A direct method and convergence analysis for some system of singular integro-difierential equations Iurie Caraus⁄ Zhilin Liy Abstract A class of singular integro-difierential equations in Lebesgue spaces are stud- ied. There are many applications of the singular integro-difierential equations discussed in this paper. An example in modeling the stress distribution of an elastic medium with holes is discussed in the paper. Direct numerical schemes using a collocation method and a mechanical quadrature rule designed for the singular integro-difierential equations are proposed for arbitrary smooth closed contours. Convergence analysis of these methods are given. Numerical examples are also provided. Keywords: singularintegro-difierentialequations,elasticity,collocationmethods, mechanical quadrature rule. 1 Introduction In this paper, we study the following system of singular integro-difierential equations (SIDE) Z X” (r) 1 x (¿) (Mx ·) [A~r(t)x(r)(t)+B~r(t)…i ¿ ¡t d¿ r=0 ¡ Z (1) 1 + Kr(t;¿)¢x(r)(¿)d¿] = f(t); t 2 ¡; 2…i ¡ whereA~r(t), B~r(t), andKr(t;¿), (r = 0;”)aregivenmbymmatrixfunctions, f(t)isa (0) given m dimensional vector function, x (t) = x(t) is a m dimensional unknown vector r d x(t) function with up to ”’s continuous derivatives (r = 1;”), and ¡ is an arbitrary dtr ⁄Department of Mathematics and Informatics, Moldova State University, Mateevici 60 str., Chisinau, Moldova, MD-2009, e-mail: caraush@mail.md yCenter for Research in Scientiflc Computation & Department of Mathematics, North Carolina State University, Raleigh, NC 27695, e-mail: zhilin@math.ncsu.edu 1 Report Documentation Page Form Approved OMB No. 0704-0188 Public reporting burden for the collection of information is estimated to average 1 hour per response, including the time for reviewing instructions, searching existing data sources, gathering and maintaining the data needed, and completing and reviewing the collection of information. Send comments regarding this burden estimate or any other aspect of this collection of information, including suggestions for reducing this burden, to Washington Headquarters Services, Directorate for Information Operations and Reports, 1215 Jefferson Davis Highway, Suite 1204, Arlington VA 22202-4302. Respondents should be aware that notwithstanding any other provision of law, no person shall be subject to a penalty for failing to comply with a collection of information if it does not display a currently valid OMB control number. 1. REPORT DATE 3. DATES COVERED 2003 2. REPORT TYPE 00-00-2003 to 00-00-2003 4. TITLE AND SUBTITLE 5a. CONTRACT NUMBER A direct method and covergence analysis for some system of singular 5b. GRANT NUMBER integro-differential equations 5c. PROGRAM ELEMENT NUMBER 6. AUTHOR(S) 5d. PROJECT NUMBER 5e. TASK NUMBER 5f. WORK UNIT NUMBER 7. PERFORMING ORGANIZATION NAME(S) AND ADDRESS(ES) 8. PERFORMING ORGANIZATION North Carolina State University,Center for Research in Scientific REPORT NUMBER Computation,Raleigh,NC,27695-8205 9. SPONSORING/MONITORING AGENCY NAME(S) AND ADDRESS(ES) 10. SPONSOR/MONITOR’S ACRONYM(S) 11. SPONSOR/MONITOR’S REPORT NUMBER(S) 12. DISTRIBUTION/AVAILABILITY STATEMENT Approved for public release; distribution unlimited 13. SUPPLEMENTARY NOTES The original document contains color images. 14. ABSTRACT see report 15. SUBJECT TERMS 16. SECURITY CLASSIFICATION OF: 17. LIMITATION OF 18. NUMBER 19a. NAME OF ABSTRACT OF PAGES RESPONSIBLE PERSON a. REPORT b. ABSTRACT c. THIS PAGE 17 unclassified unclassified unclassified Standard Form 298 (Rev. 8-98) Prescribed by ANSI Std Z39-18 smooth closed contour. We wish to flnd the solution x(t) to equation (1) satisfying the condition Z 1 x(¿)¿¡k¡1d¿ = 0; k = 0; ” ¡1: (2) 2…i ¡ We will develop two direct methods to solve the system of singular integral equations (SIDE) (1)-(2) and provide the convergence analysis. Singularintegralequations(SIE)andSIDEhavebeenusedtomodelmanyphysical problems, for examples, elasticity theory [2, 9, 11, 15, 16, 17, 20], aerodynamics [11, 12, 14]. We present an example from elasticity theory in Section 2 to show one application of the SIDE (1)-(2) discussed in this paper. Since analytic solution to SIDE is rarely available, we look for an approximate solution to the SIDE using direct methods. Most of early direct methods for SIE and SIDE are designed for the case where the boundary ¡ is a unit circle or an interval [1], [3]-[7], [8, 18]. The requirement of unit circle is essential because some special polynomial interpolations are only deflned on the unit circle, see [13], for example. There are few papers that deal with arbitrary closed contours in the literature. It should be mentioned that using a conforming mapping, one can transform an arbitrary smooth closed contour to the unit circle. However, this approach may not simplify the problem due to the following: † It is not an easy job to flnd a conforming mapping. † The coe–cients, the kernels, and the right hand side of the transformed equation may be more complicated and may not be smooth anymore. † The convergence analysis may be more complicated due to the transformation of the contour. In the 1980-90’s, theoretical foundations for direct methods for SIE and systems of SIE deflned on arbitrary smooth contours in the complex plane were established in [22, 23], and maybe others. But direct methods for SIDE or system of SIDE for arbitrary contours are yet to be developed and analyzed. In this paper, we develop a collocation method for the systems of SIDE with arbi- trarysmoothcontours,andthemechanicalquadraturerule,see[23]forthedeflnition,to approximatetheintegralsinvolvedinthesystemofSIDE.Wealsogivetheconvergence analysis of our new method in Lebesgue spaces. The classical continuous function space can not be used because the singular operator of the integration is unbounded. The classical theory of projection methods for Lp spacesdoesnotapplybecausethenormofsomeprojectors(forexample, theprojectors of interpolation) is unbounded for Lebesgue spaces. Thus, it is necessary to elaborate a new theory for the collocation and quadrature method discussed in this paper. 2 The paper is organized as follows. In the next section, we present an application of SIDEtoshowtheimportanceofSIDE.InSection3, wedescribethenumericalschemes for systems of SIDE with an arbitrary smooth contour. The convergence analysis is given in Section 4. Numerical examples are provided in Section 5. 2 An application of singular integro-difierential equations We show an application of SIDE in this section to show the importance of our work on SIDE. Consider an elastic medium under conditions of plane strain or generalized plane stress which fllls all of the inflnite plane of the variable z and is weakened by a hole of arbitrary shape. By convention, this problem refers to the determination of the stress components along the contour of the hole when external forces are prescribed on the same contour. The state of stress at inflnity is given. As described in [9], the physical model is the following: 8 ‡ · >>< (cid:190)‰(1)+(cid:190)v(1) = 2 '1(»)+'1(») ; 2 ‡ · (3) >>: (cid:190)v(1)¡(cid:190)‰(1)+2i¿r(v1) = 2» !(»)'01(»)+!0(»)“1(») ; ‰2!0(») (1) (1) (1) where (cid:190)‰ ; (cid:190)v ; ¿rv are the components of the stress in the curvilinear coordinates related to the conformal mapping iv z = !(»); » = ‰e (4) 8 >>< `01(») ˆ10(») '1(») = !0(»); “1(») = !0(») (5) >>: `1(») = `(!(»)); ˆ1(») = ˆ(!(»)); and `(z); ˆ(z); as always, denote the complex potentials in the z plane. Suppose that the region § with boundary (cid:176) which is mapped by relation (4) onto the physical region S represents an inflnite region in the plane of the auxiliary variable » outside the unit circle centered at the origin. This means that the function !(») is representable in the region § as the series c1 c2 cn !(») = R»+ + +:::+ :::: (6) » »2 »n It may also be recalled that the functions `1(»); ˆ1(») admit, in the same region the representation, 8 >>>< `1(») = ¡ X +iY log»+R¡»+`⁄(»); 2…(1+•) (7) >>>: ˆ1(») = ¡•(X ¡iY) log»+R¡0»+ˆ⁄(»); 2…(1+•) 3 where `⁄; ˆ⁄ are homomorphic functions, regular at » = 1; and X;Y are the components of the resultant of the external forces applied to the boundary of the region S: Assume now (1) 0 (1) 0 (1) 0 (cid:190)‰ = (cid:190)‰+(cid:190)‰; (cid:190)v = (cid:190)v +(cid:190)v; ¿rv = ¿rv +¿rv; (8) 0 0 0 where (cid:190) ; (cid:190) ; ¿ are the components of stress in a uniform fleld characterized by the ‰ v rv constants ¡ and ¡0 (¡ = ¡„), and represent relations (3) with ‰ = 1 as follows ((cid:190) = eiv): ‡ · 2 '((cid:190))+'((cid:190)) = (cid:190)‰+(cid:190)v on (cid:176); ‡ · (9) 2 !((cid:190))'0((cid:190))+!0((cid:190))“((cid:190)) = (cid:190)2!0((cid:190))((cid:190)v ¡(cid:190)‰+2i¿‰v): Here ' and “ are two unknown functions, homomorphic in the region §. From (5) to (7), we have the following asymptotic when j»j is large, 0 A A ¡2 ¡2 '(») = +O(» ); “(») = +O(» ); (10) » » A = ¡ X +iY ; A0 = •(X ¡iY): (11) 2…R(1+•) 2…R(1+•) 0 0 0 The stresses (cid:190) ; (cid:190) ; ¿ then can be calculated from ‰ v ‰v 8 ‡ · >>< (cid:190)‰0+(cid:190)v0 = 2 '0(»)+'0(») ; 2 ‡ · (12) >>: (cid:190)v0¡(cid:190)‰0+2i¿‰0v = 2» !(»)'00(»)+!0(»)“0(») ‰2!0(») with 0 R¡ R¡ '0 = !0(»); “0(») = !0(»): (13) Compared with (8), we have '1(») = '(»)+'0(»); “1(») = “(»)+“0(»): It is well known that, whenever the region occupied by the elastic medium contains 0 the inflnitely remote point of the plane, the constants ¡, ¡ , X, Y are specifled by 0 0 0 0 the problem itself. The stresses (cid:190) ; (cid:190) ; ¿ and the constants A and A are therefore ‰ v rv known, they are deflned by formulas (11) and (12). Our problem is to determine the normal stress (cid:190)v by formulas (9) from the given values on circumference (cid:176) of the other two components, (cid:190)‰ and ¿‰v: Assume (cid:190)v +(cid:190)‰ = ›((cid:190)); ¡2((cid:190)‰¡i¿‰v) = £((cid:190)) on (cid:176) (14) 4 and rewrite (9) as 8 ‡ · < 2 '((cid:190))+'((cid:190)) = ›((cid:190)); (15) : (cid:190)2!0((cid:190))(›((cid:190))+£((cid:190)))¡2!((cid:190))'0((cid:190)) = 2!0((cid:190))“((cid:190)) on (cid:176): After the transformations we obtain the required equation for determining ›((cid:190)) as follows: Z 1 d¿ (cid:190)2!0((cid:190))[›((cid:190))+£((cid:190))]+ ¿2!0(¿)[›(¿)+£(¿)] ¡ …i (cid:176) ¿ ¡(cid:190) Z (16) ¡ 1 !(¿)¡!((cid:190))›0(¿)d¿ ¡ 4A„R = 0 on (cid:176); …i (cid:176) ¿ ¡(cid:190) (cid:190) where A„;R;£((cid:190)) are known. After some manipulation, see [10] as well, we obtain Z (cid:190)2!0((cid:190)) ›(¿) (cid:190)2!0((cid:190))›((cid:190))+ d¿ …i (cid:176) ¿ ¡(cid:190) Z Z ¡ 1 !(¿)¡!((cid:190))›0(¿)d¿ + 1 ¿2!0(¿)¡(cid:190)2!0((cid:190))›(¿)d¿ (17) …i (cid:176) ¿ ¡(cid:190) …i (cid:176) ¿ ¡(cid:190) Z 4A„R 1 ¿2!0(¿)£(¿) = ¡(cid:190)2!0((cid:190))£((cid:190))¡ d¿ (cid:190) …i (cid:176) ¿ ¡(cid:190) or Z Z X1 A((cid:190)) ›(¿) 1 (”) A((cid:190))›((cid:190))+ …i (cid:176) ¿ ¡(cid:190)d¿ + …i (cid:176)K”((cid:190);¿)› (¿)d¿ = f((cid:190)); ”=0 where ¿2!0(¿)¡(cid:190)2!0((cid:190)) !((cid:190))¡!(¿) A((cid:190)) = (cid:190)2!0((cid:190)); K0((cid:190);¿) = ¿ ¡(cid:190) ; K1((cid:190);¿) = ¿ ¡(cid:190) ; Z 4A„R 1 ¿2!0(¿)£(¿) f((cid:190)) = ¡(cid:190)2!0((cid:190))£((cid:190))¡ d¿: (cid:190) …i (cid:176) ¿ ¡(cid:190) If !((cid:190));f((cid:190));A((cid:190)) are Hõlder functions on (cid:176), then K”((cid:190);¿) = Kj~¿”¡((cid:190)(cid:190);j¿‚), and K~”((cid:190);¿), are also Hõlder functions for 0 • ‚ < 1, and ” = 0;1, see [10]. Thus the application problem can be described by the SIDE (1)-(2) with ‚ = 0. 3 A collocation and quadrature method for system of SIDE Before we describe our numerical schemes, we need some theoretical preparations. 5 3.1 Theoretical preparations Let ¡ be a smooth Jordan curve which divides the entire complex plane C into two ¡ + ¡ + parts, F and F . Assume that F contains the inflnity and F contains the origin. Therefore F¡ = C ¡ fF+ [ ¡g. Let z = ˆ(w) be an analytic function that maps conformally the exterior of the unit circle to F¡ so that ˆ(1) = 1, ˆ(0)(1) > 0. Let us consider the Riemann function t = ˆ(w), w 2 ¡0 = f» : j»j = 1g, that has at (2) least up to second derivatives. Assume that ˆ (w) satisfles the Hõlder condition on jwj = 1, that is ˆ(2)(w) 2 H„(¡0), for some parameter „, „ (0 < „ < 1). We denote the set of such contours by C(2;„), see [23] for more details. In the complex [Lp(¡)]m(1 < p < 1) space, the norm of a vector functions g(t) = (g1(t);:::;gm(t)) is deflned as, 0 1 Z 1 Xm p 1 jjgjj = jjgkjjp; jjgkjjp = @ jgkjpjd¿jA ; l k=1 ¡ where l is the length of ¡. WewilldevelopadirectmethodforthesystemsofSIDE(1)togetherwithconditions (2). 3.2 A collocation and quadrature method Using the operators P = 1(I +S), Q = I ¡P, where I is the identity operator, and S 2 is a singular Cauchy nucleus, equation (1) can be written as follows: X” ‡ (Mx ·) Ar(t)(Px(r))(t)+Br(t)(Qx(r))(t)+ r=0 1 Z 1 + Kr(t;¿)x(r)(¿)d¿A = f(t); t 2 ¡ (18) 2…i ¡ where Ar(t) = A~r(t)+B~r(t); Br(t) = A~r(t)¡B~r(t); r = 0;”: We seek an approximate solution to problem (1)-(2) of the form Xn X¡1 xn(t) = »k(n)tk+” + »k(n)tk; t 2 ¡; (19) k=0 k=¡n where »k(n) = »k, k = ¡n; n are unknown vectors of dimension m. Note that xn(t)’s constructed using (19) satisfy the condition (2). 6 In our collocation method, we choose 2n+1 difierent points on ¡ ftjg such that the SIDE can be satisfled exactly (Mxn)(tj)¡f(tj) = 0; (20) As a result, we obtain a system of linear algebraic equations for the coe–cients »k, k = ¡n; n, X” Xn (k+”)! Xn (k+r¡1)! fAr(tj) (k+” ¡r)!tk+”¡r»k +Br(tj) (¡1)r (k¡1)! ¢ r=0 k=0 k=1 Z Xn 1 (k+”)! ¢t¡j k¡r ¢»¡k + 2…i ¢ (k+” ¡r)! Kr(tj;¿)¿k+”¡rd¿ ¢»k (21) k=0 ¡ Z Xn (k+r¡1)! 1 + (¡1)r (k¡1)! ¢ 2…i Kr(tj;¿)¿¡k¡rd¿ ¢»¡kg = f(tj); k=1 ¡ where j = 0; 2n. We need to approximate the integrals in the system of linear algebraic equations above as well. This is done using the mechanical quadrature formulae described in [23], Z Z 1 1 g(¿)¿l+kd¿ … Un(¿l+1¢g(¿))¿k¡1d¿; 2…i 2…i ¡ ¡ where the operator of interpolation Un is determined by, see [23], X2n (Ung)(t) = g(ts)¢ls(t); s=0 (cid:181) ¶ (22) lj(t) = k=0Q2;nk6=j ttj¡¡ttkk ttj n · kX=n¡n⁄k(j)tk; t 2 ¡; j = 0;2n: Thus, the system of linear algebraic equations for the unknown »k of (21) has been approximated by the following system of linear algebraic equations X” Xn (k+”)! fAr(tj) (k+” ¡r)!tkj+”¡r»k r=0 k=0 Xn (k+r¡1)! +Br(tj) (¡1)r (k¡1)! t¡j k¡r ¢»¡k k=1 (23) Xn X2n (k+”)! 1+”¡r (s) + (k+” ¡r)! Kr(tj;ts)ts ⁄¡k»k k=0 s=0 +Xn (¡1)r(k(+k¡r¡1)1!)! X2n Kr(tj;ts)t¡s1¡r⁄k(s)»¡kg = f(tj); k=1 s=0 for j = 0;2n. Thus we have obtained the system of linear algebraic equations for the unknowns »k in (19). Once these coe–cients are obtained, the solution to the singular integro-difierential equations is expressed as (19). 7 4 Error estimates for the numerical method In this section, we give some error estimates for the numerical method proposed in the previous section. ” ” d (Px)(t) d (Qx)(t) As was proved in [10], a paired vector functions and can be dt” dt” represented by the integrals of Cauchy type with the same density ‡(t): Z 9 ” d (Pdtx”)(t) = 21…i ¿‡(¡¿)td¿; t 2 F+ >>>>= d”(Qdtx”)(t) = 2t¡…”i Z¡ ¿‡(¡¿)td¿; t 2 F¡: >>>>; (24) ¡ Bymeansoftheaboveformulae,theproblem(1)-(2)canbereducedtoanequivalent (in terms of solving) singular integral equations system Z Z D(t) ‡(¿) 1 (R‡ ·)C(t)‡(t)+ d¿ + h(t;¿)‡(¿)d¿ = f(t); t 2 ¡; (25) …i ¿ ¡t 2…i ¡ ¡ where 1 ¡” C(t) = [A”(t)+t B”(t)]; 2 (26) 1 D(t) = [A”(t)¡t¡”B”(t)]; 2 and h(t;¿) is a matrix function belonging to [C(¡)]m space for both variables t and ¿, see [10]. The equivalence of the existence of the solution between the system of SIE (25) and the problem (1)-(2) is the result of the following lemma from [10]. Lemma 1 The system of SIE (25) and the problem (1)-(2) are equivalent in terms of the solvability. That is, for each solution ‡(t) of system (25), there is a solution x(t) of the problem (1)-(2), determined by the following formulae Z ˆ (cid:181) ¶ ! (¡1)” t ”X¡1 (Px)(t) = 2…i(” ¡1)! ‡(¿) (¿ ¡t)”¡1log 1¡ ¿ + fi~k¿”¡k¡1tk d¿ (27) k=1 ¡ Z ˆ (cid:181) ¶ ! (¡1)” ¿ ”X¡2 (Qx)(t) = 2…i(” ¡1)! ‡(¿)¿¡” (¿ ¡t)”¡1log 1¡ t + fl~k¿”¡k¡1tk d¿; k=1 ¡ wherefi~k andfl~k, arerealnumbers. Ontheotherhand, foreachsolutionx(t)ofproblem (1)-(2), there is a solution ‡(t) ” ” d (Px)(t) d (Qx)(t) ” ‡(t) = +t ; dt” dt” to the system (25). Furthermore, given a set of linear-independent solutions ‡(t) to the system (25), there are corresponding set of linear-independent solutions to the problem (1)-(2) from (27) and vise versa. 8 The proof can be found in [10]. Before we discuss the main convergence theorem of our method, we need several additional lemmas from [19]. Those lemmas will be used in our discussions in this section. Deflne the space n o – (”) [Wp ]m(¡) = g; 9g(r) 2 [C(¡)]m; r = 0; ” ¡1; g(”) 2 [Lp(¡)]m : – (”) – (”) For any g 2 [Wp ]m, g satisfles equation (2). The norm in [Wp ]m is deflned by jjgjjp;” = jjg(”)jj[Lp]m: ¡” We shall denote the image of space [Lp]m under the mapping P +t Q by [Lp;”]m with the same norm as in [Lp]m. – (”) Lemma 2 The difierential operator D” : [Wp ]m ! [Lp;”]m, (D”g)(t) = g(”)(t) is – (”) continuously reversible and its reverse operator D¡” : [Lp;”]m ! [Wp ]m is determined by the equality ¡” + ¡ (D g)(t) = (N g)(t)+(N g)(t); where Z (cid:181) ¶ (¡1)” t (N+g)(t) = (Pg)(¿)(¿ ¡t)”¡1log 1¡ d¿; 2…i(” ¡1)! ¿ ¡ Z (cid:181) ¶ (¡1)”¡1 ¿ (N¡g)(t) = (Qg)(¿)(¿ ¡t)”¡1log 1¡ d¿: 2…i(” ¡1)! t ¡ The proof can be found in [19]. – (”) Lemma 3 The operator B : [Wp ]m ! [Lp]m, of the form B = (P + t”Q)D” is reversible and ¡1 ¡” ¡” B = D (P +t Q): The proof can also be found in [19]. The existence of the solution to the systems of linear algebraic equations (21) and (23) is given in the following theorems. Theorem 4 Assume the following conditions are satisfled: 1. the contour ¡ 2 C(2;„); 0 < „ < 1, 9

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