[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-85459-en":3,"doc-seo-85459-105":29,"detail-sidebar-cat-0-en-105":90},{"code":4,"msg":5,"data":6},0,"success",{"doc_id":7,"user_id":8,"nickname":9,"user_avatar":10,"doc_module":4,"category_id":11,"category_name":12,"doc_title":13,"doc_description":14,"doc_content":15,"file_id":16,"file_url":17,"file_type":18,"file_size":19,"view_count":4,"is_deleted":4,"is_public":20,"is_downloadable":20,"audit_status":20,"page_count":21,"language":22,"language_code":23,"site_id":24,"html_lang":23,"table_of_contents":25,"faqs":26,"seo_title":13,"seo_description":14,"update_tm":27,"read_time":28},85459,549758146520,"Patrick","https://ap-avatar.wpscdn.com/avatar/80002397d8c0411e94?_k=1775819394049821470",8,"Research & Report","A Regularized B-Spline–Heaviside Collocation Method for Cauchy Singular Integral Equations with Piecewise Hölder Solutions","A B-spline–Heaviside collocation method is developed for Cauchy singular integral equations on smooth closed C2 contours when the exact solution is piecewise Hölder continuous with finitely many prescribed jumps. The approach addresses logarithmic terms caused by discontinuities by mapping Cauchy operators into a logarithmically enlarged piecewise Hölder space with lateral Hölder logarithmic coefficients. Discontinuities are represented via a nonredundant normalized relative Heaviside basis adapted to the contour, with collocation using spline nodes separated from the jump set and logarithmic-coefficient functionals at jumps. Under stability and consistency assumptions, existence, uniqueness, and an error estimate are proven, supported by matrix formulation and numerical experiments.","arXiv :2510 .24984v3 [math .NA] 12 Jul 2026  \nA REGULARIZED B-SPLINE–HEAVISIDE COLLOCATION METHOD FOR CAUCHY SINGULAR INTEGRAL EQUATIONS WITH PIECEWISE  \nHÖLDER SOLUTIONS  \nMARIA CAPCELEA AND TITU CAPCELEA  \nAbstract. We develop a B-spline–Heaviside collocation method for Cauchy singular integral equations on a smooth closed C2 contour when the exact solution is piecewise Hölder continuous with finitely many prescribed jumps. Since the Cauchy singular integral of a discontinuous function generally has logarithmic terms at the jump points, we study  \nM = cI + dS + K : Xα → Yα , Xα = PH α (Γ , D), Yα = PHlαog , ∗ (Γ , D),  \nwhere Yα is a logarithmically enlarged piecewise Hölder space with lateral Hölder logarithmic coefficients. The discontinuous component is represented by a nonredundant system of normalized relative Heaviside functions adapted to the closed contour. Collocation uses point evaluations at spline nodes separated from the jump set together with logarithmic-coefficient functionals at the jumps. Assuming continuous stability of M : Xβ → Yβ , mesh-uniform scaled discrete stability of the regularized collocation operators, and a scaled consistency estimate for exact-jump approximants, we prove existence and uniqueness for sufficiently fine meshes and the error bound  \nkϕ − ϕHnB kX β ≤ ChαB−β kϕkXα , 0 \u003C β \u003C α \u003C 1.  \nWe give a matrix realization of the regularized scheme, including the logarithmic and pointcollocation blocks, singularity-subtracted evaluation of the Cauchy action on splines, principalvalue-safe arc formulas for the Heaviside terms, and an implementation algorithm. An abstract perturbation result shows that the same rate is preserved under sufficiently accurate quadrature.  \nNumerical experiments with arcwise errors and finite-dimensional stability and consistency indicators support the theoretical assumptions over the tested meshes.  \n1. Introduction  \nCauchy singular integral equations on closed contours form a traditional class of equations in complex analysis, boundary integral methods, and mathematical physics. They arise in boundary formulations of elliptic problems, in interface and transmission models, in crack problems, in plane elasticity, in potential theory, and in many models where a boundary condition changes from one part of the contour to another. For equations with smooth coefficients and smooth right-hand sides, polynomial, trigonometric, spline, Nyström, and Galerkin discretizations are well understood and can provide stable and high-order approximations; see, for example, the classical and numerical references [1, 2, 3, 4, 5] .  \nThe numerical situation changes substantially when the coefficients, the data, or the solution are only piecewise regular. In many applications the boundary is divided into finitely many arcs, and the physical model prescribes different coefficients or different boundary conditions on different arcs. The resulting unknown may be Hölder continuous on each arc but may have finite jumps at the interface points. Spline collocation and Galerkin methods for singular integral equations with piecewise continuous coefficients have been studied in related settings in [6, 7] . Smooth global approximation spaces are not structurally adapted to such functions. If a discontinuity is approximated indirectly by polynomials, trigonometric polynomials, or globally continuous splines, spurious oscillations and a loss of strong convergence near the jump points are typical. A natural remedy is to build the discontinuity structure into the trial space itself.  \n2020 Mathematics Subject Classification. Primary 65R20, 45E05; Secondary 65N35, 41A15, 41A25 .  \nKey words and phrases. Cauchy singular integral equation, B-spline collocation, Heaviside enrichment, piecewise Hölder solution, logarithmic singularity, closed contour.  \n2 MARIA CAPCELEA AND TITU CAPCELEA  \nThe approximation space used in this paper combines periodic B-splines with finitely many step functions. The B-spline part approximates the cont","cbCainGDDm3sPdfS","https://ap.wps.com/l/cbCainGDDm3sPdfS","pdf",1160244,1,35,"English","en",105,"# Introduction\n## Piecewise regularity and jump-induced singularities\n## Enriched trial spaces and logarithmically enlarged targets","[{\"question\":\"How does the method handle discontinuities in piecewise Hölder solutions?\",\"answer\":\"Discontinuities are built into the approximation space using finitely many step/Heaviside functions, while B-splines approximate the continuous Hölder component. The Heaviside basis is adapted to the known jump set to explicitly represent jumps.\"},{\"question\":\"Why is a logarithmically enlarged function space required?\",\"answer\":\"For piecewise Hölder solutions with nonzero jumps, the Cauchy singular integral typically generates logarithmic singularities at the jump points. Therefore the operator naturally maps into a space enlarged to include endpoint logarithmic terms with lateral Hölder logarithmic coefficients.\"},{\"question\":\"What kind of convergence guarantee is proved?\",\"answer\":\"Assuming continuous stability, mesh-uniform scaled discrete stability, and a scaled consistency estimate for exact-jump approximants, the paper proves existence and uniqueness for sufficiently fine meshes and establishes an error bound of the form ||φ−φ_{HnB}||_{Xβ} ≤ C h^{α−β} ||φ||_{Xα} for 0\\u003cβ\\u003cα\\u003c1.\"}]",1784203706,88,{"code":4,"msg":30,"data":31},"ok",{"site_id":24,"language":23,"slug":32,"title":13,"keywords":33,"description":14,"schema_data":34,"social_meta":85,"head_meta":87,"extra_data":89,"updated_unix":27},"a-regularized-b-splineheaviside-collocation-method-for-cauchy-singular-integral-equations-with-piecewise-holder-solutions","",{"@graph":35,"@context":84},[36,53,67],{"@type":37,"itemListElement":38},"BreadcrumbList",[39,43,47,50],{"item":40,"name":41,"@type":42,"position":20},"https://docshare.wps.com","Home","ListItem",{"item":44,"name":45,"@type":42,"position":46},"https://docshare.wps.com/document/","Document",2,{"item":48,"name":12,"@type":42,"position":49},"https://docshare.wps.com/document/research-report/",3,{"item":51,"name":13,"@type":42,"position":52},"https://docshare.wps.com/document/a-regularized-b-splineheaviside-collocation-method-for-cauchy-singular-integral-equations-with-piecewise-holder-solutions/85459/",4,{"url":51,"name":13,"@type":54,"author":55,"headline":13,"publisher":57,"fileFormat":60,"inLanguage":23,"description":14,"dateModified":61,"datePublished":61,"encodingFormat":60,"isAccessibleForFree":62,"interactionStatistic":63},"DigitalDocument",{"name":9,"@type":56},"Person",{"url":40,"name":58,"@type":59},"DocShare","Organization","application/pdf","2026-07-16",true,{"@type":64,"interactionType":65,"userInteractionCount":4},"InteractionCounter",{"@type":66},"ViewAction",{"@type":68,"mainEntity":69},"FAQPage",[70,76,80],{"name":71,"@type":72,"acceptedAnswer":73},"How does the method handle discontinuities in piecewise Hölder solutions?","Question",{"text":74,"@type":75},"Discontinuities are built into the approximation space using finitely many step/Heaviside functions, while B-splines approximate the continuous Hölder component. The Heaviside basis is adapted to the known jump set to explicitly represent jumps.","Answer",{"name":77,"@type":72,"acceptedAnswer":78},"Why is a logarithmically enlarged function space required?",{"text":79,"@type":75},"For piecewise Hölder solutions with nonzero jumps, the Cauchy singular integral typically generates logarithmic singularities at the jump points. Therefore the operator naturally maps into a space enlarged to include endpoint logarithmic terms with lateral Hölder logarithmic coefficients.",{"name":81,"@type":72,"acceptedAnswer":82},"What kind of convergence guarantee is proved?",{"text":83,"@type":75},"Assuming continuous stability, mesh-uniform scaled discrete stability, and a scaled consistency estimate for exact-jump approximants, the paper proves existence and uniqueness for sufficiently fine meshes and establishes an error bound of the form ||φ−φ_{HnB}||_{Xβ} ≤ C h^{α−β} ||φ||_{Xα} for 0\u003Cβ\u003Cα\u003C1.","https://schema.org",{"og:url":51,"og:type":86,"og:title":13,"og:site_name":58,"og:description":14},"article",{"robots":88,"canonical":51},"index,follow",{"doc_id":7,"site_id":24},{"code":4,"msg":5,"data":91},[92,96,100,104,109,114,119,122,127,130,134],{"id":20,"doc_module":4,"doc_module_name":45,"category_name":93,"show_sort_weight":94,"slug":95},"Story & Novel",90,"story-novel",{"id":46,"doc_module":4,"doc_module_name":45,"category_name":97,"show_sort_weight":98,"slug":99},"Literature",80,"literature",{"id":52,"doc_module":4,"doc_module_name":45,"category_name":101,"show_sort_weight":102,"slug":103},"Exam",70,"exam",{"id":105,"doc_module":4,"doc_module_name":45,"category_name":106,"show_sort_weight":107,"slug":108},5,"Comic",60,"comic",{"id":110,"doc_module":4,"doc_module_name":45,"category_name":111,"show_sort_weight":112,"slug":113},6,"Technology",50,"technology",{"id":115,"doc_module":4,"doc_module_name":45,"category_name":116,"show_sort_weight":117,"slug":118},7,"Healthcare",40,"healthcare",{"id":11,"doc_module":4,"doc_module_name":45,"category_name":12,"show_sort_weight":120,"slug":121},30,"research-report",{"id":123,"doc_module":4,"doc_module_name":45,"category_name":124,"show_sort_weight":125,"slug":126},9,"Religion & Spirituality",20,"religion-spirituality",{"id":125,"doc_module":4,"doc_module_name":45,"category_name":128,"show_sort_weight":125,"slug":129},"World Cup","world-cup",{"id":131,"doc_module":4,"doc_module_name":45,"category_name":132,"show_sort_weight":131,"slug":133},10,"Lifestyle","lifestyle",{"id":135,"doc_module":4,"doc_module_name":45,"category_name":136,"show_sort_weight":105,"slug":137},19,"General","general"]