- Research
- Open Access
Some classes of singular integral equations of convolution type in the class of exponentially increasing functions
- Pingrun Li^{1}Email author
https://doi.org/10.1186/s13660-017-1580-z
© The Author(s) 2017
- Received: 19 July 2017
- Accepted: 3 December 2017
- Published: 16 December 2017
Abstract
In this article, we study some classes of singular integral equations of convolution type with Cauchy kernels in the class of exponentially increasing functions. Such equations are transformed into Riemann boundary value problems on either a straight line or two parallel straight lines by Fourier transformation. We propose one method different from the classical one for the study of such problems and obtain the general solutions and the conditions of solvability. Thus, the result in this paper improves the theory of integral equations and the classical boundary value problems for analytic functions.
Keywords
- singular integral equations
- Riemann boundary value problems
- dual type
- convolution kernel
- the class of exponentially increasing functions
MSC
- 45E05
- 45E10
- 30E25
1 Introduction
It is well known that singular integral equations (SIEs) and integral equations of convolution type are two basic kinds of equations in the theory of integral equations. There have been many papers studying singular integral equations and a relatively complete theoretical system is almost formed (see, e.g., [1–6]). These equations play important roles in other subjects and practical applications, such as engineering mechanics, physics, fracture mechanics, and elastic mechanics. For operators containing both the Cauchy principal value integral and convolution, Karapetiants-Samko [7] studied the conditions of their Noethericity in the more general case. In recent decades, many mathematicians studied some SIEs of convolution type. Litvinchuk [8] studied a class of Wiener-Hopf type integral equations with convolution and Cauchy kernel and proved the solvability of the equation. Giang-Tuan [9] studied the Noether theory of convolution type SIEs with constant coefficients. Nakazi-Yamamoto [10] proposed a class of convolution SIEs with discontinuous coefficients and transformed the equations into a Riemann boundary value problem (RBVP) by Fourier transform, and given the general solutions of the equation. Later on, Li [11] discussed the SIEs with convolution kernels and periodicity, which can be transformed into a discrete jump problems by discrete Fourier transformation, and the solvable conditions and the explicit expressions of general solutions were obtained.
The purpose of this article is to extend the theory to some classes of singular integral equations of convolution type with Cauchy kernels in the class of exponentially increasing functions. Such equations can be transformed to RBVPs with either an unknown function on a straight line or two unknown functions on two parallel straight lines by Fourier transformation. We prove the existence of the solution for the equations; moreover, the general solutions and the conditions of solvability are obtained under some conditions. Therefore, the result in this paper further generalizes the results of [7–11].
The Fourier transforms used in this paper are understood to be performed in \(L^{2}(\mathbb{R})\) and the functions involved certainly belong to this space.
2 Definitions and lemmas
Definition 2.1
A function \(F(x)\) belongs to class \(\{\{0\}\}\), if the following two conditions are fulfilled:
(1) \(F(x)\in\widehat{H}\), that is, it satisfies the Hölder condition on the whole real domain \(\mathbb{R}\), including ∞, i.e. ±∞;
(2) \(F(x)\in L^{2}(\mathbb{R})\), that is, \(L^{2}(\mathbb{R})=\{F(x)\mid\int ^{+\infty}_{-\infty}| F(x)|^{2}\,dx<+\infty\}\).
Definition 2.2
Definition 2.3
Definition 2.4
If there exists a real constant τ such that \(f(t)e^{-\tau t}\in\{0\}\), we say that \(f(t)\) belongs to the class of exponentially increasing functions, and we denote it as \(f(t)\in\{\tau\} \), where τ is called the order of the exponential increase.
If \(f^{+}(t)\in\{\tau\}\), \(f^{-}(t)\in\{\sigma\}\), then we call \(f(t)\in \{\tau,\sigma\}\), where τ and σ are real constants.
Definition 2.5
We have the following lemmas.
Lemma 2.1
([4])
If \(f(t)\in\{0\}\), \(F(0)=0\), then \(V(Tf(t))=-F(x)\operatorname{sgn}x\), and \(Tf(t)\in\{0\}\).
Lemma 2.2
Proof
Lemmas 2.3-2.4 are obvious facts and we omit their proofs here. □
Lemma 2.3
Let \(f(t)\in\{p\}\), \(f(t)\in\{q\}\), then \(f(t)\in\{ \min (p,q), \max (p,q)\}\).
Lemma 2.4
We introduce the following two lemmas (see [4]).
Lemma 2.5
If \(f(t)\), \(g(t)\in\{0\}\), then \(f*g(t)\in\{0\}\), and \(V[f*g(t)]=F(x)G(x)\), where \(F(x)=Vf(t)\), \(G(x)=Vg(t)\).
Lemma 2.6
Let \(F(x)=Vf(t)\), if \(f(t)\in\{p,q\}\), \(F(ip)=F(iq)=0\), then \(Tf(t)\in\{p,q\}\).
3 Presentation of the problem
In this section we consider the following several classes of SIEs in the class of exponentially increasing functions, and we shall transform these equations into the generalized RBVPs.
(2) Wiener-Hopf type: \(af^{+}(t)+b Tf^{+}(t)+h*f^{+}(t)=g(t)\), \(t\in \mathbb{R}^{+}\).
(3) One convolution kernel: \(af(t)+b Tf(t)+h*f(t)=g(t)\), \(t\in\mathbb{R}\).
(4) Two convolution kernels: \(af(t)+ b Tf(t)+h_{1}*f^{+}(t)+h_{2}*f^{-}(t)=g(t)\), \(t\in\mathbb{R}\), where a, b (\(b\neq0\)), \(a_{j}\), \(b_{j}\) are constants and \(b_{j}\) are not equal to zero simultaneously. In the literature [2, 7], equations (1)-(4) were discussed in class \(\{0\}\) and the general solutions and the conditions of solvability were obtained. In this paper we extend the results of [2, 7] to the class of exponentially increasing functions. Without loss of generality, we mainly study the SIEs of dual type. The method mentioned in this paper may also be applied to solving the other classes of equations.
Since the relationship of the size among \(\sigma_{1}\), \(\sigma_{2}\), \(\tau_{1}\) and \(\tau_{2}\) has 24 kinds of permutations, each permutation shall determine a different class of exponentially increasing function, and then cause different boundary value problems (BVPs). As a whole, after taking the Fourier transform for (3.2), we can obtain the following four cases.
Case 1: Equation (3.2) (or (3.1)) is transformed into the RBVP on one straight line.
Case 2: Equation (3.2) is transformed into the RBVP with two unknown functions on two parallel straight lines.
Case 3: Equation (3.2) is transformed into one-sided BVP.
Case 4: In class \(\{p, q\}\), (3.2) is not solvable.
But in Case 3 it is possible that a solution of (3.1) exists under some additional conditions. By using the method of analytic continuation, this case can be transformed into Case 2, therefore, in this paper we only discuss Case 1 and Case 2.
4 Methods of solutions of (3.4) and (3.12)
4.1 On the solutions of (3.4)
Since \(b_{1}\), \(b_{2}\) are not equal to zero simultaneously, thus (3.4) is the RBVP with discontinuous coefficients and nodal point \(i\tau_{1}\) on \(\operatorname{Im}\xi=\tau_{1}\), and it can be described as follows: we want to get a function \(\Phi(z)\) such that it is analytic in \(\operatorname{Im} \xi>\tau_{1}\), \(\operatorname{Im} \xi<\tau_{1}\), respectively, and satisfies the boundary value condition (3.4).
Since the \(X^{\pm}(t)\) are bounded and \(X^{\pm}(t)\neq0\), it is easy to prove that \(\Phi^{\pm}(t)\), \(\Phi(t)\in\{\{0\}\}\). Putting (4.6) into (3.3), we can obtain \(F(x)\), thus a solution \(f(t)\) of (3.4) is given by \(f(t)=V^{-1}[F(x)]\).
Above all, we have the following.
4.2 The solutions and the solvability conditions of (3.12)
Next, we come to discuss the solvability conditions for equation (3.1).
(1) If \(C_{1}(z)C_{2}(z)\neq0\) in \(l_{2}<\operatorname{Im}z<l_{1}\), we can obtain \(\Phi ^{\pm}(z)\) and \(\Psi(z)\) from (4.15).
Thus, we have the following conclusion as regards the solution of equation (3.1).
Theorem 4.2
Under condition (b), in the case of normal type, if \(C_{1}(z)C_{2}(z)\neq0\) in \(l_{2}<\operatorname{Im}z<l_{1}\), then the solvable condition of (3.1) is (4.20). If \(C_{1}^{*}(z)\), \(C_{2}(z)\) have some common zero-points \(z_{1}^{**},z_{2}^{**},\ldots,z_{\nu}^{**}\) in \(l_{2}<\operatorname{Im}z<l_{1}\), then (4.21) and (4.22) must be augmented. Thus, a solution \(f(t)\) of (3.1) is given by (4.19), in which \(F^{+}(\xi)\), \(F^{-}(\xi)\) are obtained by (4.17) and (4.18), respectively. It is easy to verify that \(f(t)\in\{ p,q\}\).
5 Results and discussion
In this article, some classes of SIEs of convolution type with Cauchy kernels are solved in the class of exponentially increasing functions. By Fourier transform, such equations are transformed into RBVPs on either a straight line or two parallel straight lines. The exact solutions of equation (3.1), denoted by integrals and the conditions of solvability are obtained. Here, our method is different from the ones for the classical boundary value problem, and it is novel and effective. Thus, the result in this paper generalizes the theory of classical boundary value problems and singular integral equations. Similarly, the above equations can also be solved in Clifford analysis (see [14–17]). Further discussion is omitted here.
6 Conclusions
In this paper, we mainly study the singular integral equations of dual type in the class of exponentially increasing functions. This class of equations (that is, equation (3.1)) have important applications in practical problems, such as elastic mechanics, heat conduction, and electrostatics. Hence, the study of equation (3.1) is of significance not only in applications but also in the theory of resolving the equation itself. To many problems, such as piezoelectric material, voltage magnetic materials and functional gradient materials, one can often attribute the problem to finding solutions for this classes of equations. Hence, the result in this paper improves some results in Refs. [2, 4, 9–11, 18, 19], and it supplies a theoretical basis for solving the physics problems involved.
Declarations
Acknowledgements
The author would like to express his gratitude to the anonymous referees for their invaluable comments and suggestions, which helped to improve the quality of the paper. This work is supported financially by the National Natural Science Foundation of China (11501318) and the Natural Science Foundation of Shandong Province of China (ZR2017MA045).
Authors’ contributions
All authors read and approved the final manuscript.
Competing interests
The author declares there to be no conflicts of interest.
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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