Research Article The Modified Trapezoidal Rule for Computing Hypersingular Integral on Interval

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1 Hindwi Pulishing Corportion Journl of Applied Mthemtics Volume 03, Article ID , 9 pges Reserch Article The Modified Trpezoidl Rule for Computing Hypersingulr Integrl on Intervl Jin Li, nd Xiuzhen Li School of Science, Shndong Jinzhu University, Jinn 500, Chin School of Mthemtics, Shndong University, Jinn 5000, Chin Correspondence should e ddressed to Jin Li; lijin@lsec.cc.c.cn Received 3 August 03; Revised Octoer 03; Accepted Octoer 03 Acdemic Editor: Kzutke Komori Copyright 03 J. Li nd X. Li. This is n open ccess rticle distriuted under the Cretive Commons Attriution License, which permits unrestricted use, distriution, nd reproduction in ny medium, provided the originl work is properly cited. The modified trpezoidl rule for the computtion of hypersingulr integrls in oundry element methods is discussed. When the specil function of the error functionl equls zero, the convergence rte is one order higher thn the generl cse. A new qudrture rule is presented nd the symptotic expnsion of error function is otined. Bsed on the error expnsion, not only do we otin high order of ccurcy, ut lso posteriori error estimte is conveniently derived. Some numericl results re lso reported to confirm the theoreticl results nd show the efficiency of the lgorithms.. Introduction Consider the following integrl: f (x) I(f,s):, s (, ), p,, () p (x s) where denotes Hdmrd finite-prt integrl (p is clled hypersingulr integrl nd p is clled supersingulr integrl) nd s is the singulr point. The formultion of these clsses of oundry vlue prolems in terms of hypersingulr integrl equtions hs drwn lots of interest. Mny scientific nd engineering prolems, such s coustics, electromgnetic scttering, nd frcture mechnics, cn e reduced to oundry integrl equtions with hypersingulr kernels. There exist severl definitions, equivlent mthemticlly, for such kind of integrls in some litertures. We mention the following one: f (x) s ε { ε 0 lim sε f (x) f (x) f (s) }, s (, ). (x s) ε () Accurte clcultion of oundry element methods (BEM) rising in oundry integrl equtions hs een suject of intensive reserch in recent yers. The hypersingulr integrls hve certin properties different from regulr nd wek singulr integrls. One of the mjor prolems rising from oundry element method, for solving such integrl equtions, is how to evlute the hypersingulr integrls on the intervl or on the circle efficiently. Hypersingulr integrl must e considered in Hdmrd finite-prt sense. Numerous works [ 8] hve een devoted towrds developing efficient qudrture formuls. In 983, the series expnsion of hypersingulr integrl kernel on circle ws firstly suggested y Yu [9]. He solved the hrmonic nd ihrmonic nturl oundry integrl equtions successfully. The Newton-Cotes methods to compute the hypersingulr integrl on intervl were firstly studied y Linz [0]withgenerlized trpezoidl nd Simpson rules which fil ltogether when the singulr point s is close to mesh point. In order to

2 Journl of Applied Mthemtics mkethemesheselectedinsuchwythts flls ner the center of suintervl, two shorter suintervls t the end of the intervl were llowed. Then Yu [] gve new qudrture formule to compute the cse of singulr point coinciding with the mesh point which presented tht the error estimte is O(h ln h ). In 999, Wu nd Yu []presentedsimple,esy to e implemented methods not ffected y the loction of singulr point with clcultion of doule. In recent yers, the cse of singulr point coincided with the mesh point, nd Wu et l. [3] presented modified trpezoidl rule nd proved the O(h) convergence rte. In this pper, for the cse of singulr point coinciding with the mesh point new qudrture rule is introduced. Bsed on the expnsion of the error functionl, the error estimte is presented nd posteriori error estimte is given. Then not only do we otin high order of ccurcy, ut lso posteriori error estimte is conveniently derived. The rest of this pper is orgnized s follows. In Section, fter introducing some sic formuls of the generl (composite) trpezoidl rule nd nottions, we present our min result. In Section 3, the corresponding theoreticl nlysis is given. Finlly, severl numericl exmples re given to vlidte our nlysis.. Min Result Let x 0 <x < <x n <x n e uniform prtition of the intervl [, ] with mesh size h ( )/n nd set x i h, i,,...,n; (3) 6 then we get the new prtition: x 00 <x 0 < < <x 0,n. (4) We define f L (x), the modified trpezoidl interpoltion for f(x),s f L (x) x f(x ) x f(x 0i ), 0i nd liner trnsformtion x [, ], 0 i n, x (τ) : (τ) x 0i, i,...,n, τ [, ], from the reference element [, ] to the suintervl [, ].Forthetwosuintervls[, x 0 ] nd [,]ner the end of the intervl [, ], τ vlues in [/3, ] nd [, /3], respectively. Replcing f(x) in ()withf L (x) gives the new composite trpezoidl rule: n f I n (f, s) : L (x) ω j (s) f( ) I(f,s) E n (f), i0 (5) (6) (7) where ω i (s) is the Cotes coefficients: δ ω i (s) i0 x ln 0i s x 0,i x 0,i x 0,i s δ i,n s ln s δ i0 x 00 s δ i,n, s x 0,n (8) 0 i n, δ ij denotes the Kronecker delt, nd E n (f) denotes the error functionl. Theorem. Assume f(x) C α [, ], α [0,).Forthe trpezoidl rule I n (f, s) defined in (7), thereexistspositive constnt C,independentofh nd s,suchtht where E n (f) C( ln h ln γ (τ) )hα, (9) γ (τ) min s x i 0 i n h τ, τ (, ). (0) Proof. Let R(x) f(x) f L (x);thenwehve R(x) Ch α, s I(f,s) I n (f, s) x 0,m n i0,i m. () For the first prt of (), since R(x) C α [, ], y Tylor expnsion, we hve Ch α i, i0,,. () By the definition of finite-prt integrl, f (x) ( ) f (s) (s )( s) f (s) ln s s we hve x 0,m f (x) f(s) f (s)(x s), hr (s) (s )(x 0,m s) R (s) ln x 0,m s s x 0,m R(s) R (s)(x s). (3) (4)

3 Journl of Applied Mthemtics 3 Now, we estimte the right hnd side of (4) term y term. Since R( )0,wehve hr (s) h[r(s) R(x 0m )] (s )(x 0,m s) (s )(x 0,m s) hr m (ξ m) (s x 0,m ) Ch α, ξ m (,s), x 0,m R (s) ln x 0,m s s C ln γ (τ) hα, R(s) R (s)(x s) (x s) Ch α, η m (,x 0,m ). For the second prt of (), x 0 ( n i,i m x 0 ( ) x 0,m x 0,m x 0 ( ) n ) i,i m,m,m x0,m x 0,m n i,i m,m,m R( ) x 0,m x 0,m R(x 0,m ) x 0,m x 0 ( n i,i m,m,m R (ξ 0m )(x ) x 0,m ) x 0,m R (ξ 0,m )(x x 0,m ) x 0,m Ch α x 0 ( n i,i m,m,m (5) Ch α ( x 0,m ) x s x0,m x 0,m C[ ln γ (τ) ln h ]hα. x s ) (6) Comining (4), (5), nd (6) ledsto(9) ndtheproof is completed. Firstly, we set then we hve { φ (x) { { τ dτ, x <, τ x τ dτ, x >; τ x φ (x) { τ dτ, x <, (τ x) { { τ dτ, x >. (τ x) By stright clcultion, we get (7) (8) φ x (x) xlog. (9) x Let J:(, ) (, ) (, ) nd the opertor W: C(J) C(, ) e defined s Wf (τ) : f (τ) i [f (i τ) f( i τ)], i τ (, ). (0) Oviously, the opertor W is liner opertor. Then we set S (τ) Wφ (τ) φ (τ) [φ (i τ) φ ( i τ)]. () Now we present our min results elow. The proof will e given in the next section. Theorem. Assume f(x) C 3 [, ]. Forthetrpezoidl rule I n (f, s) defined in (7), there exists positive constnt C, independent of h nd s,suchtht E n (f) f (s) h S (τ) R f (s), () where s (τ)h/, i,,...,n,nd R f (s) C(η(s) ln h ln γ (τ) )h (3) (γ(τ) defined s (0))nd η (s) mx { s, }. (4) s

4 4 Journl of Applied Mthemtics 3. Proof of Min Results 3.. Preliminries. In the following section, C denotes certin constnt independent of h nd s, nd its vlue vries with plces. Lemm 3. Assume tht f(x) C 3 [, ] nd f L (x) re defined y (5); there holds where f (s) (x x 0,i )(x ) (5) R i (x) R i (x), R i (x) (x )(x ) 6( ) R i (x) f(3) (β 0i ) ξ 0i,η 0i,β 0i (, ),nd [f (3) (ξ 0i )(x ) f (3) (η 0i )(x ) ], (x )(x ) (x s), (6) R i (x) Ch3. (7) Proof. Tking Tylor expnsion for f( ), f( ) t x, there holds f (x)! (x )(x ) (x )(x ) 6( ) : f (s) [f (3) (ξ 0i )(x ) f (3) (η 0i )(x ) ] (x )(x )R i (x) R i (x) ; y pplying Tylor expnsion to f (x) in (8)ts,wehve (8) f (x) f (s) f (3) (η i ) (x s). (9) Therefore, (5) cneotineddirectlyfrom(8) nd(9). Lemm 4. Assume tht s (, ) nd c i (s )/h, i n ; there holds φ (c { h x0,i (x )(x ), i m, i) { x0,i { h (x )(x ), i m. (30) Proof. According to () nd the liner trnsformtion (6)for im,wehve x 0,m (x )(x x 0,m ) c m (ε/h) {( ε 0 hlim h ) c m (ε/h) τ (τ ) dτ } (τ c m ) ε τ (τ c m ) dτ hφ (c m), (3) wherewehveuse (τ). The second identity cn e similrly otined. Lemm 5. Suppose φ (x) nd η(s) re defined y (8) nd (4), respectively; then one hs φ (i τ) im in m φ ( i τ) Cη(s) h, (3) where denotes tht the first intervl is certin prt of the reference element. Proof. By strightforwrd clcultion, we hve φ (x) C dτ τ x. (33) Noting tht s ((τ )/)h (m (τ/))h (h/3), we hve (s )/h 3τ 6m nd φ (i τ) im C[ /3 dt i τ t im dt iτ t ] C 3τ6m x C 3τ 6m Ch s. (34) On the other hnd, since nh,wehve6( s)/h 6(n m) (3τ ) nd in m φ (τ i) dt i τt /3 C[ in m dt i τt ] C 6(n m) (3τ ) x C 6 (n m) (3τ ) Ch s. (35)

5 Journl of Applied Mthemtics 5 Comining (34)nd(35), we get (3). Set H m (x) f(x) f L (x) f (s) (x x 0,m )(x ). (36) Lemm 6. Under the sme ssumptions of Theorem, for H m (x) in (36), there holds tht x 0,m where γ(τ) is defined in (0). H m (x) C ln γ (τ) h, (37) Proof. Since f(x) C 3 [, ], y Tylor expnsion, we hve H(i) m (x) Ch3 i, i0,,. (38) By the definition of finite-prt integrl (3), we hve x 0,m H m (x) hh m (s) (s )(x 0,m s) H m (s) ln x 0,m s s x 0,m H m (x) H m (s) H m (s)(x s) (x s). (39) Now, we estimte the right hnd side of (39) term y term. Since H m ( )0,wehve hh m (s) h[h m (s) H m ( )] (s )(x 0,m s) (s )(x 0,m s) hh m (ξ m) (s x 0,m ) x 0,m Ch, ξ m (s,x 0,m ), (40) H m (s) ln x 0,m s s C ln γ (τ) h, (4) H m (x) H m (s) H m (s)(x s) x 0,m H m (η m) Ch, η m (,x 0,m ). (4) Comining (40), (4), nd (4)ledsto(37)ndtheproof is completed. Proof of Theorem. Consider ( ) x 0,m n i0,i m f (s) n i,i m n i0,i m x 0 f (x) n i0,i m (x s) 0i f L (x) (x )(x ) R i (x) n i0,i m By the definition of E m (x),wehve x 0,m x 0,m f (s) H m (x) x 0,m R i (x). (x )(x ). Putting (43)nd(44)together,wehve where n R x 0 (s) ( i x 0 f (s) hs (τ) R f (s), R f (s) R (s) R (s) R 3 (s), R (s) x 0 ( R 3 (s) h f (s) n x 0,m i,i m im [ n i,i m H m (x), φ (i τ) ) R i (x) ) R i (x), φ in m ( i τ)]. (43) (44) (45) (46)

6 6 Journl of Applied Mthemtics For the first prt of R (s), x 0 n ( i,i m x 0 n ( i,i x 0 ( ) R i (x) m,m,m ) R i (x) x 0,m R m (x) x0,m x 0,m R m (x) x 0,m x 0,m x 0,m x 0,m x 0 ( n i,i m,m,m R m (x) R m () ) R i (x) Rm (x) R m (x 0,m) n i,i m,m,m R m (ξ0m )(x x 0m ) x 0,m x 0,m x 0,m Ch 3 x 0 ( Ch ( x 0,m ) R i (x) Rm (ξ0,m )(x x 0,m ) n i,i m,m,m x s x0,m x 0,m ) x s ) C[ ln γ (τ) ln h ]h, (47) where ξ 0m (x 0,m, ) nd ξ 0,m (x 0,m,x 0,m ).We hve lso used the identity R m () R m (x 0,m) 0, x < x s,nd x x 0,m < x s. For the second prt of R (s), x 0 ( n i,i m Ch x 0 ( ) R i (x) n i,i m C[ ln γ (τ) ln h ]h. By Lemms 5 nd 6,wehve ) x s R f (s) R (s) R (s) C[η(s) ln h ln γ (τ) ]h. Then the proof is completed. (48) (49) 3.. The Clcultion of S (τ). Let Q n (x) e the function of the second kind ssocited with the Legendre polynomil P n (x), defined y (cf. [4]) Q 0 (x) ln x x Q (x) xq 0 (x). (50) We lso define W(f,τ):f(τ) i0 Then, y the definition of W, W(Q 0 ) (τ) ln τ τ i [f (i τ) f( i τ)], τ (, ). (ln iτ i τ ln i τ i τ ) lim iτ ln i i τ 0, i (5) W(xQ 0 ) (τ) τ τ i τ ( (i τ) i τ ( i τ) ) It follows tht which mens kn n k n lim k(/) (τ/) π tn πτ. (5) S(Q,τ)W(Q 0 xq 0,τ) π tn τπ, (53) S(Q,τ) π tn τπ πτ τ ln cos C. (54) Wht remins is to determine the constnt C. Byusing the identities (cf. [4,Chpter, Section.]), x cot x j k ( ) k (x) k B k (k)!, ln ( cos x) cos (jx), x (0, π), j where B k denote the Bernoulli numers, we hve k ( ) k (x) k B k (k )! (55) xln (sin x) [ (ln ) x jj sin (jx) ]. [ ] (56)

7 Journl of Applied Mthemtics 7 Tle : Errors of the mod-trpezoidl rule with sx [0n/4] (τ)h/. n τ0 τ/3 τ /3 τ/ e e e e e e e e e e e e e e e e e e e e e e e e 004 h α Tle : Errors of the mod-trpezoidl rule with s (τ)h/. n τ0 τ/3 τ /3 τ/ 3.57e e 00.30e e e e e e e e e e e e e e e e e e e e e e 003 h α Setting xπ/gives Then we hve k W(Q ;0) ( ) k (π) k B k ln. (57) (k )! i Q (i) (k )(i) k i k k ( ) k (π) k B k ln, (k )! (58) where we hve used the formule (cf. [4,Chpter,Section.]) Then we hve Q (x), x >, (k ) xk k i ( )k k B k (k)! k π k. i (59) S (τ) ln cos τπ. (60) By Theorem, we get the following error expnsion: E n (f) f (s) h log cos πτ O(h ). (6) Let T(h l )I l n 0 (f,s), l,,..., (6) where n 0 is the strting meshes, l is the refining numers, nd h l ( )/ l n 0. Then we hve the following. Corollry 7. Under the sme ssumption of Theorem nd (6), there holds 4. Numericl Exmples T(h l ) T(h l )O(h l ). (63) In this section, computtionl results re reported to confirm our theoreticl nlysis. Exmple. Consider the hypersingulr integrl 0 3s3 s 3s log s. (64) s We exmine the dynmic point sx [0n/4] (τ)h/,in Tle show tht when the locl coordinte of singulr point τ±/3, the qudrture rech the convergence rte of O(h ) s for the nonsupersingulr point, there re no convergence rte which gree with our theoremticlly nlysis. For the cse of s (τ )h/, Tle shows tht there is no superconvergence phenomenon ecuse of the influence of η(s) which coincides with our theoreticl nlysis. x 3 Exmple. Consider the hypersingulr integrl x 4 0 4s s 4 3 s s (s ) 4s3 log s. s (65)

8 8 Journl of Applied Mthemtics Tle 3: Errors of the mod-trpezoidl rule. n s /4 h α s/ h α s5/8 h α 3.e e e e e e e e e e e e e e e e e e Tle 4: A posteriori error estimte of the mod-trpezoidl rule. n s /4 h α s/ h α s5/8 h α e e e e e e e e e e e e e e e Tle 5: Errors of the mod-trpezoidl rule. n i h α i0 h α i h α 3.90e e e e e e e e e e e e e e e e e e e e e e 00 Tle 6: A posteriori error estimte of the mod-trpezoidl rule. n i h α i0 h α i h α e e e e e e e e e e e e e e e e e e e 00 The numericl results show tht the convergence rte reches (h ) when the singulr point coincides with the mesh point in Tle 3. In Tle 4, posteriori error estimte is presented nd the convergence rte is lso O(h ) which grees with our theoreticl nlysis. Exmple 3. Now we consider n exmple of less regulrity. Let, s0,nd f (x) F i (x) : x (sign (x)) x i0.5, (66) i,0,. Oviously, F i (x) C 3 i0.5 [, ] (i, 0, ). The exct vlue of the integrl is I (F i (x),0) 4 4i 3 i. (67) The numericl results re presented in Tles 5 nd 6. When the density function f(x) is smooth enough i, theerroroundiso(h ), nd if the density function hs less regulrity i 0,, there is no hyperconvergence phenomenon, which mens the regulrity of density function cnnot e reduced. Acknowledgments The work of Jin Li ws supported y the Ntionl Nturl Science Foundtion of Chin (nos. 047, 009, nd 046), Chin Postdoctorl Science Foundtion Fund Project(no.03M54054),theShndongProvincilNturl Science Foundtion of Chin (no. ZR0AQ00), nd Project of Shndong Province Higher Eductionl Science nd Technology Progrm (no. JLE8).

9 Journl of Applied Mthemtics 9 References [] G. Monegto, Numericl evlution of hypersingulr integrls, Journl of Computtionl nd Applied Mthemtics, vol. 50, no. 3, pp. 9 3, 994. [] U. J. Choi, S. W. Kim, nd B. I. Yun, Improvement of the symptotic ehviour of the Euler-Mclurin formul for Cuchy principl vlue nd Hdmrd finite-prt integrls, Interntionl Journl for Numericl Methods in Engineering,vol. 6,no.4,pp ,004. [3] J. T. Chen nd H. K. Hong, Review of dul oundry element methods with emphsis on hypersingulr integrls nd divergent series, Applied Mechnics Reviews, vol. 5, no., pp. 7 3, 999. [4] A. Frngi nd M. Guiggini, A direct pproch for oundry integrl equtions with high-order singulrities, Interntionl Journl for Numericl Methods in Engineering,vol.49,no.7,pp , 000. [5] T. Hsegw, Uniform pproximtions to finite Hilert trnsform nd its derivtive, Journl of Computtionl nd Applied Mthemtics,vol.63,no.,pp.7 38,004. [6] C. Y. Hui nd D. Shi, Evlutions of hypersingulr integrls using Gussin qudrture, Interntionl Journl for Numericl Methods in Engineering,vol.44,no.,pp.05 4,999. [7] N. I. Iokimidis, On the uniform convergence of Gussin qudrture rules for Cuchy principl vlue integrls nd their derivtives, Mthemtics of Computtion, vol. 44, no. 69, pp. 9 98, 985. [8] P. Kim nd U. J. Choi, Two trigonometric qudrture formule for evluting hypersingulr integrls, Interntionl Journl for Numericl Methods in Engineering, vol. 56, no. 3, pp , 003. [9] J. Li, J. M. Wu, nd D. H. Yu, Generlized extrpoltion for computtion of hypersingulr integrls in oundry element methods, Computer Modeling in Engineering & Sciences, vol. 4, no., pp. 5 75, 009. [0] J. Li nd D. H. Yu, The superconvergence of certin twodimensionl Cuchy principl vlue integrls, Computer Modeling in Engineering & Sciences, vol. 7, no. 4, pp , 0. [] J. Li nd D. H. Yu, The superconvergence of certin twodimensionl Hilert singulr integrls, Computer Modeling in Engineering & Sciences, vol. 8, no. 3-4, pp. 33 5, 0. [] J. Li nd D. H. Yu, The error estimte of Newton-Cotes methods to compute hypersingulr integrls, Mthemtic Numeric Sinic,vol.33,no.,pp.77 86,0. [3] J. Li, X. Zhng, nd D. Yu, Superconvergence nd ultrconvergence of Newton-Cotes rules for supersingulr integrls, Journl of Computtionl nd Applied Mthemtics,vol.33,no., pp , 00. [4] J.Li,X.Zhng,ndD.Yu, Extrpoltionmethodstocompute hypersingulr integrl in oundry element methods, Science Chin Mthemtics,vol.56,no.8,pp ,03. [5]M.N.J.Moore,L.J.Gry,ndT.Kpln, Evlutionof supersingulr integrls: second-order oundry derivtives, Interntionl Journl for Numericl Methods in Engineering,vol. 69,no.9,pp ,007. [6] Q. K. Du, Evlutions of certin hypersingulr integrls on intervl, Interntionl Journl for Numericl Methods in Engineering,vol.5,no.0,pp.95 0,00. [7] J. Wu nd W. Sun, The superconvergence of the composite trpezoidl rule for Hdmrd finite prt integrls, Numerische Mthemtik,vol.0,no.,pp ,005. [8] X. Zhng, J. Wu, nd D. Yu, Superconvergence of the composite Simpson s rule for certin finite-prt integrl nd its pplictions, Journl of Computtionl nd Applied Mthemtics, vol. 3, no., pp , 009. [9] D. H. Yu, Nturl Boundry Integrl Method nd Its Applictions, Kluwer Acdemic Pulishers, 00. [0] P. Linz, On the pproximte computtion of certin strongly singulr integrls, Computing, vol.35,no.3-4,pp , 985. [] D. H. Yu, The pproximte computtion of hypersingulr integrls on intervl, Numericl Mthemtics, vol.,no.,pp. 4 7, 99. [] J. M. Wu nd D. H. Yu, An pproximte computtion of hypersingulr integrls on n intervl, Journl on Numericl Methods nd Computer Applictions, vol.9,no.,pp.8 6, 998. [3] J. Wu, Y. Wng, W. Li, nd W. Sun, Toeplitz-type pproximtions to the Hdmrd integrl opertor nd their pplictions to electromgnetic cvity prolems, Applied Numericl Mthemtics,vol.58,no.,pp.0,008. [4] L. C. Andrews, Specil Functions of Mthemtics for Engineers, McGrw-Hill, New York, NY, USA, nd edition, 99.

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