{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,6,18]],"date-time":"2026-06-18T07:43:38Z","timestamp":1781768618450,"version":"3.54.5"},"reference-count":29,"publisher":"Springer Science and Business Media LLC","issue":"1","license":[{"start":{"date-parts":[[2023,12,15]],"date-time":"2023-12-15T00:00:00Z","timestamp":1702598400000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"},{"start":{"date-parts":[[2023,12,15]],"date-time":"2023-12-15T00:00:00Z","timestamp":1702598400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"}],"funder":[{"DOI":"10.13039\/501100000266","name":"Engineering and Physical Sciences Research Council","doi-asserted-by":"publisher","award":["EP\/V053868\/1"],"award-info":[{"award-number":["EP\/V053868\/1"]}],"id":[{"id":"10.13039\/501100000266","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100000266","name":"Engineering and Physical Sciences Research Council,United Kingdom","doi-asserted-by":"crossref","award":["EP\/R014604\/1"],"award-info":[{"award-number":["EP\/R014604\/1"]}],"id":[{"id":"10.13039\/501100000266","id-type":"DOI","asserted-by":"crossref"}]},{"DOI":"10.13039\/501100000608","name":"London Mathematical Society","doi-asserted-by":"publisher","award":["URB-2022-12"],"award-info":[{"award-number":["URB-2022-12"]}],"id":[{"id":"10.13039\/501100000608","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Numer Algor"],"published-print":{"date-parts":[[2024,9]]},"abstract":"<jats:title>Abstract<\/jats:title><jats:p>We consider the numerical evaluation of a class of double integrals with respect to a pair of self-similar measures over a self-similar fractal set (the attractor of an iterated function system), with a weakly singular integrand of logarithmic or algebraic type. In a recent paper (Gibbs et al. Numer. Algorithms <jats:bold>92<\/jats:bold>, 2071\u20132124 2023), it was shown that when the fractal set is \u201cdisjoint\u201d in a certain sense (an example being the Cantor set), the self-similarity of the measures, combined with the homogeneity properties of the integrand, can be exploited to express the singular integral exactly in terms of regular integrals, which can be readily approximated numerically. In this paper, we present a methodology for extending these results to cases where the fractal is non-disjoint but non-overlapping (in the sense that the open set condition holds). Our approach applies to many well-known examples including the Sierpinski triangle, the Vicsek fractal, the Sierpinski carpet, and the Koch snowflake.<\/jats:p>","DOI":"10.1007\/s11075-023-01705-8","type":"journal-article","created":{"date-parts":[[2023,12,15]],"date-time":"2023-12-15T05:02:02Z","timestamp":1702616522000},"page":"311-343","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":4,"title":["Numerical evaluation of singular integrals on non-disjoint self-similar fractal sets"],"prefix":"10.1007","volume":"97","author":[{"given":"A.","family":"Gibbs","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"D. P.","family":"Hewett","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"B.","family":"Major","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"297","published-online":{"date-parts":[[2023,12,15]]},"reference":[{"key":"1705_CR1","doi-asserted-by":"publisher","first-page":"299","DOI":"10.1007\/s13373-013-0041-3","volume":"3","author":"M Barnsley","year":"2013","unstructured":"Barnsley, M., Vince, A.: Developments in fractal geometry. Bull. Math. Sci. 3, 299\u2013348 (2013)","journal-title":"Bull. Math. Sci."},{"key":"1705_CR2","first-page":"243","volume":"399","author":"MF Barnsley","year":"1985","unstructured":"Barnsley, M.F., Demko, S.: Iterated function systems and the global construction of fractals. Proc. Roy. Soc. A. Math. Phys. Sci. 399, 243\u2013275 (1985)","journal-title":"Proc. Roy. Soc. A. Math. Phys. Sci."},{"key":"1705_CR3","doi-asserted-by":"publisher","first-page":"920","DOI":"10.1103\/PhysRevA.36.920","volume":"36","author":"D Bessis","year":"1987","unstructured":"Bessis, D., Fournier, J., Servizi, G., Turchetti, G., Vaienti, S.: Mellin transforms of correlation integrals and generalized dimension of strange sets. Phys. Rev. A 36, 920 (1987)","journal-title":"Phys. Rev. A"},{"key":"1705_CR4","doi-asserted-by":"crossref","unstructured":"Bogachev, V.I.: Measure theory (Volume 1). Springer (2007)","DOI":"10.1007\/978-3-540-34514-5"},{"key":"1705_CR5","doi-asserted-by":"publisher","first-page":"75","DOI":"10.1007\/s00607-004-0076-0","volume":"74","author":"S B\u00f6rm","year":"2005","unstructured":"B\u00f6rm, S., Hackbusch, W.: Hierarchical quadrature for singular integrals. Computing 74, 75\u2013100 (2005)","journal-title":"Computing"},{"key":"1705_CR6","doi-asserted-by":"crossref","unstructured":"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. arXiv:2212.06594 (2022)","DOI":"10.1007\/s00211-021-01182-y"},{"key":"1705_CR7","unstructured":"Caetano, A.M., Chandler-Wilde, S.N., Gibbs, A., Hewett, D.P.: Properties of IFS attractors with non-empty interiors and associated function spaces and scattering problems. In preparation"},{"key":"1705_CR8","doi-asserted-by":"publisher","first-page":"120","DOI":"10.1016\/j.cam.2008.10.022","volume":"229","author":"F Calabr\u00f2","year":"2009","unstructured":"Calabr\u00f2, F., Corbo Esposito, A.: An evaluation of Clenshaw-Curtis quadrature rule for integration w.r.t. singular measures. J. Comput. Appl. Math. 229, 120\u2013128 (2009)","journal-title":"J. Comput. Appl. Math."},{"key":"1705_CR9","unstructured":"Falconer, K.: Fractal geometry: Mathematical foundations and applications. Wiley, 3rd ed. (2014)"},{"key":"1705_CR10","doi-asserted-by":"publisher","first-page":"878","DOI":"10.1137\/S0036141096306911","volume":"29","author":"B Forte","year":"1998","unstructured":"Forte, B., Mendivil, F., Vrscay, E.: Chaos games for iterated function systems with grey level maps. SIAM J. Math. Anal. 29, 878\u2013890 (1998)","journal-title":"SIAM J. Math. Anal."},{"key":"1705_CR11","doi-asserted-by":"crossref","unstructured":"Gautschi, W.: Computational aspects of orthogonal polynomials. In: Nevai, P. (ed.) Orthogonal polynomials: Theory and Practice, NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci., Kluwer Acad. Publ., Dordrecht, pp.\u00a0181\u2013216 (1990)","DOI":"10.1007\/978-94-009-0501-6_9"},{"key":"1705_CR12","doi-asserted-by":"crossref","unstructured":"Gautschi, W.: Orthogonal polynomials: computation and approximation. OUP, (2004)","DOI":"10.1093\/oso\/9780198506720.001.0001"},{"key":"1705_CR13","doi-asserted-by":"publisher","first-page":"2071","DOI":"10.1007\/s11075-022-01378-9","volume":"92","author":"A Gibbs","year":"2023","unstructured":"Gibbs, A., Hewett, D., Moiola, A.: Numerical quadrature for singular integrals on fractals. Numer. Algorithms 92, 2071\u20132124 (2023)","journal-title":"Numer. Algorithms"},{"key":"1705_CR14","doi-asserted-by":"publisher","first-page":"A652","DOI":"10.1137\/120889873","volume":"35","author":"N Hale","year":"2013","unstructured":"Hale, N., Townsend, A.: Fast and accurate computation of Gauss-Legendre and Gauss-Jacobi quadrature nodes and weights. SIAM J. Sci. Comput. 35, A652\u2013A674 (2013)","journal-title":"SIAM J. Sci. Comput."},{"key":"1705_CR15","doi-asserted-by":"publisher","first-page":"713","DOI":"10.1512\/iumj.1981.30.30055","volume":"30","author":"JE Hutchinson","year":"1981","unstructured":"Hutchinson, J.E.: Fractals and self-similarity. Indiana Univ. Math. J. 30, 713\u2013747 (1981)","journal-title":"Indiana Univ. Math. J."},{"key":"1705_CR16","doi-asserted-by":"crossref","unstructured":"Kunze, H., La\u00a0Torre, D., Mendivil, F., Vrscay, E.R.: Fractal-based methods in analysis. Springer, (2011)","DOI":"10.1007\/978-1-4614-1891-7"},{"key":"1705_CR17","doi-asserted-by":"publisher","first-page":"509","DOI":"10.1007\/BF02437506","volume":"12","author":"G Mantica","year":"1996","unstructured":"Mantica, G.: A stable Stieltjes technique for computing orthogonal polynomials and Jacobi matrices associated with a class of singular measures. Constr. Approx. 12, 509\u2013530 (1996)","journal-title":"Constr. Approx."},{"key":"1705_CR18","doi-asserted-by":"crossref","unstructured":"Mantica, G.: On computing Jacobi matrices associated with recurrent and M\u00f6bius iterated function systems. In: Proceedings of the 8th International Congress on Computational and Applied Mathematics, ICCAM-98 (Leuven), vol.\u00a0115(1-2), pp.\u00a0419\u2013431 (2000)","DOI":"10.1016\/S0377-0427(99)00188-0"},{"key":"1705_CR19","doi-asserted-by":"publisher","first-page":"265","DOI":"10.1007\/s00023-006-0308-2","volume":"8","author":"G Mantica","year":"2007","unstructured":"Mantica, G., Vaienti, S.: The asymptotic behaviour of the Fourier transforms of orthogonal polynomials I: Mellin transform techniques. Ann. Henri Poincar\u00e9 8, 265\u2013300 (2007)","journal-title":"Ann. Henri Poincar\u00e9"},{"key":"1705_CR20","doi-asserted-by":"crossref","unstructured":"Mattila, P.: Fourier analysis and Hausdorff dimension. CUP, (2015)","DOI":"10.1017\/CBO9781316227619"},{"key":"1705_CR21","doi-asserted-by":"publisher","first-page":"91","DOI":"10.1515\/jnum.2010.004","volume":"18","author":"P Meszmer","year":"2010","unstructured":"Meszmer, P.: Hierarchical quadrature for multidimensional singular integrals. J. Numer. Math. 18, 91\u2013117 (2010)","journal-title":"J. Numer. Math."},{"key":"1705_CR22","doi-asserted-by":"publisher","first-page":"33","DOI":"10.1515\/jnum-2014-0002","volume":"22","author":"P Meszmer","year":"2014","unstructured":"Meszmer, P.: Hierarchical quadrature for multidimensional singular integrals - part II. J. Numer. Math. 22, 33\u201360 (2014)","journal-title":"J. Numer. Math."},{"key":"1705_CR23","doi-asserted-by":"publisher","first-page":"2297","DOI":"10.1090\/S0002-9947-98-02218-1","volume":"350","author":"M Mor\u00e1n","year":"1998","unstructured":"Mor\u00e1n, M., Rey, J.-M.: Singularity of self-similar measures with respect to Hausdorff measures. T. Am. Math. Soc. 350, 2297\u20132310 (1998)","journal-title":"T. Am. Math. Soc."},{"key":"1705_CR24","doi-asserted-by":"crossref","unstructured":"Strichartz, R.S.: Self-similar measures and their Fourier transforms I. Indiana U. Math. J. 797\u2013817 (1990)","DOI":"10.1512\/iumj.1990.39.39038"},{"key":"1705_CR25","doi-asserted-by":"publisher","first-page":"316","DOI":"10.1080\/00029890.2000.12005199","volume":"107","author":"RS Strichartz","year":"2000","unstructured":"Strichartz, R.S.: Evaluating integrals using self-similarity. Am. Math. Mon. 107, 316\u2013326 (2000)","journal-title":"Am. Math. Mon."},{"key":"1705_CR26","first-page":"337","volume":"36","author":"A Townsend","year":"2016","unstructured":"Townsend, A., Trogdon, T., Olver, S.: Fast computation of Gauss quadrature nodes and weights on the whole real line. IMA J. Numer. Anal. 36, 337\u2013358 (2016)","journal-title":"IMA J. Numer. Anal."},{"key":"1705_CR27","doi-asserted-by":"publisher","first-page":"67","DOI":"10.1137\/060659831","volume":"50","author":"LN Trefethen","year":"2008","unstructured":"Trefethen, L.N.: Is Gauss quadrature better than Clenshaw-Curtis? SIAM Rev. 50, 67\u201387 (2008)","journal-title":"SIAM Rev."},{"key":"1705_CR28","doi-asserted-by":"crossref","unstructured":"Trefethen, L.N.: Ten digit problems. In: Schleicher, D., Lackmann, M. (eds.) An invitation to mathematics: from competitions to research, pp.\u00a0119\u2013136. Springer, (2011)","DOI":"10.1007\/978-3-642-19533-4_9"},{"key":"1705_CR29","unstructured":"Trefethen, L.N.: Approximation theory and approximation practice. SIAM (2013)"}],"container-title":["Numerical Algorithms"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s11075-023-01705-8.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s11075-023-01705-8\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s11075-023-01705-8.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2024,8,6]],"date-time":"2024-08-06T09:25:16Z","timestamp":1722936316000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s11075-023-01705-8"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2023,12,15]]},"references-count":29,"journal-issue":{"issue":"1","published-print":{"date-parts":[[2024,9]]}},"alternative-id":["1705"],"URL":"https:\/\/doi.org\/10.1007\/s11075-023-01705-8","relation":{},"ISSN":["1017-1398","1572-9265"],"issn-type":[{"value":"1017-1398","type":"print"},{"value":"1572-9265","type":"electronic"}],"subject":[],"published":{"date-parts":[[2023,12,15]]},"assertion":[{"value":"23 March 2023","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"10 November 2023","order":2,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"15 December 2023","order":3,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}},{"order":1,"name":"Ethics","group":{"name":"EthicsHeading","label":"Declarations"}},{"value":"Not applicable","order":2,"name":"Ethics","group":{"name":"EthicsHeading","label":"Ethical approval"}},{"value":"The authors declare no competing interests.","order":3,"name":"Ethics","group":{"name":"EthicsHeading","label":"Conflict of interest"}}]}}