A new stable numerical method for Mellin integral equations in weighted spaces with uniform norm
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A new stable numerical method for Mellin integral equations in weighted spaces with uniform norm Concetta Laurita1 Received: 27 January 2020 / Revised: 23 July 2020 / Accepted: 25 July 2020 © Istituto di Informatica e Telematica (IIT) 2020
Abstract In this paper a new modified Nyström method is proposed to solve linear integral equations of the second kind with fixed singularities of Mellin convolution type. It is based on the Gauss–Radau quadrature formula with a suitable Jacobi weight. The stability and convergence of the method is proved in weighted spaces with uniform norm. Moreover, an error estimate of the numerical solution is given under certain assumptions on the Mellin kernel. The efficiency of the method is shown through some examples. The numerical results also confirm that the error estimate is sharp. Keywords Mellin kernel · Integral equations of Mellin type · Nyström method · Gauss–Radau rule Mathematics Subject Classification 65R20 · 45E99
1 Introduction The aim of the present paper is to propose a numerical method for solving singular integral equations having the following form
f (y) +
∫0
1
k(x, y)f (x)dx +
∫0
1
h(x, y)f (x)dx = g(y),
(1)
y ∈ (0, 1],
where k(x, y) is continuous for all x, y ∈ [0, 1] such that x + y > 0 and has a fixed singularities at the origin of Mellin-type, i.e. ( ) 1 y k(x, y) = k̄ (2) x x
* Concetta Laurita [email protected] 1
Department of Mathematics, Computer Science and Economics, University of Basilicata, Via dell’Ateneo Lucano 10, 85100 Potenza, Italy
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for a given function k̄ on [0, +∞) satisfying certain assumptions, while the kernel h(x, y) and the right-hand side g(y) are known functions on [0, 1] × [0, 1] and [0, 1], respectively, satisfying suitable conditions which will be specified later. The function f(y) represents the unknown. Such class of equations often occurs in the solution of PDEs on nonsmooth domains by boundary integral equation methods. For instance, the representation in the form of a single layer potential of the solution of the exterior Neumann problem for the Laplace equation in planar domains with corners leads to solve an integral equation of type (1), with the Mellin kernel defined by
sin 𝛼 ̄ = 1 , k(t) 𝜋 1 − 2t cos 𝛼 + t2 being 𝛼 the interior angle at the corner point. In what follows we will assume that the Mellin kernel defined in (2) fulfills the following condition
∫0
∞
̄ t−1+𝜎 |k(t)|dt < ∞,
(3)
for some 0 < 𝜎 < 1. In the recent literature several papers were dedicated to the numerical treatment of integral equations of type (1) (see [2–8, 10, 11, 13, 15] and the references therein). The presence of a fixed singularity in the Mellin kernel makes the stability proof for the numerical methods a rather delicate issue, being the noncompactness of the corresponding integral operator the main theoretical difficulty. Moreover, under the assumption (3), the solution f(y) of equation (1) is singular at the point y = 0 . To overcome these drawbacks, suitable m
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