Numerical quadrature for singular integrals on fractals
We present and analyse numerical quadrature rules for evaluating regular and singular integrals on self-similar fractal sets. The integration domain Γ ⊂ R n is assumed to be the compact attractor of an iterated function system of contracting similarities satisfying the open set condition. Integratio...
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| Abstract | We present and analyse numerical quadrature rules for evaluating regular and singular integrals on self-similar fractal sets. The integration domain
Γ
⊂
R
n
is assumed to be the compact attractor of an iterated function system of contracting similarities satisfying the open set condition. Integration is with respect to any “invariant” (also known as “balanced” or “self-similar”) measure supported on
Γ
, including in particular the Hausdorff measure
H
d
restricted to
Γ
, where
d
is the Hausdorff dimension of
Γ
. Both single and double integrals are considered. Our focus is on composite quadrature rules in which integrals over
Γ
are decomposed into sums of integrals over suitable partitions of
Γ
into self-similar subsets. For certain singular integrands of logarithmic or algebraic type, we show how in the context of such a partitioning the invariance property of the measure can be exploited to express the singular integral exactly in terms of regular integrals. For the evaluation of these regular integrals, we adopt a composite barycentre rule, which for sufficiently regular integrands exhibits second-order convergence with respect to the maximum diameter of the subsets. As an application we show how this approach, combined with a singularity-subtraction technique, can be used to accurately evaluate the singular double integrals that arise in Hausdorff-measure Galerkin boundary element methods for acoustic wave scattering by fractal screens. |
|---|---|
| AbstractList | We present and analyse numerical quadrature rules for evaluating regular and singular integrals on self-similar fractal sets. The integration domain
Γ
⊂
R
n
is assumed to be the compact attractor of an iterated function system of contracting similarities satisfying the open set condition. Integration is with respect to any “invariant” (also known as “balanced” or “self-similar”) measure supported on
Γ
, including in particular the Hausdorff measure
H
d
restricted to
Γ
, where
d
is the Hausdorff dimension of
Γ
. Both single and double integrals are considered. Our focus is on composite quadrature rules in which integrals over
Γ
are decomposed into sums of integrals over suitable partitions of
Γ
into self-similar subsets. For certain singular integrands of logarithmic or algebraic type, we show how in the context of such a partitioning the invariance property of the measure can be exploited to express the singular integral exactly in terms of regular integrals. For the evaluation of these regular integrals, we adopt a composite barycentre rule, which for sufficiently regular integrands exhibits second-order convergence with respect to the maximum diameter of the subsets. As an application we show how this approach, combined with a singularity-subtraction technique, can be used to accurately evaluate the singular double integrals that arise in Hausdorff-measure Galerkin boundary element methods for acoustic wave scattering by fractal screens. We present and analyse numerical quadrature rules for evaluating regular and singular integrals on self-similar fractal sets. The integration domain $$\Gamma \subset \mathbb {R}^n$$ Γ ⊂ R n is assumed to be the compact attractor of an iterated function system of contracting similarities satisfying the open set condition. Integration is with respect to any “invariant” (also known as “balanced” or “self-similar”) measure supported on $$\Gamma$$ Γ , including in particular the Hausdorff measure $$\mathcal {H}^d$$ H d restricted to $$\Gamma$$ Γ , where d is the Hausdorff dimension of $$\Gamma$$ Γ . Both single and double integrals are considered. Our focus is on composite quadrature rules in which integrals over $$\Gamma$$ Γ are decomposed into sums of integrals over suitable partitions of $$\Gamma$$ Γ into self-similar subsets. For certain singular integrands of logarithmic or algebraic type, we show how in the context of such a partitioning the invariance property of the measure can be exploited to express the singular integral exactly in terms of regular integrals. For the evaluation of these regular integrals, we adopt a composite barycentre rule, which for sufficiently regular integrands exhibits second-order convergence with respect to the maximum diameter of the subsets. As an application we show how this approach, combined with a singularity-subtraction technique, can be used to accurately evaluate the singular double integrals that arise in Hausdorff-measure Galerkin boundary element methods for acoustic wave scattering by fractal screens. We present and analyse numerical quadrature rules for evaluating regular and singular integrals on self-similar fractal sets. The integration domain Γ⊂Rn is assumed to be the compact attractor of an iterated function system of contracting similarities satisfying the open set condition. Integration is with respect to any “invariant” (also known as “balanced” or “self-similar”) measure supported on Γ, including in particular the Hausdorff measure Hd restricted to Γ, where d is the Hausdorff dimension of Γ. Both single and double integrals are considered. Our focus is on composite quadrature rules in which integrals over Γ are decomposed into sums of integrals over suitable partitions of Γ into self-similar subsets. For certain singular integrands of logarithmic or algebraic type, we show how in the context of such a partitioning the invariance property of the measure can be exploited to express the singular integral exactly in terms of regular integrals. For the evaluation of these regular integrals, we adopt a composite barycentre rule, which for sufficiently regular integrands exhibits second-order convergence with respect to the maximum diameter of the subsets. As an application we show how this approach, combined with a singularity-subtraction technique, can be used to accurately evaluate the singular double integrals that arise in Hausdorff-measure Galerkin boundary element methods for acoustic wave scattering by fractal screens. |
| Author | Moiola, Andrea Gibbs, Andrew Hewett, David |
| Author_xml | – sequence: 1 givenname: Andrew orcidid: 0000-0002-2934-008X surname: Gibbs fullname: Gibbs, Andrew email: andrew.gibbs@ucl.ac.uk organization: Department of Mathematics, University College London – sequence: 2 givenname: David surname: Hewett fullname: Hewett, David organization: Department of Mathematics, University College London – sequence: 3 givenname: Andrea surname: Moiola fullname: Moiola, Andrea organization: Dipartimento di Matematica, Università degli studi di Pavia |
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| Cites_doi | 10.1007/BF02437506 10.1007/s13373-013-0041-3 10.1007/s00041-011-9202-5 10.4236/wet.2011.22015 10.1090/S0002-9947-98-02218-1 10.1137/17M1131933 10.1007/s00211-021-01182-y 10.1109/8.664115 10.1512/iumj.1981.30.30055 10.1103/PhysRevA.36.920 10.1007/BF02476031 10.1090/S0025-5718-1982-0669644-3 10.1017/S0334270000002757 10.1016/j.jmaa.2004.10.059 10.1090/S0002-9947-99-01982-0 10.1137/S0036141096306911 10.1007/s11075-007-9111-5 10.1016/j.jmaa.2019.01.036 10.1109/8.475113 10.1063/1.1664597 10.1007/BF02878985 10.1007/978-3-0348-0034-1 10.1080/02786820117868 10.1007/s00023-006-0308-2 10.1142/S0218348X09004296 10.1017/CBO9780511623813 10.1007/978-1-4899-6886-9_5 10.1007/s00041-001-4001-z 10.1007/BF01889599 |
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| References | BessisDFournierJServiziGTurchettiGVaientiSMellin transforms of correlation integrals and generalized dimension of strange setsPhysical Review A19873692090172210.1103/PhysRevA.36.920 ManticaGFractal measures and polynomial sampling: I.F.S.-Gaussian integrationNumer. Algorithms200745269281235598810.1007/s11075-007-9111-51131.65027 MarionJMesure de Hausdorff d’un fractal à similitude interneAnn. Sci. Math. Québec19861051848411200613.28007 CarvalhoACaetanoAOn the Hausdorff dimension of continuous functions belonging to Hölder and Besov spaces on fractal d\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$d$$\end{document}-setsJ. Fourier Anal. Appl.201218386409289873410.1007/s00041-011-9202-51253.28003 FalconerKJThe Geometry of Fractal Sets1986CambridgeCambridge University Press0587.28004 GrahamIGGalerkin methods for second kind integral equations with singularitiesMath. Comp.19823951953366964410.1090/S0025-5718-1982-0669644-30496.65068 Mantica, G., Vaienti, S.: The asymptotic behaviour of the Fourier transforms of orthogonal polynomials I: Mellin transform techniques 8, 265–300 (2007) EltonJHYanZApproximation of measures by Markov processes and homogeneous affine iterated function systemsConstr. Approx.19895698798272510.1007/BF018895990658.60093 InfusinoMVolčičAUniform distribution on fractalsUnif. Distrib. Theory20094475825594121249.11070 JonssonAWavelets on fractals and Besov spacesJ. Fourier Anal. Appl.19984329340165098010.1007/BF024760310912.42025 SrivatsunGRaniSSKrishnanGSA self-similar fractal Cantor antenna for MICS band wireless applicationsWireless Eng. Tech.2011210711110.4236/wet.2011.22015 Geronimo, J., Hardin, D.: An exact formula for the measure dimensions associated with a class of piecewise linear maps. In: Constructive Approximation, pp. 89–98. Springer (1989) MóraPEstimate of the Hausdorff measure of the Sierpinski triangleFractals200917137148253545010.1142/S0218348X090042961178.28007 Puente-BaliardaCRomeuJPousRCardamaAOn the behavior of the Sierpinski multiband fractal antennaIEEE T. Antenn. Propag.199846517524161599310.1109/8.6641150944.78529 NIST Digital Library of Mathematical Functions. http://dlmf.nist.gov/, release 1.1.3 of 2021-09-15 Jonsson, A., Wallin, H.: Function Spaces on Subsets of Rn\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb{R}^{n}$$\end{document}. Math. Rep., 2 (1984) ManticaGA stable Stieltjes technique for computing orthogonal polynomials and Jacobi matrices associated with a class of singular measuresConstr. Approx.199612509530141219710.1007/BF024375060878.42014 BarnsleyMFDemkoSIterated function systems and the global construction of fractalsProc. Roy. Soc. A. Math. Phys. Sci.19853992432757991110588.28002 MoránMReyJ-MSingularity of self-similar measures with respect to Hausdorff measuresT. Am. Math. Soc.199835022972310147569110.1090/S0002-9947-98-02218-10899.28002 Bezanson, J., Karpinski, S., Shah, V.B., Edelman, A.: Julia: A fast dynamic language for technical computing. arXiv:1209.5145 (2012) FalconerKFractal Geometry: Mathematical Foundations and Applications20143New YorkWiley1285.28011 Mattila, P.: Geometry of Sets and Measures in Euclidean Spaces: Fractals and Rectifiability, CUP (1995) TriebelHFractals and Spectra1997BaselBirkhäuser10.1007/978-3-0348-0034-10898.46030 MarionJMesures de Hausdorff d’ensembles fractalsAnn. Sci. Math. Québec1987111119121660624.28003 XiongYZhouJThe Hausdorff measure of a class of Sierpinski carpetsJ. Math. Anal. Appl.2005305121129212811610.1016/j.jmaa.2004.10.0591077.28011 ZhouZWuMThe Hausdorff measure of a Sierpinski carpetScience in China Series A: Math.199942673680171700110.1007/BF028789850984.28005 SorensenCLight scattering by fractal aggregates: a reviewAerosol Science & Technology20013564868710.1080/02786820117868 DrmotaMInfusinoMOn the discrepancy of some generalized Kakutani’s sequences of partitionsUnif. Distrib. Theory201277510429431621313.11084 Falconer, K.: Personal communication AnselonePMSingularity subtraction in the numerical solution of integral equationsJ. Austral. Math. Soc. Ser. B19812240841862693210.1017/S03342700000027570477.65095 ForteBMendivilFVrscayE“Chaos games” for iterated function systems with grey level mapsSIAM J. Math. Anal.199829878890161772210.1137/S00361410963069110923.58032 KunzeHLa TorreDMendivilFVrscayERFractal-based Methods in Analysis2011New YorkSpringer1237.28002 StrichartzRSelf-similarity in harmonic analysisJ. Fourier Anal. Appl.19941137130706710.1007/s00041-001-4001-z0851.42001 Chandler-Wilde, S.N., Hewett, D.P.: Well-posed PDE and integral equation formulations for scattering by fractal screens. SIAM J. Math. Anal., 50 (2018) DubucSHamzaouiROn the diameter of the attractor of an IFSC. R. Math. Rep. Acad. Sci. Canada199416859012884300813.58034 HutchinsonJEFractals and self-similarityIndiana Univ. Math. J.19813071374762560010.1512/iumj.1981.30.300550598.28011 Caetano, A.M., Chandler-Wilde, S.N., Gibbs, A., Hewett, D., Moiola, A.: A Hausdorff measure boundary element method for acoustic scattering by fractal screens, In preparation (2022) SchlittDNumerical solution of a singular integral equation encountered in polymer physicsJ. Math. Phys.1968943643910.1063/1.16645970187.12403 ZubermanLExact Hausdorff and packing measure of certain Cantor sets, not necessarily self-similar or homogeneousJ. Math. Anal. Appl.2019474143156391289510.1016/j.jmaa.2019.01.0361414.28012 BarnsleyMVinceADevelopments in fractal geometryBull. Math. Sci.20133299348308401010.1007/s13373-013-0041-31275.28006 AyerEStrichartzRSExact Hausdorff measure and intervals of maximum density for Cantor setsTrans. Amer. Math. Soc.199935137253741143311010.1090/S0002-9947-99-01982-00933.28003 AmariSBornemannJEfficient numerical computation of singular integrals with applications to electromagneticsIEEE T. Antenn. Propag.1995431343134810.1109/8.475113 Chandler-WildeSNHewettDPMoiolaABessonJBoundary element methods for acoustic scattering by fractal screensNumer. Math.2021147785837424546210.1007/s00211-021-01182-y1475.65209 M Barnsley (1378_CR5) 2013; 3 Y Xiong (1378_CR41) 2005; 305 S Dubuc (1378_CR14) 1994; 16 J Marion (1378_CR31) 1987; 11 MF Barnsley (1378_CR6) 1985; 399 1378_CR16 S Amari (1378_CR2) 1995; 43 L Zuberman (1378_CR43) 2019; 474 A Jonsson (1378_CR24) 1998; 4 KJ Falconer (1378_CR17) 1986 SN Chandler-Wilde (1378_CR12) 2021; 147 Z Zhou (1378_CR42) 1999; 42 PM Anselone (1378_CR3) 1981; 22 C Sorensen (1378_CR37) 2001; 35 J Marion (1378_CR30) 1986; 10 IG Graham (1378_CR21) 1982; 39 R Strichartz (1378_CR39) 1994; 1 D Bessis (1378_CR7) 1987; 36 1378_CR11 E Ayer (1378_CR4) 1999; 351 H Kunze (1378_CR26) 2011 1378_CR32 C Puente-Baliarda (1378_CR35) 1998; 46 G Srivatsun (1378_CR38) 2011; 2 M Morán (1378_CR34) 1998; 350 P Móra (1378_CR33) 2009; 17 1378_CR29 K Falconer (1378_CR18) 2014 JE Hutchinson (1378_CR22) 1981; 30 M Drmota (1378_CR13) 2012; 7 B Forte (1378_CR19) 1998; 29 1378_CR1 G Mantica (1378_CR27) 1996; 12 G Mantica (1378_CR28) 2007; 45 A Carvalho (1378_CR10) 2012; 18 H Triebel (1378_CR40) 1997 1378_CR25 1378_CR20 1378_CR9 1378_CR8 JH Elton (1378_CR15) 1989; 5 M Infusino (1378_CR23) 2009; 4 D Schlitt (1378_CR36) 1968; 9 |
| References_xml | – reference: Chandler-WildeSNHewettDPMoiolaABessonJBoundary element methods for acoustic scattering by fractal screensNumer. Math.2021147785837424546210.1007/s00211-021-01182-y1475.65209 – reference: Falconer, K.: Personal communication – reference: EltonJHYanZApproximation of measures by Markov processes and homogeneous affine iterated function systemsConstr. Approx.19895698798272510.1007/BF018895990658.60093 – reference: TriebelHFractals and Spectra1997BaselBirkhäuser10.1007/978-3-0348-0034-10898.46030 – reference: InfusinoMVolčičAUniform distribution on fractalsUnif. Distrib. Theory20094475825594121249.11070 – reference: AnselonePMSingularity subtraction in the numerical solution of integral equationsJ. Austral. Math. Soc. Ser. B19812240841862693210.1017/S03342700000027570477.65095 – reference: Mantica, G., Vaienti, S.: The asymptotic behaviour of the Fourier transforms of orthogonal polynomials I: Mellin transform techniques 8, 265–300 (2007) – reference: FalconerKJThe Geometry of Fractal Sets1986CambridgeCambridge University Press0587.28004 – reference: DrmotaMInfusinoMOn the discrepancy of some generalized Kakutani’s sequences of partitionsUnif. Distrib. Theory201277510429431621313.11084 – reference: ManticaGFractal measures and polynomial sampling: I.F.S.-Gaussian integrationNumer. Algorithms200745269281235598810.1007/s11075-007-9111-51131.65027 – reference: MóraPEstimate of the Hausdorff measure of the Sierpinski triangleFractals200917137148253545010.1142/S0218348X090042961178.28007 – reference: Bezanson, J., Karpinski, S., Shah, V.B., Edelman, A.: Julia: A fast dynamic language for technical computing. arXiv:1209.5145 (2012) – reference: BarnsleyMFDemkoSIterated function systems and the global construction of fractalsProc. Roy. Soc. A. Math. Phys. Sci.19853992432757991110588.28002 – reference: CarvalhoACaetanoAOn the Hausdorff dimension of continuous functions belonging to Hölder and Besov spaces on fractal d\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$d$$\end{document}-setsJ. Fourier Anal. Appl.201218386409289873410.1007/s00041-011-9202-51253.28003 – reference: DubucSHamzaouiROn the diameter of the attractor of an IFSC. R. Math. Rep. Acad. Sci. Canada199416859012884300813.58034 – reference: JonssonAWavelets on fractals and Besov spacesJ. Fourier Anal. Appl.19984329340165098010.1007/BF024760310912.42025 – reference: BessisDFournierJServiziGTurchettiGVaientiSMellin transforms of correlation integrals and generalized dimension of strange setsPhysical Review A19873692090172210.1103/PhysRevA.36.920 – reference: Jonsson, A., Wallin, H.: Function Spaces on Subsets of Rn\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb{R}^{n}$$\end{document}. Math. Rep., 2 (1984) – reference: ManticaGA stable Stieltjes technique for computing orthogonal polynomials and Jacobi matrices associated with a class of singular measuresConstr. Approx.199612509530141219710.1007/BF024375060878.42014 – reference: FalconerKFractal Geometry: Mathematical Foundations and Applications20143New YorkWiley1285.28011 – reference: GrahamIGGalerkin methods for second kind integral equations with singularitiesMath. Comp.19823951953366964410.1090/S0025-5718-1982-0669644-30496.65068 – reference: NIST Digital Library of Mathematical Functions. http://dlmf.nist.gov/, release 1.1.3 of 2021-09-15 – reference: HutchinsonJEFractals and self-similarityIndiana Univ. Math. J.19813071374762560010.1512/iumj.1981.30.300550598.28011 – reference: BarnsleyMVinceADevelopments in fractal geometryBull. Math. Sci.20133299348308401010.1007/s13373-013-0041-31275.28006 – reference: SrivatsunGRaniSSKrishnanGSA self-similar fractal Cantor antenna for MICS band wireless applicationsWireless Eng. Tech.2011210711110.4236/wet.2011.22015 – reference: MoránMReyJ-MSingularity of self-similar measures with respect to Hausdorff measuresT. Am. Math. Soc.199835022972310147569110.1090/S0002-9947-98-02218-10899.28002 – reference: SorensenCLight scattering by fractal aggregates: a reviewAerosol Science & Technology20013564868710.1080/02786820117868 – reference: KunzeHLa TorreDMendivilFVrscayERFractal-based Methods in Analysis2011New YorkSpringer1237.28002 – reference: ZhouZWuMThe Hausdorff measure of a Sierpinski carpetScience in China Series A: Math.199942673680171700110.1007/BF028789850984.28005 – reference: StrichartzRSelf-similarity in harmonic analysisJ. Fourier Anal. Appl.19941137130706710.1007/s00041-001-4001-z0851.42001 – reference: MarionJMesure de Hausdorff d’un fractal à similitude interneAnn. Sci. Math. Québec19861051848411200613.28007 – reference: SchlittDNumerical solution of a singular integral equation encountered in polymer physicsJ. Math. Phys.1968943643910.1063/1.16645970187.12403 – reference: XiongYZhouJThe Hausdorff measure of a class of Sierpinski carpetsJ. Math. Anal. Appl.2005305121129212811610.1016/j.jmaa.2004.10.0591077.28011 – reference: MarionJMesures de Hausdorff d’ensembles fractalsAnn. Sci. Math. Québec1987111119121660624.28003 – reference: Chandler-Wilde, S.N., Hewett, D.P.: Well-posed PDE and integral equation formulations for scattering by fractal screens. SIAM J. Math. Anal., 50 (2018) – reference: ForteBMendivilFVrscayE“Chaos games” for iterated function systems with grey level mapsSIAM J. Math. Anal.199829878890161772210.1137/S00361410963069110923.58032 – reference: Puente-BaliardaCRomeuJPousRCardamaAOn the behavior of the Sierpinski multiband fractal antennaIEEE T. Antenn. Propag.199846517524161599310.1109/8.6641150944.78529 – reference: Caetano, A.M., Chandler-Wilde, S.N., Gibbs, A., Hewett, D., Moiola, A.: A Hausdorff measure boundary element method for acoustic scattering by fractal screens, In preparation (2022) – reference: Geronimo, J., Hardin, D.: An exact formula for the measure dimensions associated with a class of piecewise linear maps. In: Constructive Approximation, pp. 89–98. Springer (1989) – reference: Mattila, P.: Geometry of Sets and Measures in Euclidean Spaces: Fractals and Rectifiability, CUP (1995) – reference: ZubermanLExact Hausdorff and packing measure of certain Cantor sets, not necessarily self-similar or homogeneousJ. Math. Anal. Appl.2019474143156391289510.1016/j.jmaa.2019.01.0361414.28012 – reference: AmariSBornemannJEfficient numerical computation of singular integrals with applications to electromagneticsIEEE T. Antenn. Propag.1995431343134810.1109/8.475113 – reference: AyerEStrichartzRSExact Hausdorff measure and intervals of maximum density for Cantor setsTrans. Amer. Math. Soc.199935137253741143311010.1090/S0002-9947-99-01982-00933.28003 – volume: 12 start-page: 509 year: 1996 ident: 1378_CR27 publication-title: Constr. Approx. doi: 10.1007/BF02437506 – volume: 11 start-page: 111 year: 1987 ident: 1378_CR31 publication-title: Ann. Sci. Math. Québec – ident: 1378_CR1 – volume: 3 start-page: 299 year: 2013 ident: 1378_CR5 publication-title: Bull. Math. Sci. doi: 10.1007/s13373-013-0041-3 – volume-title: The Geometry of Fractal Sets year: 1986 ident: 1378_CR17 – volume: 399 start-page: 243 year: 1985 ident: 1378_CR6 publication-title: Proc. Roy. Soc. A. Math. Phys. Sci. – volume: 18 start-page: 386 year: 2012 ident: 1378_CR10 publication-title: J. Fourier Anal. Appl. doi: 10.1007/s00041-011-9202-5 – volume: 2 start-page: 107 year: 2011 ident: 1378_CR38 publication-title: Wireless Eng. Tech. doi: 10.4236/wet.2011.22015 – volume: 350 start-page: 2297 year: 1998 ident: 1378_CR34 publication-title: T. Am. Math. Soc. doi: 10.1090/S0002-9947-98-02218-1 – ident: 1378_CR11 doi: 10.1137/17M1131933 – volume: 7 start-page: 75 year: 2012 ident: 1378_CR13 publication-title: Unif. Distrib. Theory – ident: 1378_CR9 doi: 10.1007/s00211-021-01182-y – volume: 46 start-page: 517 year: 1998 ident: 1378_CR35 publication-title: IEEE T. Antenn. Propag. doi: 10.1109/8.664115 – volume: 16 start-page: 85 year: 1994 ident: 1378_CR14 publication-title: C. R. Math. Rep. Acad. Sci. Canada – volume: 30 start-page: 713 year: 1981 ident: 1378_CR22 publication-title: Indiana Univ. Math. J. doi: 10.1512/iumj.1981.30.30055 – volume: 147 start-page: 785 year: 2021 ident: 1378_CR12 publication-title: Numer. Math. doi: 10.1007/s00211-021-01182-y – volume: 36 start-page: 920 year: 1987 ident: 1378_CR7 publication-title: Physical Review A doi: 10.1103/PhysRevA.36.920 – volume: 4 start-page: 329 year: 1998 ident: 1378_CR24 publication-title: J. Fourier Anal. Appl. doi: 10.1007/BF02476031 – volume-title: Fractal-based Methods in Analysis year: 2011 ident: 1378_CR26 – volume-title: Fractal Geometry: Mathematical Foundations and Applications year: 2014 ident: 1378_CR18 – volume: 39 start-page: 519 year: 1982 ident: 1378_CR21 publication-title: Math. Comp. doi: 10.1090/S0025-5718-1982-0669644-3 – volume: 22 start-page: 408 year: 1981 ident: 1378_CR3 publication-title: J. Austral. Math. Soc. Ser. B doi: 10.1017/S0334270000002757 – volume: 305 start-page: 121 year: 2005 ident: 1378_CR41 publication-title: J. Math. Anal. Appl. doi: 10.1016/j.jmaa.2004.10.059 – ident: 1378_CR8 – ident: 1378_CR25 – volume: 351 start-page: 3725 year: 1999 ident: 1378_CR4 publication-title: Trans. Amer. Math. Soc. doi: 10.1090/S0002-9947-99-01982-0 – volume: 29 start-page: 878 year: 1998 ident: 1378_CR19 publication-title: SIAM J. Math. Anal. doi: 10.1137/S0036141096306911 – volume: 45 start-page: 269 year: 2007 ident: 1378_CR28 publication-title: Numer. Algorithms doi: 10.1007/s11075-007-9111-5 – volume: 474 start-page: 143 year: 2019 ident: 1378_CR43 publication-title: J. Math. Anal. Appl. doi: 10.1016/j.jmaa.2019.01.036 – volume: 43 start-page: 1343 year: 1995 ident: 1378_CR2 publication-title: IEEE T. Antenn. Propag. doi: 10.1109/8.475113 – volume: 9 start-page: 436 year: 1968 ident: 1378_CR36 publication-title: J. Math. Phys. doi: 10.1063/1.1664597 – volume: 42 start-page: 673 year: 1999 ident: 1378_CR42 publication-title: Science in China Series A: Math. doi: 10.1007/BF02878985 – volume: 10 start-page: 51 year: 1986 ident: 1378_CR30 publication-title: Ann. Sci. Math. Québec – volume-title: Fractals and Spectra year: 1997 ident: 1378_CR40 doi: 10.1007/978-3-0348-0034-1 – volume: 35 start-page: 648 year: 2001 ident: 1378_CR37 publication-title: Aerosol Science & Technology doi: 10.1080/02786820117868 – ident: 1378_CR29 doi: 10.1007/s00023-006-0308-2 – volume: 17 start-page: 137 year: 2009 ident: 1378_CR33 publication-title: Fractals doi: 10.1142/S0218348X09004296 – volume: 4 start-page: 47 year: 2009 ident: 1378_CR23 publication-title: Unif. Distrib. Theory – ident: 1378_CR16 – ident: 1378_CR32 doi: 10.1017/CBO9780511623813 – ident: 1378_CR20 doi: 10.1007/978-1-4899-6886-9_5 – volume: 1 start-page: 1 year: 1994 ident: 1378_CR39 publication-title: J. Fourier Anal. Appl. doi: 10.1007/s00041-001-4001-z – volume: 5 start-page: 69 year: 1989 ident: 1378_CR15 publication-title: Constr. Approx. doi: 10.1007/BF01889599 |
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| Snippet | We present and analyse numerical quadrature rules for evaluating regular and singular integrals on self-similar fractal sets. The integration domain
Γ
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is... We present and analyse numerical quadrature rules for evaluating regular and singular integrals on self-similar fractal sets. The integration domain $$\Gamma... We present and analyse numerical quadrature rules for evaluating regular and singular integrals on self-similar fractal sets. The integration domain Γ⊂Rn is... |
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| SubjectTerms | Acoustic waves Algebra Algorithms Approximation Computer Science Diameters Fractal analysis Fractals Galerkin method Helmholtz equations Integral equations Integrals Numeric Computing Numerical Analysis Original Paper Quadratures Self-similarity Subtraction Theory of Computation Wave scattering |
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