Singularity Subtraction for Nonlinear Weakly Singular Integral Equations of the Second Kind

Mario Ahues,Paulo B Vasconcelos,Filomena D D’Almeida,Rosário Fernandes

doi:10.1007/978-3-030-16077-7_1

Mario Ahues, Paulo B Vasconcelos + Show 2 more

Open Access

https://doi.org/10.1007/978-3-030-16077-7_1

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Publication Date: Jan 1, 2019
Citations: 1	License type: cc-by

Affiliation: University of Porto, University of Minho

Abstract

The singularity subtraction technique for computing an approximate solution of a linear weakly singular Fredholm integral equation of the second kind is generalized to the case of a nonlinear integral equation of the same kind. Convergence of the sequence of approximate solutions to the exact one is proved under mild standard hypotheses on the nonlinear factor, and on the sequence of quadrature rules used to build an approximate equation. A numerical example is provided with a Hammerstein operator to illustrate some practical aspects of effective computations.

Highlights

To be consistent with [An81] and [AhEtAl01], we assume that g is a decreasing function on ]0, b − a]
We have extended to nonlinear integral operators, the singularity subtraction technique for approching linear weakly singular integral operators in the framework of real valued continuous functions
The singularity subtraction technique cannot be settled in Lebesgue spaces

Summary

Introduction

The reference Banach space is the set X := C0([a, b], R) with the supremum norm. We consider the operator K defined by b. When N(s,t, x(t)) := κ(s,t) x(t) for some continuous function κ : [a, b]×[a, b] → R, K is a linear bounded operator from X into itself. The singularity subtraction method builds an approximation of K as it is written in (1.1), and, as described in [An81] for the linear case, it is a double approximation scheme consisting of truncation and numerical integration. In the sequence of singularity subtraction approximations, the role of δ is played by a sequence (an)n≥2 in ]0, b − a[ leading to the function gn defined by gn(s) := g(an) for s ∈ [0, an], g(s) for s ∈ ]an, b − a]. Numerical integration: To proceed with the singularity subtraction idea – like in the linear case – we define a general grid with n ≥ 2 points on [a, b]:. This grid is called the basic grid, and it determines n − 1 subintervals of [a, b]

Singularity Subtraction for Nonlinear Weakly Singular Integral Equations

Singularity subtraction

Convergence

Numerics

Conclusions