Some rapidly convergent methods for nonlinear fredholm integral equation

Abstract

Many problems in modelling can be reduced to the solution of a nonlinear equation F(x) = 0, where F is a Frechet‐differentiable (as many times as necessary) mapping between Banach spaces X and Y. For solving this equation we consider high order iteration methods of the type x_k +1 =x_k ‐ Q(x_k, Aⁱ _k ), i ∈ I, I = {1,…, r}, r ≥ 1, k = 0, 1, …, where Q(x, Aⁱ _k ) is an operator from X into itself and Aⁱ _k, i ∈ I, are some approximations to the inverse operator(s) occurring in the associated exact method. In particular, this set of methods contains methods with successive approximation of the inverse operator(s) and those based on the use of iterative methods to obtain a cheap solution of limited accuracy for corresponding linear equation(s) at each iteration step. A convergence theorem is proved and computational aspects of the methods under consideration are examined. The solution of nonlinear Fredholm integral equation by means of methods with convergence order p ≥ 2 are considered and possibilities of organizing parallel computation in iteration process are also briefly discussed.

Keletas greitojo konvergavimo metodų netiesinei Fredholmo integraliniai lygčiai

Daug modeliavimo problemų galima suformuluoti netiesinės lygties F(x) = 0 pavidalu. Čia F yra Banacho erdvės X atvaizdavimas į Banacho erdvę Y, turintis visas reikalingas Freshe išvestines. Lygčiai F(x) = 0 spręsti taikomas aukštosios eilės iteracinis procesas tokio tipo  x_k + 1 =x_k ‐ Q(x_k, Aⁱ _k ), i ∈ {1,…, r}, k = 0, 1, …,  čia Q(x, Aⁱ _k ) yra tam tikras operatorius X → X, Aⁱ _k , yra atvirkštinio atvaizdavimo aproksimacijos. Įrodyta konvergavimo teorema ir išnagrineti metodų taikymo skaičiavimo aspektai. Aptariamos skaičiavimų lygiagretinimo galimybės, taikant siūlomus metodus netiesinei Fredholmo integralinei lygčiai.

First Published Online: 14 Oct 2010

Keyword : Banach space, Hilbert space, Fredholm integral equations, methods with the high order convergence, parallel computation