On the regularization method for Fredholm integral equations with odd weakly singular kernel
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On the regularization method for Fredholm integral equations with odd weakly singular kernel Noureddine Benrabia1 · Hamza Guebbai2
Received: 27 January 2018 / Revised: 2 April 2018 / Accepted: 12 April 2018 © SBMAC - Sociedade Brasileira de Matemática Aplicada e Computacional 2018
Abstract In this paper, we propose a numerical method to approach the solution of a Fredholm integral equation with a weakly singular kernel by applying the convolution product as a regularization operator and the Fourier series as a projection. Preliminary numerical results show that the order of convergence of the method is better than the one of conventional projection methods. Keywords Integral equations · Weak singularity · Convolution · Fourier series Mathematics Subject Classification 45B05 · 45E99 · 65R20 · 65T40
1 Introduction The theory of integral equations has been an active search domain for many years and is based on analysis, function theory and functional analysis. On the other hand, integral equations are of practical interest because of the approximation theory of Fredholm integral equations, which allows the application of the notions of operator theory, spectral theory and especially those of projection and functional approximation (Debbar et al. 2016; Guebbai and Grammont 2014; Ahues et al. 2009; Amosov and Youssef 2016; Lemita and Guebbai 2017).
Communicated by Delfim F. M. Torres.
B
Hamza Guebbai [email protected] Noureddine Benrabia [email protected]
1
Département de Mathématiques, Université Mohamed-Chérif Messaadia Souk Ahras, B.P. 1553, 41000 Souk Ahras, Algeria
2
Département de Mathématiques, Université 8 Mai 1945 Guelma, B.P. 401, 24000 Guelma, Algeria
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N. Benrabia, H. Guebbai
The approximation of Fredholm weakly singular equations of the following form: For all f ∈ L 1 (0, 1), find u ∈ L 1 (0, 1), such that 1 ∀s ∈ [0, 1] , u(s) = g (|s − t|) u(t)dt + f (s), 0
where, g, the source of the weak singularity, is supposed to be in L 1 ([0, 1] , R), positive, decreasing and such that lim g (t) = +∞,
t→0+
knows an impassable error given by: ∃C1 , C2 > 0, for n ∈ N large enough, u − u n L 1 (0,1) ≤ C1
1 n
g (t) dt ≤ C2
0
γ 1 , n
where γ depends on g and belongs to (0, 1) for some interesting cases like Abel’s kernel. Recently, Debbar et al, have improved this bound, by adding rather acceptable conditions on the solution u, to obtain a convergence order strictly greater than γ and at most equal to 1. In this paper, we are interested in the same integral equation in the case of an odd kernel i.e. g verifies (1) g ∈ L 1 ([−1, 1] , R), (2) g(−t) = −g(t), t ∈ [−1, 1] , (3) g(t) ≥ 0, t ∈ [0, 1] , (4) g is decreasing over [0, 1] , (5) lim g(t) = +∞. t→0+
This version was studied numerically in Guebbai and Grammont (2014) and the authors obtained a convergence order equal to γ . Our goal is to show that the method developed in Debbar et al. (2016) is applicable in this case and that we can improve the order of convergence obtained before.
2 Setting the problem and studying its
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