A fast algorithm for the numerical solution of an integral equation with logarithmic kernel | Zendy

Katharina Flemming | Zendy; Peter Junghanns | Zendy

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A fast algorithm for the numerical solution of an integral equation with logarithmic kernel

Author(s) -

Katharina Flemming,

Peter Junghanns

Publication year - 2014

Publication title -

publications de l institut mathematique

Language(s) - English

Resource type - Journals

SCImago Journal Rank - 0.246

H-Index - 17

eISSN - 1820-7405

pISSN - 0350-1302

DOI - 10.2298/pim1410143f

Subject(s) - mathematics , logarithm , quadrature (astronomy) , integral equation , kernel (algebra) , gaussian quadrature , rate of convergence , collocation (remote sensing) , nyström method , algorithm , sobolev space , convergence (economics) , collocation method , mathematical analysis , discrete mathematics , differential equation , computer science , physics , key (lock) , ordinary differential equation , machine learning , economic growth , optics , economics , computer security

We describe an algorithm for the numerical solution of an integral equation of the form − 1/π ∫1−1 [(y − x) ln |y − x| − h(x, y)] u(y) dy/√1−y2 = f(x), −1 < x < 1, which is based on a collocation-quadrature method and which has the same convergence rate as this method, but only O(n log n) complexity. This integral equation turns out to be an ill-posed problem in (the best possible choice of) a pair of non-periodic Sobolev-like spaces. The present paper presents the technique, how to overcome this peculiarity in the investigation of the fast algorithm.

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