Abstract
In this paper, we first define the Teodorescu operator \(T_{\psi,\alpha }\) related to the Helmholtz equation and discuss its properties in quaternion analysis. Then we propose the Riemann boundary value problem related to the Helmholtz equation. Finally we give the integral representation of the boundary value problem by using the previously defined operator.
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1 Introduction
It is well known that the Helmholtz equation is an elliptic partial differential equation describing the electromagnetic wave, which has important applications in geophysics, medicine, engineering application, and many other fields. Many problems associated with the Helmholtz equation have been studied by many scholars, for example [1–5]. The boundary value problem for partial differential equations is an important and meaningful research topic. The singular integral operator is the core component of the solution of the boundary value problem for a partial differential system. The Teodorescu operator is a generalized solution of the inhomogeneous Dirac equation, which plays an important role in the integral representation of the general solution for the boundary value problem. Many experts and scholars have studied the properties of the Teodorescu operator. For example, Vekua [6] first discussed some properties of the Teodorescu operator on the plane and its application in thin shell theory and gas dynamics. Hile [7] and Gilbert [8] studied some properties of the Teodorescu operator in n-dimensional Euclid space and high complex space, respectively. Yang [9] and Gu [10] studied the boundary value problem associated with the Teodorescu operator in quaternion analysis and Clifford analysis, respectively. Wang [11–15] studied the properties of many Teodorescu operators and related boundary value problems.
In this paper, we will study some properties of the singular integral operator and the Riemann boundary value problem associated to the Helmholtz equation using the quaternion analysis method. The structure of this paper is as follows: in Section 2, we review some basic knowledge of quaternion analysis. In Section 3, we first construct a singular integral operator \(T_{\psi,\alpha}\) related to the Helmholtz equation and study some of its properties. In Section 4, we propose the Riemann boundary value problem related to the Helmholtz equation. Finally we give the integral representation of the boundary value problem by using the previously defined operator.
2 Preliminaries
Let \(\{i_{1},i_{2},i_{3}\}\) be an orthogonal basis of the Euclidean space \(R^{3}\) and \(\mathbb{H}(\mathbb{C})\) be the set of complex quaternions with basis
where \(i_{0}\) is the unit and \(i_{1}\), \(i_{2}\), \(i_{3}\) are the quaternionic imaginary units with the following properties:
Then an arbitrary quaternion a can be written as \(a=\sum_{k=0}^{3}a_{k}i_{k}\), \(a_{k}\in\mathbb{C}\). The quaternionic conjugation is defined by \(\bar{a}=a_{0}-\sum_{k=1}^{3}a_{k}\cdot i_{k}\). The norm for an element \(a\in\mathbb{H}(\mathbb{C})\) is taken to be \(|a|=\sqrt{\sum_{k=0}^{3}|a_{k}|^{2}}\). Moreover, if \(a\bar{a}=\bar {a}a=|a|^{2}\) and \(|a|\neq0\), then we say that a is reversible. Obviously, its inverse element can be written as \(a^{-1}=\frac{\bar {a}}{|a|^{2}}\).
Let \(\lambda\in\mathbb{C}\backslash\{0\}\) and let α be its complex square root: \(\alpha\in\mathbb{C}\), \(\alpha ^{2}=\lambda\). Suppose \(\Omega\subset R^{3}\) is a domain and ∂Ω is its boundary. We shall consider functions f defined in \(\Omega\subset R^{3}\) with values in \(\mathbb{H}(\mathbb {C})\). Then f can be expressed as \(f=\sum_{k=0}^{3}f_{k}(x)i_{k}\). Here \(f_{k}(x)\) (\(k=0,1,2,3\)) are complex functions defined on Ω.
Let \(C^{(m)}(\Omega,\mathbb{H}(\mathbb{C}))=\{f\mid f:\Omega \rightarrow\mathbb{H}(\mathbb{C}), f(x)=\sum_{k=0}^{3}f_{k}(x)i_{k}, f_{k}(x)\in C^{m}(\Omega,\mathbb{C})\}\). We define the operators as follows:
where \(\psi=\{\psi_{1},\psi_{2},\psi_{3}\}=\{i_{1},i_{2},i_{3}\}\).
For the above stated α, let us introduce the following operators:
f will be called a left (right)-\((\psi,\alpha)\)-hyperholomorphic in the domain Ω, if \({{}^{\psi}D}_{\alpha}[f]=0\) (\({{}_{\alpha }D}^{\psi}[f]=0\)) in Ω. Let \(\alpha\in\mathbb{C}\backslash \{0\}\) and \(\operatorname{Im}\alpha\neq0\). For \(x\in R^{3}\backslash\{0\}\), we introduce the following notation:
In both cases it is a fundamental solution of the Helmholtz equation with \(\lambda=\alpha^{2}\). Then the fundamental solution to the operator \({{}^{\psi}D}_{\alpha}\), \(\mathcal{K}_{\psi,\alpha}\) is given by
If \(f(x)\in L^{p,\sigma}(R^{3},\mathbb{H}(\mathbb{C}))\) means that \(f(x)\in L^{p}(B,\mathbb{H}(\mathbb{C}))\), \(f^{(\sigma )}(x)=|x|^{-\sigma}f(\frac{\overline{x}}{|x|^{2}})\in L^{p}(B,\mathbb{H}(\mathbb{C}))\), in which \(B=\{x\mid |x|<1\}\), σ is a real number, \(\|f\|_{p,\sigma}=\|f\|_{L^{p}(B)}+\|f^{(\sigma )}\|_{L^{p}(B)}\), \(p\geq1\).
Definition 2.1
Suppose that the functions u, v, φ are defined in Ω with values in \(\mathbb{H}(\mathbb{C})\) and \(u, v\in L^{1}(\Omega,\mathbb{H}(\mathbb{C}))\). If, for arbitrary \(\varphi\in C_{0}^{\infty}(\Omega,\mathbb{H}(\mathbb{C}))\), u, v satisfy
then u is called a generalized derivative of the function v, where we denote \(u={{}^{\psi}D}_{\alpha}[v]\).
Lemma 2.1
([16])
If \(\sigma_{1},\sigma_{2}>0\), \(0\leq \gamma\leq1\), then we have
Lemma 2.2
([17])
Suppose Ω is a bounded domain in \(R^{3}\) and let \(\alpha'\), \(\beta'\) satisfy \(0<\alpha', \beta'<3\), \(\alpha'+\beta'>3\). Then, for all \(x_{1},x_{2}\in R^{3}\) and \(x_{1}\neq x_{2}\), we have
Lemma 2.3
([18])
Let Ω, ∂Ω be as stated above. If \(f \in C^{(m)}(\overline{\Omega},\mathbb{H}(\mathbb {C}))\) (\(m\geq1\)), then we have
3 Some properties of the singular integral operator \(T_{\psi ,\alpha}\) for the Helmholtz equation
In this section, we will discuss some properties of the singular integral operators as follows:
where \(B=\{x\mid |x|<1\}\), \(\alpha=a+ib\), \(b>0\).
Theorem 3.1
Assume B to be as stated above, \(\alpha=a+ib\), \(b>0\). If \(f\in L^{p}(B,\mathbb{H}(\mathbb{C}))\), \(3< p<+\infty\), then
-
(1)
\(|(T_{\psi,\alpha}^{(1)}[f])(x)|\leq M_{1}(p)\|f\| _{L^{p}(B)}\), \(x\in R^{3}\),
-
(2)
\(|(T_{\psi,\alpha}^{(1)}[f])(x_{1})-(T_{\psi ,\alpha}^{(1)}[f])(x_{2})|\leq M_{2}(p)\|f\|_{L^{p}(B)}|x_{ 1}-x_{2}|+ M_{3}(p)\|f\|_{L^{p}(B)}|x_{1}-x_{2}|^{\beta}\), \(x_{ 1}, x_{2}\in R^{3}\),
-
(3)
\({{}^{\psi}D}_{\alpha}(T_{\psi,\alpha }^{(1)}[f])(x)=f(x)\), \(x\in B\), \({{}^{\psi}D}_{\alpha}(T_{\psi,\alpha }^{(1)}[f])(x)=0\), \(x\in R^{3}\backslash\overline{B}\),
where \(0<\beta=\frac{p-3}{p}<1\).
Proof
(1)
(i) By the Taylor series, we have \(|e^{i\alpha |y-x|}|=|e^{i(a+ib)|y-x|}|=e^{-b|y-x|}\leq\frac{1}{b|y-x|}\). By the Hölder inequality, we have
When \(x\in\overline{B}\), because \(p>3\), \(\frac{1}{p}+\frac {1}{q}=1\). Then \(1< q<\frac{3}{2}\). Thus \(\int_{B}\frac{1}{|y-x|^{2q}}\,dv_{y}\) is bounded. Hence we suppose
When \(x\in R^{3}\backslash\overline{B}\), by Lemma 2.1 and the generalized spherical coordinate, we have
where \(\rho=|y-x|\), \(d_{0}=d(x,B)\). Therefore, for arbitrary \(x\in R^{3}\), we obtain
where \(M_{1}^{(1)}(p)=\max\{J_{1}J_{2}^{\frac {1}{q}},J_{1}J_{4}^{\frac{1}{q}}\}\).
(ii) Obviously, \(e^{-b|y-x|}\leq1\). By the Hölder inequality, we have
Then, by inequality (3.3) and (3.4), we have
where \(M_{1}^{(2)}(p)=\max\{J_{5}J_{2}^{\frac {1}{q}},J_{5}J_{4}^{\frac{1}{q}}\}\).
(iii) This case is similar to (ii). We obtain
By inequalities (3.5)-(3.7), we obtain
where \(M_{1}(p)=M_{1}^{(1)}(p)+M_{1}^{(2)}(p)+M_{1}^{(3)}(p)\).
Let us consider \(e^{i\alpha|y-x|}\). For arbitrary \(x\in R^{3}\), it is easy to prove \(|e^{i\alpha|y-x|}|\leq1\) and satisfy \(|e^{i\alpha|y-x_{1}|}-e^{i\alpha|y-x_{2}|}|\leq c|x_{1}-x_{2}|\).
(i) For arbitrary \(x_{1}, x_{2}\in R^{3}\), by the Hölder inequality, we have
As \(1< q<\frac{3}{2}\), \(\int_{B}\frac {1}{|y-x_{1}|^{q}|y-x_{2}|^{q}}\,dv_{y}\) and \(\int_{B}\frac {1}{|y-x_{1}|^{q}}\,dv_{y}\) are bounded. Hence
By the Hölder inequality, we have
As \(1< q<\frac{3}{2}\), \(\int_{B}\frac{1}{|y-x_{1}|^{2q}}\,dv_{y}\) is bounded. So we have
By the Hölder inequality and the Hile lemma, we have
We suppose \(\alpha'=q\), \(\beta'=2q\). As \(1< q<\frac{3}{2}\), we have \(\alpha'=q<3\), \(\beta'=2q<3\), \(\alpha'+\beta'=3q>3\). Hence, by Lemma 2.2, we have
So we have
where \(0<\beta=\frac{p-3}{p}<1\). By inequality (3.9) and (3.10), we have
Similar to \(I_{5}^{(1)}\), we have
By the Hölder inequality and the Hile lemma, we have
As \(1< q<\frac{3}{2}\), \(\int_{B}\frac {1}{|y-x_{1}|^{q}|y-x_{2}|^{q}}\,dv_{y}\) is bounded. So we have
By inequalities (3.12) and (3.13), we have
where \(M_{2}^{(2)}(p)=J_{12}+J_{14}\). By inequalities (3.8), (3.11) and (3.14), we have
where \(M_{2}(p)=M_{2}^{(1)}(p)+J_{9}+M_{2}^{(2) }(p)\), \(M_{3}(p)=J_{11}\).
(3) When \(x\in B\), for arbitrary \(\varphi\in C_{0}^{\infty}(B,\mathbb{H}(\mathbb{C}))\), by Lemma 2.3 and the Fubini theorem, we have
Hence, in the sense of generalized derivatives, \({{}^{\psi}D}_{\alpha }(T_{\psi,\alpha}^{(1)}[f])(x)=f(x)\), \(x\in B\). When \(x\in R^{3}\backslash\overline{B}\), it is easy to see \({{}^{\psi}D}_{\alpha }(T_{\psi,\alpha}^{(1)}[f])(x)=0\). □
Theorem 3.2
Assume B to be as stated above and \(\alpha=a+ib\), \(b>0\).If \(f\in L^{p,3}(B,\mathbb{H}(\mathbb{C}))\), \(3< p<+\infty\), then
-
(1)
\(|(T_{\psi,\alpha}^{(2)}[f])(x)|\leq M_{4}(p)\| f^{(3)}\|_{L^{p}(B)}\), \(x\in R^{3}\),
-
(2)
\(|(T_{\psi,\alpha}^{(2)}[f])(x_{1})-(T_{ \psi,\alpha}^{(2)}[f])(x_{2})|\leq M_{5}(p)\|f^{(3)}\|_{ L^{p}(B)}|x_{1}-x_{2}|+ M_{6}(p)\|f^{(3)}\|_{L^{p}(B)}|x_{1}-x_{2}|^{\beta }\), \(x_{1}, x_{2}\in R^{3}\),
-
(3)
\({{}^{\psi}D}_{\alpha}(T_{\psi,\alpha }^{(2)}[f])(x)=0\), \(x\in B\), \({{}^{\psi}D}_{\alpha}(T_{\psi,\alpha }^{(2)}[f])(x)=f(x)\), \(x\in R^{3}\backslash\overline{B}\),
where \(0<\beta=\frac{p-3}{p}<1\).
Proof
(1)
As the first step, by the Hölder inequality, we have
where \(\frac{1}{p}+\frac{1}{q}=1\). Next we discuss \(O_{1}(x)\) in two cases.
(i) When \(|x|\geq\frac{1}{2}\), since
we have
We suppose \(\alpha'=2q\), \(\beta'=q\). As \(1< q<\frac{3}{2}\), we have \(0<\alpha'<3\), \(0<\beta'<3\), \(\alpha'+\beta'=3q>3\). Thus, by Lemma 2.2, we have
(ii) When \(|x|<\frac{1}{2}\), by \(|y|<1\), we have \(|1-yx|\geq \frac{1}{2}\), thus
Therefore, by (3.15)-(3.17), we have
where \(M_{4}^{(1)}(p)=\max\{C_{1}C_{3}^{\frac {1}{q}},C_{1}C_{6}^{\frac{1}{q}}\}\).
As the second step, by the Hölder inequality, we have
Similar to \(O_{1}(x)\), we find that \(O_{2}(x)\) is bounded. Suppose \(O_{2}(x)\leq C_{8}\). Then
As the third step, similar to \(I_{7}\), we have
By inequalities (3.18), (3.20), and (3.21),
where \(M_{4}(p)=M_{4}^{(1)}(p)+M_{4}^{(2)}(p)+M_{4}^{(3) }(p)\).
Firstly, we discuss \(I_{10}\). We have
By the Hölder inequality, we have
By (3.16) and (3.17), we have \(O_{1}(x)\leq\max\{C_{3},C_{6}\} \). Therefore
where \(C_{10}=\max\{C_{9}C_{3}^{\frac{1}{q}},C_{9}C_{6}^{\frac {1}{q}}\}\).
By the Taylor series, we have \(|e^{i\alpha|\frac{\overline {y}}{|y|^{2}}-x_{2}|}|= |e^{-b|\frac{\overline {y}}{|y|^{2}}-x_{2}|}|\leq\frac{1}{b|\frac{\overline{y}}{|y|^{2}}-x_{2}|}\). Therefore
Since
we have
By (3.23), we have
In the following, we discuss \(O_{4}(x)\) in four cases.
(i) When \(|x_{1}|\leq\frac{1}{2}\), \(|x_{2}|\leq\frac{1}{2}\), as \(|y|\leq1\), we have \(|1-yx_{1}|\geq\frac{1}{2}\), \(|1-yx_{2}|\geq \frac{1}{2}\), \(|x_{1}-x_{2}|\leq1\). Hence
As \(|x_{1}-x_{2}|\leq1\), \(0<\beta=\frac{p-3}{p}<1\), we have \(|x_{1}-x_{2}|\leq|x_{1}-x_{2}|^{\beta}\). Therefore, by (3.24), we have
(ii) When \(|x_{1}|\geq\frac{1}{2}\), \(|x_{2}|\leq\frac{1}{2}\), we have \(|1-yx_{2}|\geq\frac{1}{2}\), \(\frac{1}{|x_{1}|}\leq2\), \(\frac {|x_{2}|}{|x_{1}|}\leq1\). Thus
Again, since
we have \(|x_{1}|^{-q}\leq C_{19}|x_{1}-x_{2}|^{(\beta-1)q}\). Again from the notion that \(1< q<\frac{3}{2}\), we know \(\int_{B}\frac {1}{|y-\frac{\overline{x}_{1}}{|x_{1}|^{2}}|^{q}}\,dv_{y}\) is bounded. Hence,we obtain
Therefore, by (3.24), we have
(iii) When \(|x_{1}|\leq\frac{1}{2}\), \(|x_{2}|\geq\frac {1}{2}\), similar to (ii), we have
(iv) When \(|x_{1}|\geq\frac{1}{2}\), \(|x_{2}|\geq\frac{1}{2}\), we have \(\frac{1}{|x_{1}|}\leq2\), \(\frac{1}{|x_{2}|}\leq2\). Since
We have
Suppose \(\alpha'=q\), \(\beta'=2q\). Then \(0<\alpha'<3\), \(0<\beta '<3\), \(\alpha'+\beta'=3q>3\). Thus, by Lemma 2.2, we have
Therefore, by (3.24), we have
where \(0<\beta=\frac{p-3}{p}<1\). From (3.25)-(3.28), we obtain
where \(M_{6}^{(1)}(p)=\max\{C_{15},C_{21},C_{22},C_{28}\}\).
Secondly, we discuss \(I_{11}\). We have
Similar to \(I_{10}^{(1)}\), we get
By the Hölder inequality and the Hile lemma, we have
This is similar to \(I_{10}^{(2)}\) and it is easy to prove the following:
Therefore, we obtain
Finally, we discuss \(I_{12}\). We have
Similar to \(I_{10}^{(1)}\), we get
By the Hile lemma and the Hölder inequality, we have
Therefore
by (3.35) and (3.36), so we have
By (3.30), (3.34), and (3.37), we have
where \(M_{5}(p)=C_{10}+C_{29}+C_{36}\), \(M_{6}(p)=M_{6}^{(1) }(p)+C_{35}+C_{38}\).
(3) This case is similar to Theorem 3.1, and it is easy to prove. □
Remark 3.1
Assume B to be as stated above and \(\alpha=a+ib, b>0\). If \(f\in L^{p,3}(B,\mathbb{H}(\mathbb{C}))\), \(3< p<+\infty\), then
-
(1)
\(|(T_{\psi,\alpha}[f])(x)|\leq M_{7}(p)\|f\| _{p,3}\), \(x\in R^{3}\),
-
(2)
\(|(T_{\psi,\alpha}[f])(x_{1})-(T_{\psi,\alpha }[f])(x_{2})|\leq M_{8}(p)\|f\|_{p,3}|x_{1}-x_{2}|+ M_{9}(p)\|f\|_{p,3}|x_{1}-x_{2}|^{\beta}\), \(x_{1},x_{2}\in R^{3}\),
-
(3)
\({{}^{\psi}D}_{\alpha}(T_{\psi,\alpha }[f])(x)=f(x)\), \(x\in R^{3}\backslash\partial B\),
where \(0<\beta=\frac{p-3}{p}<1\).
4 Integral representation of solution of Riemann boundary problem to inhomogeneous partial differential system
In this section, we will discuss the inhomogeneous partial differential system of first order equations as follows:
where \(w_{j}(x)\), \(c_{j}(x)\) (\(j=0,1,2,3\)) are real-value functions.
Problem P
Let \(B\subset R^{3}\) be as stated above. The Riemann boundary value problem for system (4.1) is to find a solution \(w(x)\) of (4.1) that satisfies the boundary condition
where \(w^{\pm}(\tau)=\lim_{x\in B^{\pm}, x\rightarrow\tau}w(x)\), \(B^{+}=B\), \(B^{-}=R^{3}\backslash\overline{B}\), G is a quaternion constant, \(G^{-1}\) exists, and \(f\in H^{\nu}_{\partial B}\) (\(0<\nu<1\)).
In fact,
Let
By (4.2) and (4.3), the inhomogeneous partial differential system (4.1) can be transformed to the following equation:
Therefore Problem P as stated above can be transformed to Problem Q.
Problem Q
Let \(B\subset R^{3}\) be as stated above. The Riemann boundary value problem for system (4.1) is to find a solution \(w(x)\) of (4.4) that satisfies the boundary condition
where \(w^{\pm}(\tau)=\lim_{x\in B^{\pm}, x\rightarrow\tau}w(x)\), \(B^{+}=B\), \(B^{-}=R^{3}\backslash\overline{B}\), G is a quaternion constant, \(G^{-1}\) exists, and \(f\in H^{\nu}_{\partial B}\) (\(0<\nu<1\)).
Theorem 4.1
Let B be as stated above. Find a quaternion-valued function \(u(x)\) satisfying the system \({{}^{\psi}D}_{\alpha}[u]=0(x\in R^{3}\backslash \partial B)\) and vanishing at infinity with the boundary condition
where \(u^{\pm}(\tau)=\lim_{x\in B^{\pm}, x\rightarrow\tau}u(x)\), G is a quaternion constant, \(G^{-1}\) exists, and \(f\in H^{\lambda }_{\partial B}\) (\(0<\lambda<1\)). Then the solution can be expressed as
Proof
Define
Then it is obvious that \({{}^{\psi}D}_{\alpha}[\varphi]=0\) (\(x\in R^{3}\backslash\partial B\)) and the Riemann boundary condition (4.5) is equivalent to
Suppose \(\Psi(x)=\int_{\partial B}\mathcal{K}_{\psi,\alpha }(y-x)\, d\sigma_{y}f(y)\). Then \({{}^{\psi}D}_{\alpha}[\Psi]=0\) (\(x\in R^{3}\backslash\partial B\)). By the Plemelj formula, we have
Hence \(\varphi^{+}(\tau)-\Psi^{+}(\tau)=\varphi^{-}(\tau)-\Psi ^{-}(\tau)\) (\(\tau\in\partial B\)). Thus \({{}^{\psi}D}_{\alpha}[\varphi -\Psi]=0\) and by Theorem 3.12 in [10] we obtain \(\varphi(x)=\Psi(x)\). So the solution can be expressed as
□
Theorem 4.2
Let B be as stated above and \(g(x)\in L^{p,3}(R^{3},\mathbb {H}(\mathbb{C}))\), \(3< p<+\infty\). Find a quaternion-valued function \(w(x)\) satisfying the system \({{}^{\psi}D}_{\alpha}[w](x)=g(x)\) (\(x\in R^{3}\backslash\partial B\)) and vanishing at infinity with the boundary condition
where \(w^{\pm}(\tau)=\lim_{x\in B^{\pm}, x\rightarrow\tau}w(x)\), G is a quaternion constant, \(G^{-1}\) exists, and \(f\in H^{\lambda }_{\partial B}\) (\(0<\lambda<1\)). Then the solution has the form
in which \({{}^{\psi}D}_{\alpha}[\Psi]=0\) and
where \(\tilde{f}=f+(T_{\psi,\alpha}[g])(G-1)\).
Proof
By Remark 3.1, we know \({{}^{\psi}D}_{\alpha}[w]={{}^{\psi}D}_{\alpha }[\Psi(x)+(T_{\psi,\alpha}[g])(x)]=g(x)\). The boundary condition (4.6) is equivalent to
Again from Remark 3.1, we know that \((T_{\psi,\alpha}[g])(x)\) has continuity in \(R^{3}\). Thus \((T_{\psi,\alpha}[g])^{+}=(T_{\psi ,\alpha}[g])^{-}=T_{\psi,\alpha}[g]\), so (4.7) is equivalent to
Suppose \(\tilde{f}=f+(T_{\psi,\alpha}[g])(G-1)\). Then (4.8) has the following form:
Again from Theorem 4.1, the solutions which satisfy the system \({{}^{\psi }D_{\alpha}}[\Psi]=0\) and boundary condition (4.9) can be represented in the form
where \(\tilde{f}=f+(T_{\psi,\alpha}[g])(G-1)\). □
Remark 4.1
By Theorem 4.2, the solution of problem P can be expressed as
in which \({{}^{\psi}D}_{\alpha}[\Psi]=0\) and
where \(\tilde{f}=f+(T_{\psi,\alpha}[g])(G-1)\).
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Acknowledgements
This work was supported by the National Science Foundation of China (No. 11401162, No. 11571089), the Natural Science Foundation of Hebei Province (No. A2015205012, No. A2016205218), and Hebei Normal University Dr. Fund (No. L2015B03).
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Yang, P., Wang, L. & Gao, L. Some properties and applications of the Teodorescu operator associated to the Helmholtz equation. J Inequal Appl 2017, 264 (2017). https://doi.org/10.1186/s13660-017-1537-2
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DOI: https://doi.org/10.1186/s13660-017-1537-2