Fast and quasi-fast solvers for weakly singular Fredholm integral equations of the second kind
Date
2020-07-09
Authors
Journal Title
Journal ISSN
Volume Title
Publisher
Abstract
Doktoritöös käsitletakse lineaarsete Fredholmi teist liiki integraalvõrrandite ligikaudse lahendamisega seotud probleeme situatsioonis, kus võrrandi tuum võib argumentide kokkulangemise korral olla iseärane (nõrgalt singulaarne). Tuuma iseärasus toob reeglina kaasa integraalvõrrandi lahendi iseärase käitumise integreerimispiirkonna raja lähedal ning raskused kiirete lahendusmeetodite konstrueerimisel niisuguste võrrandite jaoks. Töö põhitulemuseks on kiirete ja kvaasikiirete meetodite väljatöötamine nimetatud võrrandite korral. Kiire meetod tähendab siin meetodit võrrandi lähislahendite leidmiseks, mis antud ülesannete klassi korral annab lähislahenditele optimaalset järku täpsuse võimalikult väikese aritmeetiliste tehete arvu korral. Vajalikud veahinnangud on saadud lähteülesande periodiseerimise kaudu, mille puhul integraalvõrrandi lähislahendite leidmine taandub perioodiliste funktsioonide aproksimeerimisele trigonomeetriliste polünoomide abil.
In the present thesis the bounds of fast solving Fredholm integral equations of the second kind with a possible weak diagonal singularity of the kernel and certain boundary singularities of the derivatives of the free term has been discussed in a situation when the information about the smooth coefficient functions in the kernel and about the free term is restricted to a given number of their sample values. In a fast solver, the conditions of optimal accuracy and minimal arithmetical operations (complexity of the solver) are met. We mean the order optimality and order minimal work on a class of problems; the class of problems is defined by the smoothness conditions which have been set on the kernel and free term of the underlying problem.
In the present thesis the bounds of fast solving Fredholm integral equations of the second kind with a possible weak diagonal singularity of the kernel and certain boundary singularities of the derivatives of the free term has been discussed in a situation when the information about the smooth coefficient functions in the kernel and about the free term is restricted to a given number of their sample values. In a fast solver, the conditions of optimal accuracy and minimal arithmetical operations (complexity of the solver) are met. We mean the order optimality and order minimal work on a class of problems; the class of problems is defined by the smoothness conditions which have been set on the kernel and free term of the underlying problem.
Description
Keywords
integral equations, integral equations