# Singular Integral Equations of Convolution Type with Cosecant Kernels and Periodic Coefficients

Authors: Pingrun Li
Journal: Mathematical Problems in Engineering
 (2017)

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## Abstract

We study singular integral equations of convolution type with cosecant kernels and periodic coefficients in classL2[-π,π]. Such equations are transformed into a discrete jump problem or a discrete system of linear algebraic equations by using discrete Fourier transform. The conditions of Noethericity and the explicit solutions are obtained by means of the theory of classical boundary value problem and of the Fourier analysis theory. This paper will be of great significance for the study of improving and developing complex analysis, integral equations, and boundary value problems.


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## Body

## 1. Introduction

It is well-known that the boundary value problems for analytic functions have been widely used in many fields, such as engineering mechanics, physics, engineering technology, and fracture mechanics. Various types of boundary value problems for analytic functions and singular integral equations have been deeply studied and widely applied to practical problem (see [1–3]). In the theory of integral equations, the convolution type integral equations and singular integral equations are two important classes of equations, which had been studied by many mathematical workers and there were already rather complete theoretical systems (see [4, 5]). These theories have been widely used in practical applications, such as engineering mechanics, fracture mechanics, and elastic mechanics (see [6, 7]).The main aim of this paper is to extend further the theory to singular integral equations of convolution type with periodic coefficients and cosecant kernel in classL2[-π,π] (see [8] for the definition of L2[-π,π]). By using discrete Fourier transform, such equations are transformed into a discrete jump problem or a discrete system of equations. The explicit expressions of general solution and the conditions of solvability are obtained in class L2[-π,π]. Therefore, this paper generalizes some results for [1–5].


## 2. Definitions and Lemmas

In this section we present some definitions and lemmas.Definition 1.
Supposeφ={φk}={…,φ-1,φ0,φ1,…} is an infinite dimensional vector consisting of two-way infinite sequence; then all of ∑k=-∞+∞φkp which satisfy ∑k=-∞+∞|φk|p<+∞ constitute a linear space; we denote it as φ∈lp.Remark 2.
From Definition1, we know that if φ∈lp, then cφ∈lp for any constant c. Moreover, if {φk}∈lp, {ψk}∈lq, then {φkψk}∈l1, where p-1+q-1=1(p>1,q>1). In this paper we mainly consider the case p=2.Definition 3.
LetΦ(θ), Ψ(θ) be periodic functions with period 2π, and Φ(θ), Ψ(θ)∈L2[-π,π]; then(1)Φ∗Ψθ=12π∫-ππΦθ-νΨνdν,θ∈-π,πis called the convolution of Φ(θ) and Ψ(θ).Remark 4.
With regard to the convolution(Φ∗Ψ)(θ), we can extend the value of θ from [-π,π] to R by Definition 3.Definition 5.
Letφ={φk}k=-∞+∞∈l2; the transforms V and V-1 defined by(2)Vφ=∑k=-∞+∞φkeikθ=Φθ,V-1Φθ=12π∫-ππΦθe-ikθdθ=φk,k=0,±1,±2,…,are called the discrete Fourier transform and the inverse transform, respectively. For simplification, in (2), we denote them as Vφ=Φ(θ) and V-1[Φ(θ)]=φk, respectively.Definition 6.
Letφ={φk}k=-∞+∞ be a number sequence and M and η be positive constants with 0<η≤1. If |k1+ηφk|≤M holds for any integer k, we say that φ={φk} belongs to class {1}.Definition 7.
Ifφ={φk}k=-∞+∞∈{1}, then we say Φ(θ)∈{0~}.The following lemma plays an important role, and it is proposed firstly in this paper.Lemma 8.
LetΦ(θ)∈L2[-π,π],SΦ(θ)=(1/π)∫-ππΦ(ν)csc(ν-θ)dν,(3)S~Φθ=1π∫-ππΦνcscν+θdν.Then(1)
V-1SΦθ=2φkisgn⁡kρk, V-1(S~Φ(θ))=-2φkisgn⁡kρk;(2)
SΦ(θ)∈L2[-π,π], V-1(SΦ(θ))∈l2[-π,π],where SΦ(θ), S~Φ(θ) are singular integrals with cosecant kernels and (4)ρk=1,k=2n+1;0,k=2n,n∈Z.Proof.
(1) Since(5)SΦθ=1π∫-ππΦνcscν-θdν=1π∫-ππΦν2ieiν-θ-e-iν-θdν=2iπ∫-ππΦνeiθ+νe2iv-e2iθ,by Definition 5, we have(6)V-1SΦθ=i·1π2∫-ππΦνeiν∫-ππe-ik-1θe2iν-e2iθdθdν.It follows from the extended Residue theorem (see [9]) that(7)∫-ππe-ik-1θe2iν-e2iθdθ=πeik+1νρksgn⁡k.This implies(8)V-1SΦθ=i1π2πρksgn⁡k∫-ππ1eik+1νΦνeiνdν=2iφkρksgn⁡k.The other equality can be proved similarly.
(2) By using Riesz’s theory (see [9]), we have SΦ(θ)2≤CΦ(θ)2, where C is a constant. Therefore, it follows from Φ(θ)∈L2[-π,π] that SΦ(θ)∈L2[-π,π].
It is also obvious for the second conclusion in (2).Lemma 9.
Ifφ={φk}k=-∞+∞ belongs to class {1}, then Φ(θ)=Vφ∈H, where H is the class of Hölder continuous function.Proof.
By Definition6, we know that ∑k=-∞,k≠0+∞(M/k|1+η converges; therefore, ∑k=-∞∞φk is convergent absolutely. For any two points θ1,θ2∈[-π,π], we have(9)Φθ1-Φθ2=∑k=-∞∞φkeikθ1-eikθ2≤∑k=-∞∞φkeikθ1-eikθ2≤M~θ1-θ2,where M~=∑k=-∞∞|φk|; thus Φ(θ)∈H.Lemma 10.
LetΦ(θ)=∑k=-∞∞φkeikθ and φ∈l1; then Φ(θ)∈L2[-π,π] if and only if φ∈l2.Proof.
SinceΦ(θ)∈L2[-π,π] and φ∈l1, then ∑k=-∞∞φk is convergent absolutely and uniformly. It is easy to see that (10)∫-ππΦθ2dθ=∫-ππΦθΦθ¯dθ=∫-ππ∑φkeikθ∑φj¯e-ijθdθ=∫-ππ∑k,j=-∞∞φkφj¯eik-jθdθ=2π∑k=-∞∞φk2.The proof of Lemma 10 is complete.Lemma11 is an obvious fact.Lemma 11 (convolution theorem).
IfΦ(θ) and Ψ(θ)∈L2[-π,π], then Φ∗Ψ(θ)∈L2[-π,π] and V-1[Φ∗Ψ(θ)]=φkψk, where ψk=V-1[Ψ(θ)].


## 3. Problem Presentation and Methods of Solution

Consider the following singular integral equation of convolution type with periodicity and cosecant kernels:(11)ξ1θaΦθ+bSΦθ+K1∗Φθ+M1∗SΦθ-cGθ+ξ2θaΦθ+bSΦθ+K2∗Φθ+M2∗SΦθ-cGθ=0,θ∈R,where (12)ξ1θ=c1cos⁡mθ2+d1sin⁡mθ2,ξ2θ=a1cos⁡mθ2+b1sin⁡mθ2,M∗SΦθ=12π∫-ππMθ-νSΦνdν.SΦ(θ) is as in Lemma 8. a,b,c,a1,b1,c1,d1 are real constants with a2+b2≠0,a1d1-b1c1≠0. The functions ξj(θ)(j=1,2) are periodic coefficients; m is a positive integer. Kj(θ), Mj(θ), Φ(θ), SΦ(θ), and G(θ) are periodic functions with period 2π. Once Φ(θ) is obtained in [-π,π], we can make the periodic extension of Φ(θ) with period 2π and then obtain the solution Φ∗(θ) in R. Therefore, in the following discussion we are restricted to θ∈[-π,π].Equation (11) is one important class of equations in the theory of integral equations, and it has important applications in physics, air dynamics, and electronic optical (e.g., see [10–14]). Hence, the study of (11) is meaningful not only in application but also in the method of solution.Sinceeimθ=cos⁡mθ+isin⁡mθ, we write (11) as(13)A+Φθ+B+SΦθ+β+K1∗Φθ+α+K2∗Φθ+β+M1∗Φθ+α+M2∗SΦθ-F+Gθ+eimθA-Φθ+B-SΦθ+β-K1∗Φθ+α-K2∗Φθ+β-M1∗Φθ+α-M2∗SΦθ-F-Gθ=0,-π≤θ≤π,where(14)α±=a1±ib1,β±=c1±id1,A±=aa1+c1±ib1+d1,B±=ba1+c1±ib1+d1,F±=ca1+c1±ib1+d1,and ξj(θ)(j=1,2) can be rewritten as(15)ξ1θ=12β-eimθ/2+β+e-imθ/2,ξ2θ=12α-eimθ/2+α+e-imθ/2.Let(16)A-Φθ+B-SΦθ+β-K1∗Φθ+α-K2∗Φθ+β-M1∗SΦθ+α-M2∗SΦθ-F-Gθ=Tθ,then,(17)A+Φθ+B+SΦθ+β+K1∗Φθ+α+K2∗Φθ+β+M1∗SΦθ+α+M2∗SΦθ-F+Gθ=-eimθTθ;Applying V-1 to both sides of (16) and (17), respectively, by Lemma 8 we have(18)τk=A-+2B-isgn⁡kρk+β-Bk1+α-Bk2+2β-Ck1isgn⁡kρk+2α-Ck2isgn⁡kρkφk-F-gk;-τk-m=A++2B+isgn⁡kρk+β+Bk1+α+Bk2+2β+Ck1isgn⁡kρk+2α+Ck2isgn⁡kρkφk-F+gk,where(19)τk=V-1T,φk=V-1Φ,Bkj=V-1Kj,Ckj=V-1Mjj=1,2,gk=V-1G.Denote(20)πk±=A±+2B±isgn⁡kρk+β±Bk1+α±Bk2+2β±Ck1isgn⁡kρk+2α±Ck2isgn⁡kρk.Then (18) can be reduced as(21)πk-φk-F-gk=τk;πk+φk-F+gk=-τk-m.Since Bkjk=-∞∞∈l2, Ckjk=-∞∞∈l2(j=1,2), then Bk(j)→0, Ck(j)→0, and πk±→A±±2iB±≠0 or πk±→A±≠0 when |k|→∞. Therefore there exists k0 (|k|>k0) such that |πk±|>μ when k>k0 for certain positive constant μ(0<μ<1).Consider first the case|k|≤k0.(1) Whenπk±=0 for k=n1,n2,…,np1(-k0≤nj≤k0,1≤j≤p1), by (21) we know that (22)τnj=-F-gnj,τnj-m=F+gnjmust be fulfilled. Now φnj(1≤j≤p1) take arbitrary constants.(2) Whenπk+=0, πk-≠0 for k=n1′,n2′,…,np2′(-k0≤nj′≤k0,1≤j≤p2), by (21) we have (23)φnj′=F-gnj′+τnj′πnj′-,where (24)τnj′=F+gnj′+m.(3) Whenπk+≠0, πk-=0 for k=n1′′,n2′′,…,np3′′(-k0≤nj′′≤k0,1≤j≤p3), by (21) we get (25)φnj′′=F+gnj′′-τnj′′-mπnj′′+,where (26)τnj′′=-F-gnj′′.(4) Whenπk±≠0(-k0≤k≤k0), φk are given by (21) and this situation is similar to the situation |k|>k0.Next we consider the case|k|>k0. Since |πk±|>μ for |k|>k0, then πk±≠0. By (21) we obtain (27)φk=1πk-τk+F-gk=1πk+-τk-m+F+gk;thus (28)τk+Rkτk-m=Hkgk,where (29)Rk=πk-πk+,Hk=2ib1c1-a1d1πk+Bk2-Bk1+2Ck2-Ck1isgn⁡kρk.Assume that(30)δ=B-B+,Rk=δ1+Yk,where (31)Yk=2ibb1c1-a1d1B-·Bk1-Bk2+2isgn⁡kρkCk1-Ck2πk+.Therefore, (28) can be written as the following discrete jump problem: (32)τk+δ1+Ykτk-m=Hkgk.Thus, we should only study (32) in place of (11). Since {Hk}∈l2, {gk}∈l2, then {Hkgk}∈l2, {τk}∈l2. Once τk is obtained, then φk can be given by (21). To solve (32), we first need to analyze the structure of 1+Yk. Assume that 1+Tk can be factorized; we can choose rk such that (33)1+Yk=rkrk-m,where μ<|rk|<μ-1; μ is as the above. Provided that μ takes sufficiently small value, we can choose rk satisfying the above requirement (33). We now give an expression of rk, taking logarithms for both sides of (33) and denoting (34)wk=ln⁡rkuk=ln⁡1+Yk;then, by (33) and (34) we obtain (35)wk-wk-m=uk.Note that we have taken a continuous branch of ln⁡(1+Yk) such that ln⁡(1+Yk)∈l2. Applying Fourier transform to both sides of (35), we can obtain (36)1-eimθWθ=Uθ,where W(θ)=Vw, U(θ)=Vu, w={wk}k=-∞+∞, and u={uk}k=-∞+∞.From the above discussions, the following conditions of solvabilityU(θj)=0 follow and that is, (37)∑ln⁡1δRkeikθj=0,ln⁡1δRk∈l2,where θj=(2j/m)π(j=0,±1,…,±m/2; [m/2] is the integer part of m/2). Therefore, (38)Wθ=Uθ1-eimθθ≠θj,j≤2m2+1.From (38), we obtain wk=V-1W(θ) and thus rk=exp⁡wk. Note (33) and denote (39)ek=τkrk,ψk=Hkgkrk.Then (32) can be rewritten as (40)ek+δek-m=ψk.Since{Hkgk}∈l2,{rk}∈l2, and μ<|rk|<μ-1, we have {ek}∈l2 and {ψk}∈l2. By taking Fourier transform V for both sides of (40), we have (41)1+δeimθEθ=Ψθ,where E(θ)=Ve, Ψ(θ)=Vψ, e={ek}k=-∞+∞, and ψ={ψk}k=-∞+∞.Since|δ|=1, we know that 1+δeimθ has finite number of zero points θ1′,θ2′,…,θn′ in [-π,π].Therefore the conditions of solvabilityΨ(θj′)=0, that is, (42)∑Hkgkrkeikθj′=0,Hkgkrk∈l2j=1,2,…,n,must be augmented. In view of (41) we get (43)Eθ=Ψθ1+δeimθθ≠θj′,j=1,2,…,n.If (37) and (42) are satisfied, then by (43) we obtain (44)ek=V-1Eθk=0,±1,….Thus τk(k=0,±1,…) can be also obtained by (39).Now we state our main result.Theorem 12.
Under (37) and (42), (11) has a solution. When πk±≠0(k=0,±1,…), φk is given by (21). When πnj±=0(j=1,2,…,p1), φnj take arbitrary constants. When πnj′+=0, πnj′-≠0(j=1,2,…,p2), φnj′, τnj′ are given by (23) and (24), respectively. When πnj′′+≠0, πnj′′-=0(j=1,2,…,p3), φnj′′, τnj′′ are given by (25) and (26), respectively. But in (23) and (25), τnj′ and τnj′′ are constants determined by (24) and (26), respectively. When k≠nj,nj′,nj′′,φk is given by (21). Therefore, the solution of (11) is of the form (45)Φθ=Vφ=∑k=-∞∞φkeikθ,and then Φ(θ)∈L2[-π,π].Proof.
From the above discussion, we only need to prove that the functionΦ(θ) obtained by (45) belongs to L2[-π,π]. Obviously, (11), (13), and (21) are equivalent to each other. Since {gk},{τk}∈l2, then {πk±}∈l2 and πk±≠0. It follows from (27) that {φk}∈l2 is a bounded sequence, and ∑k=-∞∞φkeikθ is convergent. Thus, by Lemma 10, (11) has a unique solution Φ(θ)=Vφ in class L2[-π,π].According to Theorem12, Definition 5, and Lemma 9, we obtain the following.Theorem 13.
In (11), if Kj(θ),Mj(θ)(j=1,2), G(θ)∈{0~}, then (11) is solvable in class {0~} and all conditions of solvability and its solutions are similar to those in Theorem 12.In order to illustrate that (11) has an explicit solution, we present an example and satisfy the above conditions (37) and (42). For example, suppose that (46)a=b=c=1,b1=c1=0,a1=d1≠0,K1=K2,M1=M2;then ξ1(θ)=a1sin⁡m/2θ, ξ2(θ)=a1cos⁡m/2θ, and (11) can be transformed into (47)Φ+SΦ+K1∗Φ+M1∗SΦ=G;it is easy to prove that (47) satisfies conditions (37) and (42), and the solution of (47) was obtained in the literature [1]. Therefore, we can conclude that a solution set of (11) is not empty.Remark 14.
Ifa1d1-b1c1=0, ξ2(θ)/ξ1(θ) is a constant. This case is simple, and we do not discuss it here.


## 4. Results and Discussion

In this paper, we first proposed one class of singular integral equations of convolution type with cosecant kernels and periodic coefficients. By applying discrete Fourier transform and its properties, such equation can be transformed into a discrete jump problem depending on some parameter. Here, our method is different from the ones of the classical boundary value problem, and it is novel and simple. The exact solution, denoted by series, of (11) and the conditions of solvability are obtained in L2[-π,π]. We remark that our approach is also effective in other classes of equations, such as the equations of dual type with periodicity and cosecant kernel and Wiener-Hopf type equations. Thus, this paper generalizes the classical theory of boundary value problems and singular integral equations.In this paper, we solved (11) in L2[-π,π]. Indeed, this class of equations can be also solved in Clifford analysis, which is similar to that in [15–18]. Further discussion is omitted here.


## 5. Conclusions

Equation (11) has important applications in practical problems, such as elastic mechanics, heat conduction, and electrostatics. Many problems, such as piezoelectric material, voltage magnetic materials, and functional gradient materials, can often attribute the problem to finding their solutions to (11). For the study of such equations, the present result is still rare due to lack of effective approaches. Our approach of solving the equations is novel, different from the ones in classical cases, and it is converted by using discrete Fourier transform into a discrete boundary value problem depending on some parameter; here we call it “a discrete jump problem.” The exact solutions of (11) and the conditions of solvability are obtained in class L2[-π,π].



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