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

Thesis (PhD)--Stellenbosch University, 1975. / 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.

Identifer | oai:union.ndltd.org:netd.ac.za/oai:union.ndltd.org:sun/oai:scholar.sun.ac.za:10019.1/67934 |

Date | 08 1900 |

Creators | Kutt, H. R. (Helmut Richard) |

Contributors | Goldner, S. R. F., Rosel, R., Stellenbosch University. Faculty of Science. Dept. of Mathematical Sciences. |

Publisher | Stellenbosch : Stellenbosch University |

Source Sets | South African National ETD Portal |

Language | en_ZA |

Detected Language | English |

Type | Thesis |

Format | 169 p. : ill. |

Rights | Stellenbosch University |

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