Pingrun Li

(School of Mathematical Sciences,Qufu Normal University,Qufu 273165,Shandong; School of Mathematical Sciences,University of Science and Technology of China, Hefei 230026,E-mail:lipingrun@163.com)

ON METHOD OF SOLUTION TO ONE CLASS OF SINGULAR INTEGRAL EQUATIONS WITH SINGULARITY OF ORDER ONE∗†

Pingrun Li

(School of Mathematical Sciences,Qufu Normal University,Qufu 273165,Shandong; School of Mathematical Sciences,University of Science and Technology of China, Hefei 230026,E-mail:lipingrun@163.com)

In this paper,we study one kind of singular integral equations with singularity of order one.By the generalized Plemelj formula,this class of equations are transformed into a system of linear algebraic equations.In particular,we prove the existence of solution to the equation.The general solutions and the conditions of solvability are obtained in function class H.

singular integral equation;singularity of order one;linear algebraic equations;Hölder continuous function class H;generalized Plemelj formula

2000 Mathematics Subject Classification 45E05;45E10;30E25

Ann.of Appl.Math.

31:2(2015),159-164

1　Introduction

It is well-known that singular integral equations are one class of basic equations in the theory of integral equations,and many results have been obtained,which form a relatively complete theoretical system.This class of equations play an important role in other subjects and practical applications,such as engineering mechanics,physics,fracture mechanics and elastic mechanics(see[1-6]).In this paper,we discuss some classes of singular integral equations with convolution kernels,and transform them into discrete jump problems using discrete Fourier transform and obtained the conditions of solvability and the explicit expressions of general solutions under some certain conditions(see[7-9]).

The methods of solution for the following singular integral equations have been deeply studied by many mathematical workers(see[10-14])

where C is a smooth closed contour in the complex plane,which surrounds an interior region Σ+and an exterior region Σ-,respectively,so∞∈Σ-.The known functions d(z)and g(z)are holomorphic in Σ+and satisfy Hölder condition H on Σ+(see[1]for the definition of H),m(z,η)is a holomorphic function for z∈Σ+and η∈Σ+,m(z,η)∈H forand,and the unknown function f(z)is required to belong to H on Σ+.K is a singular operator.

In this paper,we give the methods of solution to one class of singular integral equation (1.1)with the following singularity of order one

where m0(z,η)∈H onand is holomorphic in Σ+2(Σ+×Σ+)for z∈Σ+and η∈Σ+,and f(z),d(z)and g(z)are as above.

2　Lemmas and Their Proofs

The following lemmas are important to the proof of our results,which are proposed firstly in this paper.

Lemma 2.1(generalized Plemelj formula)Let C be a smooth closed contour in the complex plane and

then we have

where b(t)=m(t,t).

Therefore,by(2.2),we have

Applying Plemelj formula(see[1]),we thus obtain conclusion(2.3).

By(2.3),we have the following lemma.

Lemma 2.2 The functions m(z,s)and F(z)are as above,then,we have

and

It is easy to prove Lemma 2.2,and the detail of proof is omitted.

3　Methods of Solution to Equation(1.1)

Take kernel m(z,η)as in(2.1).Assume that f(t)is a solution to equation(1.1).Via (1.1)and(2.3),we obtain

Noting that b(t)and F+(t)have singularity of order one at t=sj(j=1,2,···,n),we have

In this paper,we only consider the case of normal type,that is,d(t)±b(t)≠0 on C. Without loss of generality,we assume the only zero-point of d(t)+b(t)in Σ+to be z=α with order u(u≥1)and the only zero-point of d(t)-b(t)in Σ+to be z=β with order v (v≥1),where u and v are integers.Obviously,α≠sjand β≠sj(j=1,2,···,n).For the case of more zero-points,we may discuss similarly.

Putting(3.1)into(2.2),we obtain

Since F+(z)/(d(z)-b(z))is meromorphic in Σ+except single pole β and belongs to H on C,m(z,s)is holomorphic for z∈Σ+and belongs to H for s,except s=sj(j=1,2,···,n).Therefore,by the extended Residue theorem(see[15,16]),we obtain

in which we have

Noticing

we denote

Then we have

where w0(s)=m0(s,s),s∈Σ+.

By(3.1)and(3.2),we obtain

We firstly consider α≠β,and take derivatives of(3.4)up to order v-1 in the neighborhood of β.Note that z=β is the zero-point of R(z)Ψ(z)with order v at least so that (R(z)Ψ(z))(α)|z=β=0,α=0,1,···,v-1,and

When k≤m≤v-1,we have

therefore,

Moreover,

Thus,z=β is a singularity of Sα(z)with order v-α at most and z=β is a zero-point of R(z)Sα(z)with order α at least,that is,

Then,we obtain

Denoting

and

By(3.6),(3.8)and(3.9),(3.7)can be rewritten as follows

and E is the unit matrix with order v.

In order that(1.1)has a solution,S and F must be fulfilled(3.10).Through the theory of system of linear equations,whenhas only one zero solution;whenfor certain k0(0≤k0≤v-1),(3.10)has nontrivial solutions.

Since equation(1.1)is solvable,we require that F(z)is analytic at z=α,therefore the following conditions must be fulfilled:

where{F+(0)(β),F+(1)(β),···,F+(v-1)(β)}are solutions to(3.10).Obviously,if(3.11)has a solution,then equation(1.1)can be solved directly.

From the above analysis,we deduce the following theorem.

Theorem 3.1Under the above supposition,equation(1.1)is solvable if and only if equation(3.11)or(3.12)is fulfilled.Assume that(3.11)or(3.12)is fulfilled,then equation (1.1)has a unique solution in class H,which is f(t)=(g(t)-2F+(t))/(d(t)-b(t)),where F+(t)is given by(3.4).And whenin(3.4),in this case,equation(1.1)has a unique solution in class H;whenfor certain k0(0≤k0≤v-1),equation(3.10)has infinite solutions,hence equation(1.1)also has infinite solutions.

We can similarly discuss the case α=β.Using the methods of this paper and in[17-20], it is possible to solve the above mentioned equation in the non-normal cases,but the analysis is complicated.

In this paper,we solve equation(1.1)in class H.Indeed,this class of equations can also be solved in Clifford analysis,similar to that in[21-23].Further discussion is omitted here.

References

[1]J.K.Lu,Boundary Value Problems for Analytic Functions,Singapore:World Sci.,2004.

[2]N.I.Muskhelishvilli,Singular Integral Equations,NauKa:Moscow Press,2002.

[3]S.G.Krantz,Function Theory of Several Complex Variables,Providence,RI,AMS Chelsea Publishing,2001.

[4]J.Y.Du,On quadrature formulae for singular integrals of arbitrary order,Acta Mathematica Scientia,24B:1(2004),9-27.

[5]J.Y.Du,N.Xu,Z.X.Zhang,Boundary behavior of Cauchy-type integrals in Clifford analysis, Acta Math.Sci.,29B:1(2009),210-224.

[6]R.Abreu Blaya,J.Bory Reyes,F.Brackx,H.De Schepper,F.Sommen,Boundary value problems for the quaternionic Hermitian in R4 analysis,Bound.Value Probl.,2012,doi:10.1687-2770-2012-74.

[7]P.R.Li,On the method of solving two kinds of convolution singular integral equations with reflection,Annals of Differential Equations,29:2(2013),159-166.

[8]P.R.Li,Integral equations containing both cosecant and convolution kernel with periodicity, J.Sys.Sci.and Math.Scis.,30:8(2010),1148-1155.(in Chinese)

[9]P.R.Li,On method of solution for two classes of equations of convolution type with harmonic singular operator,Journal of Systems Science and Mathematical Sciences,33:7(2013),854-861. (in Chinese)

[10]Y.X.Shen,One class of singular integral equation of dual type with two convolution kernel, Chin.Ann.of Math.,12A:1(1991),57-64.(in Chinese)

[11]J.K.Lu,On method of solution for some classes of singular integral equations with convolution, Chin.Ann.of Math.,8B:1(1987),97-108.

[12]P.R.Li,On method of solution of non-normal type for one kind of singular integral differential equations with both convolution kernel and Cauchy kernel,Journal of Systems Science and Mathematical Sciences,34:3(2014),302-311.(in Chinese)

[13]P.R.Li,Singular integral equations in functions class of exponential order increasing and Riemann boundary value problem,J.Sys.Sci.and Math.Scis.,35:1(2015),166-176.(in Chinese)

[14]J.K.Lu,Y.X.Shen,The singular integral equation with convolution kernel and Cauchy kernel in class{p,q},Chin.Ann.of Math.,13A:6(1992),672-680.(in Chinese)

[15]X.Li,Integral Equations,Beijing:Science Press,2008.(in Chinese)

[16]D.Eelbode,F.Sommen,The inverse Radon transform and the fundamental solution of the hyperbolic Dirac equation,Math.Z.,247(2004),733-745.

[17]P.R.Li,Singular integral differential equations with both Cauchy kernel and convolution kernel and Riemann boundary value problem,Annals of Differential Equations,30:4(2014),407-415.

[18]P.R.Li,A class of singular integral equation of convolution type with kernel,Chinese Quarterly Journal of Mathematics,29:4(2014),620-626.

[19]P.R.Li,Singular integral equation of convolution type with Hilbert kernel and periodical coefficients,Acta Mathematicae Applicatae Sinica,37:6(2014),1025-1033.

[20]P.R.Li,The method of solutions for some kinds of singular integral equations of convolution type with both reflection and translation shift,Chinese Quarterly Journal of Mathematics, 29:1(2014),107-115.

[21]S.Huang,Y.Y.Qiao,G.C.Wen,Real and Complex Clifford Analysis,Springer,New York, 2005.

[22]L.Liu,G.B.Ren,Normalized system for wave and Dunkl operators,Taiwanese J.Math., 14(2010),675-683.

[23]Q.Q.Kang,W.Wang,On Penrose integral formula and series expansion of k-regular functions on the quaternionic space,J.Geom.Phys.,64(2013),192-208.

(edited by Liangwei Huang)

∗Supported by the Qufu Normal University Youth Fund(XJ201218).

†Manuscript May 7,2014;Revised March 6,2015