English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Plus annual

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Plus monthly

Global: Zendy Tools annual

Global: Zendy Tools monthly

Global: Zendy Free Plan

We present an efficient integral equation approach to solve the heatequation, $u_t (\x) - \Delta u(\x) = F(\x,t)$, in a two-dimensional, multiplyconnected domain, and with Dirichlet boundary conditions. Instead of usingintegral equations based on the heat kernel, we take the approach ofdiscretizing in time, first. This leads to a non-homogeneous modified Helmholtzequation that is solved at each time step. The solution to this equation isformulated as a volume potential plus a double layer potential.The volumepotential is evaluated using a fast multipole-accelerated solver. The boundaryconditions are then satisfied by solving an integral equation for thehomogeneous modified Helmholtz equation. The integral equation solver is alsoaccelerated by the fast multipole method (FMM). For a total of $N$ points inthe discretization of the boundary and the domain, the total computational costper time step is $O(N)$ or $O(N\log N)$.

Fast integral equation methods for the heat equation and the modified Helmholtz equation in two dimensions