Some Properties of a Kind of Singular Integral Operator with Weierstrass Function Kernel

Size: px

Start display at page:

Download "Some Properties of a Kind of Singular Integral Operator with Weierstrass Function Kernel"

Jeffrey Adams
6 years ago
Views:

Applied Mathematics 3 4 3-35 http://dxdoiorg/436/am34866 Published Online August 3 (http://wwwscirporg/journal/am) Some Properties of a ind of Singular Integral Operator with Weierstrass Function

1 Applied Mathematics Published Online August 3 ( Some Properties of a ind of Singular Integral Operator with Weierstrass Function ernel ixia Cao Department of Information and Computing Sciences Mathematics College Northeast Petroleum University Daqing China caolixia9837@63com Received June 6 3; revised July 6 3; accepted July 5 3 Copyright 3 ixia Cao This is an open access article distributed under the Creative Commons Attribution icense which permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited ABSTRACT We considered a kind of singular integral operator with Weierstrass function kernel on a simple closed smooth curve in a fundamental period parallelogram Using the method of complex functions we established the Bertrand Poincaré formula for changing order of the corresponding integration and some important properties for this kind of singular integral operator eywords: Weierstrass Function ernel; Singular Integral Operator; Bertrand Poincaré Formula; Properties Introduction The properties of singular integral operator with Cauchy or Hilbert kernel on simple closed smooth curve or open arc have been elaborately discussed in [-3] Based on these for the boundary curve is a closed curve or an open arc the authors discussed the singular integral operators and corresponding equation with Cauchy kernel or Hilbert kernel in [-3] In recent years many authors discussed the numerical solution of a class of systems of Cauchy singular integral equations with constant coefficients Numerical methods for nonlinear singular Volterra integral equations in [4-6] In this paper we consider a kind of singular integral operator with Weierstrass function kernel on a simple closed smooth curve in a fundamental period parallelogram Our goal is to develop the Bertrand poincaré formula for changing order of the corresponding integration and some important properties of the above singular integral operator Preliminaries Definition Suppose that are complex constants with Im and P denotes the fundamental period parallelogram with vertices Then the function z z z z is called the Weierstrass -function m n denotes the sum of all except for m n Definition Suppose that is a smooth closed curve in the counterclockwise direction lying entirely in the fundamental period parallelogram P with z and the origin lying in the domain S enclosed by The following operator a t t t ( t t) t t t z d t t is called the singular integral operator with -function kernel on t H is the unknown function and t t H a t H are the given functions bt tt etting then () becomes () bt ( ) d () t t t t t t t z t a t t t t t t z t Copyright 3 SciRes

2 3 X CAO Since t is uniformly convergent in any closed bounded region lying entirely in P t t t z tt M for any t t M is some positive finite constant By noting that t t H we obtain t t t t tt t z N tt Write N is some positive finite constant bt a t t t t t t z d t o kt t t tt t tt t z k k t d t t t then () can be rewritten in the form k (3) k is a Fredholm operator and is called the characteristic operator of Now the index of is defined as D t arg π S t S t a t b t D t a t b t and for definiteness we assume that a t b t namely we assume that is an operator of normal type Now the associated operator of () takes the form or at t t tt t t t z t (4) at t b t t t t t z d t kt t t d t t (4) and so that the associated operator of becomes a t t b t t tt t z d t t In addition if we write then (4) can be rewritten as k t kt t b t bt tt t z t t bt at t t t t t zd t k t t (5) btbt tt tz D tt ( D is some finite constant) So k is a Fredholm operator and then the charac- teristic operator of operator becomes bt (6) a t t t t t t z Therefore we concluded that usually can not be established that is For convenience we write z z z the fixed nonzero point z and the origin lie in S It is not difficult to get the following results emma Suppose that f t H with the same as mentioned before then a ) d and d t f t t d f t t d t t d t f t d d f t t Copyright 3 SciRes

3 X CAO 33 b) (Poincare-Bertrand formula) tt d t ft td π ft t 3 Some Properties of Operator ) If H then H Proof Through calculation and estimation we have t tt td t t tt t M tt N t t fo r any t t M and N are all finite constant While for any t t we have t t t t t t (7) (8) t t Q t t tt t t Q is some finite constant Substituting (8) into (7) we obtain t t t t H (9) Similarly we know that d d t t t t t t t t t t z t a t t H Consequently we have H ) If are singular integral operator then is also a singular integral operator That is if then j aj tt t t t zd t j j t t t a t a t b t b t t a t t t a t t t t tt t z d t t t t t t t z tt t z t t () the sum of the former two terms in the right hand of Equation () are the characteristic operator and the remainder in that is a Fredholm operator Proof By definition we deduce that atatt att t t t t t z a t t t t t t t z t C t d C t d π t t t t t z i t t t t t z t d t t By virtue of emma (b) C t can be rewritten in the form C t b t b t t π t t t t t t t z t t t z d t t d t i Consequently () is established Now we write Copyright 3 SciRes

4 34 X CAO t t t t t t tt d t 3 4 t t t t tt t t t t t t t t d t tt tt 3 t t t t t z d t tt tt 4 t t t t tt tt t t t t By [] we know that is a Fredholm integral For 4 we know from ) t t t t H that 4 is continuous about the variable t and so that 4 t is also a Fredholm integral By noth- ing that and 3 have the same form we only need to discuss either one of them Here we consider the integral Write z ' h tz tz then ht H is analytic in P and so that ht H Consequently we read from that H therefore t t t t t H and so that is continuous on is also a Fredholm integral So far we conclude that is a singular inte gral operator 3) et 3 j denotes the indices of j j 3 then 3 Proof From ) we know and a a a bb b ab a b 3 3 S S S D D D 3 3 so 3 In addition we can see from a3 aa bb and b3 ab ab that when are normal 3 is also normal 4) 3 5) If is a singular integral operator and k is a Fredholm inte gral operator of the first kind then k and k are also Fredholm integral operators of the first kind 6) If the indies of and are and respectivel y then 7) Through careful calculation we may obtain 4) - 7) 8) Generally speaking can not be established for H Proof By definition and calculation we have Whereas et d t ta t t t tt tttz () a t t t t t t t tt tz a t t t d t t t t t t t t z d t () then by emma (a) we have W t t t t t t t z d t W t t t t d t t t z t t t tt t tz (3) Copyright 3 SciRes

5 X CAO 35 Substituting (3) into () we see that Therefore t z i (4) a t t td t t t t ) t t t cannot be established REFERENCES [] N I Muskhelishvili Singular Integral Equations World Scientific Singapore City 993 [] J u Boundary Value Problems for Analytic Functions World Scientific Singapore City 993 [3] F D Gakhov Boundary Value Problems Fizmatgiz Moscow 977 [4] M C De Bonis and C aurita Numerical Solution of Systems of Cauchy Singular Integral Equ ations with Con- stant Coefficients Applied Mathematics and Computa- tion Vol 9 No 4 pp 39-4 doi:6/jamc8 [5] T Diogo and M Rebelo Numerical Methods for Nonlinear Singular Volterra Integral Equations AIP Conference Proceedings os 9-5 September pp 6-9 [6] C aurita A Quadrature Method for Cauchy Singular Integral Equations with Index IMA Journal of Numerical Analysis Vol 3 No 3 pp 7-95 doi:93/imanum/drr3 Copyright 3 SciRes

The Variants of Energy Integral Induced by the Vibrating String

Applied Mathematical Sciences, Vol. 9, 15, no. 34, 1655-1661 HIKARI td, www.m-hikari.com http://d.doi.org/1.1988/ams.15.5168 The Variants of Energy Integral Induced by the Vibrating String Hwajoon Kim