1. Error analysis of reiterated projection methods for Hammerstein integral equations.
- Author
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Chakraborty, Samiran, Kant, Kapil, and Nelakanti, Gnaneshwar
- Subjects
INTEGRAL equations ,HAMMERSTEIN equations ,NONLINEAR integral equations ,KERNEL (Mathematics) ,NONLINEAR operators ,SINGULAR integrals ,NEWTON-Raphson method - Abstract
In this paper, we consider the Newton-iteration scheme based on Galerkin and multi-Galerkin operators to solve non-linear integral equations of the Fredholm-Hammerstein type for both smooth and weakly singular algebraic and logarithmic type kernels. By choosing X n as space of piecewise polynomials subspace of degree ≤ r − 1 , we show that for a smooth kernel, Galerkin and multi-Galerkin approximate solution in the kth Newton-iteration scheme converges with the orders O (h (k + 1) r) and O (h (2 k + 3) r) , respectively, where h is the norm of the partition. For weakly singular kernels, we show that the Galerkin approximate solution in the kth Newton-iteration scheme converges with the orders O (h r + k (1 − α)) , 0 < α < 1 for algebraic kernels and O (h r + k (log h) k) for logarithmic kernels. Also the multi-Galerkin approximate solution in the kth Newton-iteration scheme converges with the orders O (h r + (k + 1) (1 − α)) for algebraic kernels and O (h r + k + 1 (log h) k + 1) for logarithmic kernels. Numerical results are given for the justification of our theoretical results. [ABSTRACT FROM AUTHOR]
- Published
- 2022
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