Spectral Approximation of Weakly Singular Integrable Kernels Using Projections

Mario Ahues and Olivier Titaud

Chapter 4 in Integral Methods in Science and Engineering, 2002, pp 27-32 from Springer

Abstract: Abstract As theoretical framework for an integral operator $$ T:X \to X $$ defined by $$x \mapsto Tx:\tau \in \mathcal{I}: = [0,{{\tau }_{0}}] \mapsto (Tx)(\tau ): = \int_{\mathcal{I}} {g(|\tau - \tau \prime )} x(\tau \prime )d\tau \prime \in \mathbb{C},$$ , with a weakly singular kernel g, we consider $$ {\rm X}: = {L^1}(I) $$ and we suppose that (a) $$\mathop{{\lim }}\limits_{{\tau \to {{0}^{ + }}}} g(\tau ) = + \infty ;$$ (b) $$\mathop{{\lim }}\limits_{{\tau \to {{0}^{ + }}}} g(\tau ) = + \infty ;$$ (c) g is a positive decreasing function on ]0, τ0; and (d) $$\mathop{{\sup }}\limits_{{\tau \in \mathcal{I}}} {{\smallint }_{\mathcal{I}}}g(|\tau - \tau \prime |)\tau \prime

Date: 2002
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Persistent link: https://EconPapers.repec.org/RePEc:spr:sprchp:978-1-4612-0111-3_4

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DOI: 10.1007/978-1-4612-0111-3_4

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