Issue |
Wuhan Univ. J. Nat. Sci.
Volume 28, Number 3, June 2023
|
|
---|---|---|
Page(s) | 185 - 191 | |
DOI | https://doi.org/10.1051/wujns/2023283185 | |
Published online | 13 July 2023 |
Mathematics
CLC number: O175.8
A Class of Inverse Boundary Value Problems for (λ, 1) Bi-Analytic Functions
College of Information Engineering, Fujian Business University, Fuzhou 350506, Fujian, China
Received:
28
September
2022
In this paper, a class of inverse boundary value problems for (λ, 1) bi-analytic functions is given. Using the method of Riemann boundary value problem for analytic functions, the conditions of solvability and the expression of the solutions for the inverse problems are obtained.
Key words: inverse problem / Riemann boundary value problem / (λ, k) bi-analytic functions / canonical function
Biography: LIN Juan, female, Ph.D., Professor, research direction: singular integral equation and applications. E-mail: lj7862124@163.com
Fundation item: Supported by the Natural Science Foundation of Fujian Province (2020J01322)
© Wuhan University 2023
This is an Open Access article distributed under the terms of the Creative Commons Attribution License (https://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
0 Introduction
Boundary value problems for (λ, k) bi-analytic functions are discussed in Refs.[1-6]. In Ref.[4], the author used the Cauchy-type integral of (λ, 1) bi-analytic functions to study a class of inverse Riemann problems of (λ, 1) bi-analytic functions. In this article, we study the general case by using the theory of boundary value problems for analytic functions.
Let be a simple and closed smooth contour, oriented counter-clockwisely.
(
) denote respectively the interior (exterior) region bounded by
. And let
in
. Our inverse boundary value problem, simply called I-BVP, is to find functions
,
and constants
,
, satisfying the following boundary conditions
where ,
,
,
,
,
(
)
(Hölder continuous).
(
) are known constants.
1 Inverse Riemann Boundary Value Problems
In this section, we will consider a class of inverse Riemann boundary value problems, simply called I-RBVP. It is to find and
meeting the following conditions
Suppose that and
, the solvable conditions and the representation of the solutions are obtained.
Assume that (
). Let the Cauchy integral operators
and its projection operator
Let
then we have
By (6) and (2), we obtain
and
where
Now we consider the Riemann boundary value problem, simply called RBVP. It is to find meeting the conditions given in (7). Denote
, then the index of RBVP (7) is
. Suppose
, thus from Refs.[7, 8], we have the following results.
Lemma 1 If , then the solutions of RBVP (7) are
where
is the canonical,
is arbitrary polynomial whose degree does not exceed
.
If , then the RBVP (7) has a unique solution
if and only if
Hence, we obtain
Theorem 1 Under the requirement , when
, the solutions of I-RBVP (2) are
where is given in (11),
is arbitrary polynomial whose degree does not exceed
, and
When , the solutions of I-RBVP (2) are
if and only if
and
2 Inverse Boundary Value Problems for (λ, 1) Bi-Analytic Functions
In this section, we will consider the I-BVP for (λ, 1) bi-analytic functions. It is to find and constants
meeting the following conditions
where ,
(
)
, constants
(
) are known.
In the following discussion, we assume that and
. By (5), we get
Substituting (20) into the first equation in (19) and letting
where ,
, we obtain
(22) and the second equation in (7) imply is a
bi-analytic function with associate function
. By (22), we obtain
where . Let
from the inner domain
and the outer domain
, respectively, one get
and
respectively.
By the second equation and the third equation in (19), we get
and
Submitting in (24) and
in (25) into (26), we obtain
where
From (23) and (22), obviously,
Now we consider the following RBVP. It is to find meeting the conditions (28) and (30), namely,
where
and
.
is the index of RBVP (31). If
is to be finite, then we have the following results from Refs.[7, 8].
Lemma 2 1) If , then the solutions of RBVP (31) are
where is arbitrary polynomial whose degree does not exceed
,
2) If , then the RBVP (31) has a unique solution (33) (
) if and only if
3) is written as following
(26) can be changed to
Submitting in (24) into (37), one get
where is from
(let
from the inner domain
) given in Lemma 2, namely,
so, we get
where
By the fourth equations in (19), we have the following system of equations
Furthermore, we obtain
Remark 1 Take while
. Take
while
.
Summarizing the above discussion, we have the following result.
Theorem 2 For the I-BVP of (λ, 1) bi-analytic functions, the solutions are given in Theorem 1, (36) and (45), namely
( while
,
while
.)
where A(t), B(t), C(t) are given in (41), (42), (43), respectively. is from
given in Lemma 1.
is from
given in Lemma 2. For
and
, there exist four cases.
1) When and
,
and
are given in (10) and (33), respectively.
2) When and
,
is given in (10) and
is given in (33)(
) if and only if the condition (35) is satisfied.
3) When and
,
is given in (12) if and only if the condition (13) is satisfied and
is given in (33).
4) When and
,
and
are given in (12) and (33)(
) if and only if the conditions (13) and (35) are satisfied, respectively.
Remark 2 The upper conclusions may be applied to interface problems of the elastic system in plane, for instance, the welding problems and the quasi-static system of thermoelasticity, etc.Refs.[9-14].
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