## On the numerical evaluation of finite-part integrals involving an algebraic singularity

##### Date

1975-08

##### Authors

Kutt, H. R. (Helmut Richard)

##### Journal Title

##### Journal ISSN

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##### Publisher

Stellenbosch : Stellenbosch University

##### Abstract

ENGLISH ABSTRACT: Some problems of applied mathematics, for instance in the fields of
aerodynamics or electron optics, involve certain singular integrals
which do not exist classically. The problems can, however, be solved
pLovided that such integrals are interpreted as finite-part integrals.
Although the concept of a finite-part integral has existed for
about fifty years, it was possible to define it rigorously only by means
of distribution theory, developed about twenty-five years ago. But, to
the best of our knowledge, no quadrature formula for the numerical eva=
luation of finite-part integrals ha~ been given in the literature.
The main concern of this thesis is the study and discussion of.two
kinds of quadrature formulae for evaluating finite-part integrals in=
volving an algebraic singularity.
Apart from a historical introduction, the first chapter contains
some physical examples of finite-part integrals and their definition
based on distribution theory. The second chapter treats the most im=
portant properties of finite-part integrals; in particular we study
their behaviour under the most common rules for ordinary integrals.
In chapters three and four we derive a quadrature formula for equispaced
stations and one which is optimal in the sense of the Gauss-type quadra=
ture. In connection with the latter formula, we also study a new class
of orthogonal polynomials. In the fifth and.last chapter we give a
derivative-free error bound for the equispaced quadrature formula. The
error quantities which are independent of the integrand were computed
for the equispaced quadrature formula and are also given. In the case
of some examples, we compare the computed error bounds with the actual
errors.
~esides this theoretical investigation df finite-part integrals,
we also computed - for several orders of the algebraic singularity
the coefficients for both of the aforesaid quadrature formulae, in
which the number of stations ranges from three up to twenty. In the
case of the equispaced quadrature fortnu1a,we give the weights and -
for int~ger order of the singularity - the coefficients for a numerical
derivative of the integrand function. For the Gauss-type quadrature,
we give the stations, the corresponding weights and the coefficients of
the orthogonal polynomials.
These data are being published in a separate report [18] which
also contains detailed instructions on the use of the tables.

##### Description

Thesis (PhD)--Stellenbosch University, 1975.

##### Keywords

Integrals, Numerical calculations, Dissertations -- Mathematics, Theses -- Mathematics