆ’๐‘๐‘ฆ+ฮฅโˆ’(๐‘,๐œ‰,๐‘ฆ))๐‘‘๐‘โˆ’๐ฝ1(๐‘ฅโˆ’๐‘ฆ,๐‘ก).(2.109) Replacing in formula (2.97) the functions ฮจ(๐œ‰,๐‘ฆ) and ฮฅโˆ’1(๐‘,๐œ‰,0) by (2.108) and (2.107), respectively, we reduce the function ๐ฝ4(๐‘ฅ,๐‘ฆ,๐‘ก) in the form ๐ฝ41(๐‘ฅ,๐‘ฆ,๐‘ก)=โˆ’12๐œ‹๐‘–๎€œ2๐œ‹๐‘–๐‘–โˆžโˆ’๐‘–โˆž๐‘‘๐œ‰๐‘’๐œ‰๐‘ก๎€œฮจ(๐œ‰,๐‘ฆ)๐‘–โˆžโˆ’๐‘–โˆž๐‘’๐‘๐‘ฅ๐‘Œโˆ’(๐‘,๐œ‰)๐พ1๎‚ต1(๐‘)+๐œ‰โˆ’1+๐‘Œโˆ’๎‚ถ1(๐‘,๐œ‰)๐‘‘๐‘=โˆ’12๐œ‹๐‘–๎€œ2๐œ‹๐‘–๐‘–โˆžโˆ’๐‘–โˆž๐‘‘๐œ‰๐‘’๐œ‰๐‘ก๎€œ๐‘–โˆžโˆ’๐‘–โˆž๐‘’๐‘๐‘ฅ๐‘Œโˆ’(๐‘,๐œ‰)๐พ1(๎€ท๐‘’๐‘)+๐œ‰โˆ’๐‘˜(๐œ‰)๐‘ฆ๎€ธโˆ’ฮฅ(๐‘˜(๐œ‰),๐œ‰,๐‘ฆ)๐‘‘๐‘โˆ’๐ฝ2(๐‘ฅ,๐‘ฆ).(2.110) Therefore inserting into definition (2.95) expressions (2.109) and (2.110), for ๐ฝ3(๐‘ฅ,๐‘ฆ,๐‘ก) and ๐ฝ4(๐‘ฅ,๐‘ฆ,๐‘ก), respectively, we obtain the function ๐บ(๐‘ฅ,๐‘ฆ,๐‘ก) in the form 1๐บ(๐‘ฅ,๐‘ฆ,๐‘ก)=12๐œ‹๐‘–๎€œ2๐œ‹๐‘–๐‘–โˆžโˆ’๐‘–โˆž๐‘‘๐œ‰๐‘’๐œ‰๐‘ก๎€œ๐‘–โˆžโˆ’๐‘–โˆž๐‘’๐‘๐‘ฅ๐‘Œโˆ’(๐‘,๐œ‰)๐พ1ฮž(๐‘)+๐œ‰1(๐‘,๐œ‰,๐‘ฆ)๐‘‘๐‘,(2.111) where ฮž1๎€ท๐‘’(๐‘,๐œ‰,๐‘ฆ)=โˆ’๐‘๐‘ฆโˆ’๐‘’โˆ’๐‘˜(๐œ‰)๐‘ฆ๎€ธโˆ’(ฮฅโˆ’(๐‘,๐œ‰,๐‘ฆ)โˆ’ฮฅ(๐‘˜(๐œ‰),๐œ‰,๐‘ฆ)).(2.112) Also, note that since ฮฅโˆ’(๐‘,๐œ‰,๐‘ฆ)โˆ’ฮฅ(๐‘˜(๐œ‰),๐œ‰,๐‘ฆ)=lim๐‘งโ†’๐‘,Re๐‘ง>01๎€œ2๐œ‹๐‘–๐‘–โˆžโˆ’๐‘–โˆž๎‚ต1โˆ’1๐‘žโˆ’๐‘ง1๐‘žโˆ’๐‘˜(๐œ‰)๎‚ถ๎‚ต๐‘Œ+๎‚ถ๐‘’(๐‘ž,๐œ‰)โˆ’1โˆ’๐‘ž๐‘ฆ๐‘‘๐‘ž=lim๐‘งโ†’๐‘,Re๐‘ง>01๎€œ2๐œ‹๐‘–๐‘–โˆžโˆ’๐‘–โˆž๎‚ต๐‘งโˆ’๐‘˜(๐œ‰)(1๐‘žโˆ’๐‘ง)(๐‘žโˆ’๐‘˜(๐œ‰))๎‚ถ๎‚ต๐‘Œ+(๎‚ถ๐‘’๐‘ž,๐œ‰)โˆ’1โˆ’๐‘ž๐‘ฆ๐‘‘๐‘ž,(2.113) we obtain ๎€ท๐‘’โˆ’๐‘๐‘ฆโˆ’๐‘’โˆ’๐‘˜(๐œ‰)๐‘ฆ๎€ธโˆ’(ฮฅโˆ’(๐‘,๐œ‰,๐‘ฆ)โˆ’ฮฅ(๐‘˜(๐œ‰),๐œ‰,๐‘ฆ))=lim๐‘งโ†’๐‘,Re๐‘ง>01๎€œ2๐œ‹๐‘–๐‘–โˆžโˆ’๐‘–โˆž๎‚ตโˆ’1+1๐‘žโˆ’๐‘ง๐‘’๐‘žโˆ’๐‘˜(๐œ‰)๎‚ถ๎‚ธโˆ’๐‘ž๐‘ฆ+๎‚ต1๐‘Œ+๎‚ถ๐‘’(๐‘ž,๐œ‰)โˆ’1โˆ’๐‘ž๐‘ฆ๎‚น๐‘‘๐‘ž=lim๐‘งโ†’๐‘,Re๐‘ง>01๎€œ2๐œ‹๐‘–๐‘–โˆžโˆ’๐‘–โˆž๎‚ต๐‘˜(๐œ‰)โˆ’๐‘ง๎‚ถ1(๐‘žโˆ’๐‘ง)(๐‘žโˆ’๐‘˜(๐œ‰))๐‘Œ+๐‘’(๐‘ž,๐œ‰)โˆ’๐‘ž๐‘ฆ๐‘‘๐‘ž.(2.114) So, 1๐บ(๐‘ฅ,๐‘ฆ,๐‘ก)=โˆ’12๐œ‹๐‘–๎€œ2๐œ‹๐‘–๐œ€+๐‘–โˆž๐œ€โˆ’๐‘–โˆž๐‘‘๐œ‰๐‘’๐œ‰๐‘ก๎€œ๐‘–โˆžโˆ’๐‘–โˆž๐‘’๐‘๐‘ฅ๐‘Œโˆ’(๐‘,๐œ‰)(๐‘โˆ’๐‘˜(๐œ‰))๐พ1๐ผ(๐‘)+๐œ‰โˆ’(๐‘,๐œ‰,๐‘ฆ)๐‘‘๐‘,(2.115) where 1๐ผ(๐‘ง,๐œ‰,๐‘ฆ)=๎€œ2๐œ‹๐‘–๐‘–โˆžโˆ’๐‘–โˆž11(๐‘žโˆ’๐‘ง)(๐‘žโˆ’๐‘˜(๐œ‰))๐‘Œ+๐‘’(๐‘ž,๐œ‰)โˆ’๐‘ž๐‘ฆ๐‘‘๐‘ž.(2.116) Using relation ๐พ(๐‘)+๐œ‰๐พ1=๐‘Œ(๐‘)+๐œ‰+๐‘Œโˆ’,(2.117) we rewrite last formula in the following form: 1๐บ(๐‘ฅ,๐‘ฆ,๐‘ก)=โˆ’12๐œ‹๐‘–๎€œ2๐œ‹๐‘–๐œ€+๐‘–โˆž๐œ€โˆ’๐‘–โˆž๐‘‘๐œ‰๐‘’๐œ‰๐‘ก๎€œ๐‘–โˆžโˆ’๐‘–โˆž๐‘’๐‘๐‘ฅ๐‘Œ+(๐‘,๐œ‰)(๐‘โˆ’๐‘˜(๐œ‰))