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In this paper, such a method is extended to the heat equation with inhomogeneous source terms. First, the heat equation is discretised in time, then in each time step we solve a sequence of so-called modified Helmholtz equations with a parameter depending on the time step size. The modified Helmholtz equation is then split into two: a homogeneous part solved with a boundary integral method and a particular part, where the solution is obtained by evaluating a volume potential over the inhomogeneous source term over a simple domain. In this work, we introduce two components which are critical for the success of this approach: a method to efficiently compute a high-regularity extension of a function outside the domain where it is defined, and a special quadrature method to accurately evaluate singular and nearly singular integrals in the integral formulation of the modified Helmholtz equation for all time step sizes.<\/jats:p>","DOI":"10.1007\/s10444-020-09804-z","type":"journal-article","created":{"date-parts":[[2020,8,20]],"date-time":"2020-08-20T17:02:44Z","timestamp":1597942964000},"update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":10,"title":["An integral equation\u2013based numerical method for the forced heat equation on complex domains"],"prefix":"10.1007","volume":"46","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-0434-2580","authenticated-orcid":false,"given":"Fredrik","family":"Fryklund","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Mary Catherine A.","family":"Kropinski","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Anna-Karin","family":"Tornberg","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"297","published-online":{"date-parts":[[2020,8,20]]},"reference":[{"issue":"9","key":"9804_CR1","doi-asserted-by":"crossref","first-page":"2436","DOI":"10.1016\/j.camwa.2011.02.024","volume":"61","author":"MC Kropinski","year":"2011","unstructured":"Kropinski, MC, Quaife, BD: Fast integral equation methods for Rothe\u2019s method applied to the isotropic heat equation. 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