Some rapidly convergent methods for nonlinear fredholm integral equation¹

The support of the Estonian Science Foundation under grant Nr. 5006 is gratefully acknowledged.

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.

Daug modeliavimo problemu galima suformuluoti netiesines lygties F(x) = 0 pavidalu. Čia F yra Banacho erdves X atvaizdavimas i Banacho erdve Y, turintis visas reikalingas Freshe išvestines. Lygčiai F(x) = 0 spresti taikomas aukštosios eiles 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. Irodyta konvergavimo teorema ir išnagrineti metodu taikymo skaičiavimo aspektai. Aptariamos skaičiavimu lygiagretinimo galimybes, taikant si ulomus metodus netiesinei Fredholmo integralinei lygčiai.

Key words:

• Banach space
• Hilbert space
• Fredholm integral equations
• methods with the high order convergence
• parallel computation

Notes

The support of the Estonian Science Foundation under grant Nr. 5006 is gratefully acknowledged.