Discrete projection methods for Hammerstein integral equations on the half-line
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Discrete projection methods for Hammerstein integral equations on the half‑line Nilofar Nahid1 · Gnaneshwar Nelakanti1 Received: 15 March 2020 / Revised: 25 September 2020 / Accepted: 30 September 2020 © Istituto di Informatica e Telematica (IIT) 2020
Abstract In this paper, we study discrete projection methods for solving the Hammerstein integral equations on the half-line with a smooth kernel using piecewise polynomial basis functions. We show that discrete Galerkin/discrete collocation methods converge to the exact solution with order O(n−min{r,d} ), whereas iterated discrete Galerkin/iterated discrete collocation methods converge to the exact solution with order O(n−min{2r,d} ), where n−1 is the maximum norm of the graded mesh and r denotes the order of the piecewise polynomial employed and d − 1 is the degree of precision of quadrature formula. We also show that iterated discrete multi-Galerkin/iterated discrete multi-collocation methods converge to the exact solution with order O(n−min{4r,d} ) . Hence by choosing sufficiently accurate numerical quadrature rule, we show that the convergence rates in discrete projection and discrete multi-projection methods are preserved. Numerical examples are given to uphold the theoretical results. Keywords Nonlinear integral equations · Piecewise polynomials · Discrete Galerkin method · Discrete collocation method · Discrete multi-Galerkin method · Discrete multi collocation method · Superconvergence results Mathematics Subject Classification 45A05 · 45B05 · 65R20
1 Introduction We shall consider nonlinear Fredholm–Hammerstein integral equations on the halfline ℝ+ ∶= [0, ∞) of the form * Nilofar Nahid [email protected] Gnaneshwar Nelakanti [email protected] 1
Department of Mathematics, Indian Institute of Technology Kharagpur, Kharagpur 721 302, India
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x(s) − (K𝜓)(x)(s) = f (s) s ≥ 0,
(1.1)
where
(K𝜓)(x)(s) ∶=
∫0
∞
k(s, t)𝜓(t, x(t))dt,
(1.2)
is a nonlinear integral operator on X+ , the Banach space of bounded continuous function on ℝ+ with supremum norm. The kernel k(⋅, ⋅) is a smooth function and f , 𝜓 ∈ X+ are known smooth functions and x is the unknown function to be approximated. Finite-section approximations for (1.1) is given by
x𝛽 (s) − (K𝛽 𝜓)(x𝛽 )(s) = f (s), s ≥ 0,
(1.3)
with
(K𝛽 𝜓)(x)(s) ∶=
∫0
𝛽
k(s, t)𝜓(t, x(t))dt, s ∈ ℝ+ ,
(1.4)
where 𝛽 is any positive real number. Numerous application of the nonlinear integral equations of type (1.1) are well documented in [27, 30, 31, 41, 46]. Integral equations of type (1.1) arise in a variety of physical applications. These types of integral equations deal with the diffraction of the wave by a rigid disc and also have an important application on the heat conduction in a semi infinite solid with radiation at the surface. Moreover, these equations are closely related to the global solvability of initial or boundary value problems for ordinary differential equations, such as in the theory of semiconductor devices lead to
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