A collocation method for singular integral equations with cosecant kernel via Semi-trigonometric interpolation

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1 Iteratioal Joural of Mathematics Research. ISSN Volume 9 Number 1 (017) pp Iteratioal Research Publicatio House A collocatio method for sigular itegral equatios with cosecat kerel via Semi-trigoometric iterpolatio Xigtia Gog 1 ad Shuwei Yag 1 LogQiao College of Lazhou Uiversity of Fiace ad Ecoomics Lazhou Gasu Chia. School of Sciece Lazhou Uiversity of Techology Lazhou Gasu Chia. Abstract I the paper Semi-trigoometric Lagrage iterpolatio method for solvig sigular itegral equatios (SIE) with cosecat kerel is proposed. For solvig SIE difficulties lie i its sigular term. I order to remove sigular term of SIE we choose appropriate weights quadrature formula. Compared with kow ivestigatios its advatages are that the preset method requires the least computatioal effort ad accuracy i umerical computatio is higher. The fial umerical experimet illustrate the method is efficiet. Keywords: sigular itegral equatios; cosecat kerel; Semi-trigoometric; Lagrage iterpolatio; collocatio method 010 AMS Subect Classificatios: 45E99; 30E0

2 46 Xigtia Gog ad Shuwei Yag 1. INTRODUCTION This paper is cocered with umerical evaluatio for sigular itegral equatios with cosecat kerel of the form b() t t a( t) ( t) csc ( ) d K( t ) ( ) d f ( t) 0 0 t[0 ) (1) where a b H K H is a give costat to fid the solutio H. So far the study o equatio (1) is oly limited to some discussios o its characteristic equatio [3-6]there are few papers devoted to the umerical method of Equatio (1). Du ad She [1] applied the reproducig kerel method to solve this problem while Wei ad Zhog [] adopted Chebyshev polyomial method. I this paper a ew collocatio method is developed by usig Semi-trigoometric Lagrage iterpolatio polyomial. Lagrage basis for the Semi-trigoometric iterpolatio are firstly give meawhile we give umerical quadrature rules for the improper itegral with cosecat kerel usig it. Through compariso the umerical results show the accuracy ad effectiveess of this method. The work is orgaized as follows. Sectio gives basic kowledge ad a useful lemma for the followig cotet. Sectio 3 is devoted to describig the umerical solutio of the itegral equatio with cosecat kerel. I Sectio 4 the umerical results cofirm that the method is accurate ad efficiet. Sectio 5 eds this paper with a brief coclusio.. SEMI-TRIGONOMETRIC INTERPOLATION Theorem.1 The system si tcos tsi tcos t si k tcos k t () is orthogoal i the space L ([ ]) moreover it is a maximal orthogoal set Ref.[8]. Defiitio.1 For we deote by T 1 the liear space of Semi-trigoometric polyomial 1 1 q ( t) ak si k t bk cos k t k0 (3)

3 A collocatio method for sigular itegral equatios with cosecat kerel 47 with real (or complex) coefficiets a 0 a ad b 0 b. A Semi-trigoometric polyomial qt 1 () t is said to be of degree if a b 0. If f() t satisfies f ( t ) f ( t) t the we call it ati-periodic fuctio we kow f () t T 1. Theorem. There exists a uique Semi-trigoometric polyomial 1 1 q ( t) ak si k t bk cos k t k0 (4) satisfyig the iterpolatio property q ( t ) y 0 1. Its coefficiets are give by k ( )si 0 a f t k t k ( )cos 0 b f t k t where t 0 1 are equidistat subdivisio of the iterval [0 ] with grid poits of eve umber. From this we deduce that the Lagrage basis for Semi-trigoometric iterpolatio have the form 1 1 L( t) cos k t t k 0 (5) for t [0 ] ad 0 1. Lemma.3 For the Semi-trigoometric polyomial we have itegrals 0 t 1 csc si k tdt 0 t 1 csc cos k tdt 0 i the sese of Cauchy pricipal values.

4 48 Xigtia Gog ad Shuwei Yag Proof. Usig the followig Lagrage trigoometric idetities ad the itegratig it with respect to t o the iterval[0 ]. The sigular part of the itegral equatio is give by t T0 ( t) : csc ( ) d 0 0 t. (6) The Semi-trigoometric basis fuctios i( k 1 ) t f ( t) : e k are eigefuctios i.e. k T f if 0 k k (7) from the Riesz theory for compact operators we have that T0 has a bouded iverse if ad oly if T0 is iective. We ow costruct umerical quadratures for the improper itegral t Q ( t) : csc ( ) d 0 0t (8) by replacig the cotiuous ati-periodic fuctio H by its semi-trigoometric Lagrage iterpolatio polyomial described i Theorem.1. Usig the Lagrage basis we obtai 1 ( ) ( ) ( ) ( ) 0 Q t C t t (9) with the equidistat quadrature poits t ad the quadrature weights ( ) t C ( t) csc L( ) d 0 1. (10) 0 Usig Lemma.3 ad from the form (4) of the Lagrage basis we derive ( ) 1 C ( t) si k t t k (11)

5 A collocatio method for sigular itegral equatios with cosecat kerel 49 This quadrature is uiformly coverget for all semi-trigoometric polyomials sice by costructioq itegrates Semi-trigoometric polyomial of degree less tha or equal to exactly. Sice K H the Kt ( ) ( ) is periodic fuctio for variable thus we adopt composite trapezoidal rule of periodic fuctio for Kt ( ) ( ) 1 K( t ) ( ) d K( t t ) ( t ) 0 (1) 0 with the equidistat quadrature poits t NUMERICAL SOLUTION For the umerical solutio of the itegral equatios we describe the collocatio method based o Semi-trigoometric iterpolatio that is fully discreted via appropriate quadrature approximatio of itegral operators. I the collocatio method we approximate the itegrals by the quadrature formula (6) ad (9) that is we replace the (SIE) (1) by bt () a( t) ( t) C ( t) ( t ) K( t t ) ( t ) f ( t) 1 1 ( ) 0 0 (13) for t [0 ). Now solvig (10) reduces to solvig a fiite dimesioal liear system. I particular for ay solutio of (10) the values ( t ) i 01 1 trivially satisfy the liear system at the quadrature poits i i bt ( ) a t t C t K t t t f t 1 i ( ) ( i) ( i) ( i) ( i ) ( ) ( i) 0 (14) for i01 1. Ad coversely give a solutio i i01 1of the system (11) the the fuctio iterpolatio polyomial is defied by its Semi-trigoometric Lagrage

6 50 Xigtia Gog ad Shuwei Yag 1 ( t) : L ( t) 0 t [0 ). (15) 4. NUMERICAL EXAMPLES Let us cosider the SIE with cosecat kerel i Ref.[1]: 1 t t 3 ( t) csc ( ) d 5 ( ) d f ( t) 0 0 (16) where right term f ( t) 10( t ) cos( t ) 3si( t ) ad the exact solutio is ( t) si( t ). I order to imply the covergece of the method we take differet the Table 1 illustrates the covergece behaviour through the differece betwee the exact solutio () t ad the approximate solutio () t at five poits. Of course the approximatio () t is obtaied from via (1) by the correspodig Semi-trigoometric Lagrage iterpolatio. covergece is clearly exhibited. From Table 1 we ote that rapid Table 1. Absolutio error betwee the exact solutio ad approximate solutio. t 10 t 5 t t 4 5 t E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E-8 For the error ad covergece aalysis of this method we poited out that as the approximate solutio coverges uiformly to the exact solutio of itegral equatio ad the covergece order of quadrature errors for (6) ad (9) carries over to

7 A collocatio method for sigular itegral equatios with cosecat kerel 51 the error. For detailed descriptio of the quadrature error we refer to Kress [7] ad ivolved literatures therei. 5. CONCLUSION I this paper we give the approximate solutio of SIE with cosecat kerel by Semi-trigoometric iterpolatio polyomial. The fial umerical results show that this method is effective. Moreover this method ca be exteded to solve other equatios with ati-periodic itegral kerel. REFERENCES [1] H. Du ad J. She Reproducig kerel method of solvig sigular itegral equatio with cosecat kerel J. Math. Aaly. Appl. 348 (008) pp [] W. Jiag Z. Che C. Zhag Chebyshev collocatio method for solvig sigular itegral equatio with cosecat kerel Iteratioal Joural of Computer Mathematics 89:8(01) pp [3] D. Jiu ad J. Du Sigular itegral equatios with cosecat kerel i solutios with sigularities of order oe Acta Math. Sci. 6B (006) pp [4] D. JiuO the umerieal solutio for sigular itegral equatios with coseeat kerel.ph.d Thesis Wuha Uiversity005 pp [5] J. XiaoPiO paratrigoometric Lagrage iterpolatio ad its applicatio i sigular itegral equatio with cosecat kerel.ph.d Thesis Cetral Chia Normal Uiversity 007 pp [6] D. Jiyua H. Huili J. Guoxiag. O trigoometric ad paratrigoometric Hermite iterpolatio. Joural of Approximatio Theory : [7] Kress R. Liear Itegral Equatios 3rd. ed. Spriger New York 014. [8] E. Berriochoa About a system of ati-periodic trigoometric fuctios Computers Mathematics with Applicatios56(008) pp

8 5 Xigtia Gog ad Shuwei Yag