HIGH ORDER PRODUCT INTEGRATION METHODS FOR A

HIGH ORDER PRODUCT INTEGRATION METHODS FOR A

COMMUNICATIONS ON
PURE AND APPLIED ANALYSIS
Volume 3, Number 2, June 2004
Website: http://AIMsciences.org
pp. 217–235
HIGH ORDER PRODUCT INTEGRATION METHODS FOR
A VOLTERRA INTEGRAL EQUATION WITH
LOGARITHMIC SINGULAR KERNEL
T. Diogo
Centro de Matemática e Aplicações, Instituto Superior Técnico
Av. Rovisco Pais, 1049-001 Lisboa, Portugal
N.B. Franco
Laboratório Interdisciplinar de Computação Cientı́fica
Faculdades COC, 14096-160 Ribeirão Preto - SP, Brazil
P. Lima
Centro de Matemática e Aplicações, Instituto Superior Técnico
Av. Rovisco Pais, 1049-001 Lisboa, Portugal
(Communicated by Tao Tang)
Abstract. This work is concerned with the construction and analysis of high
order product integration methods for a class of Volterra integral equations
with logarithmic singular kernel. Sufficient conditions for the methods to be
convergent are derived and it is shown that optimal convergence orders are
attained if the exact solution is sufficiently smooth. The case of non-smooth
solutions is dealt with by making suitable transformations so that the new
equation possesses smooth solutions. Two particular methods are considered
and their convergence proved. A sample of numerical examples is included.
1. Introduction. We consider the Volterra integral equation of the second kind
t
F (t) +
P (t, s)F (s)ds = H(t), t ∈ [0, T ],
(1.1)
0
where
1
1
1
P (t, s) := √ π ln(t/s) s
(1.2)
and H(t) is a given function. The above equation arises in some heat conduction
problems with mixed-type boundary conditions [1]. As an example, consider
∂2u
1 ∂u
,
= 2
2
∂x
a ∂t
0 ≤ x ≤ l,
(1.3)
2000 Mathematics Subject Classification. 65R20.
Key words and phrases. Volterra integral equations, weakly singular kernel, product integration
methods, convergence.
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