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

[摘要] ENGLISH ABSTRACT: Some problems of applied mathematics, for instance in the fields ofaerodynamics or electron optics, involve certain singular integralswhich do not exist classically. The problems can, however, be solvedpLovided that such integrals are interpreted as finite-part integrals.Although the concept of a finite-part integral has existed forabout fifty years, it was possible to define it rigorously only by meansof distribution theory, developed about twenty-five years ago. But, tothe 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.twokinds of quadrature formulae for evaluating finite-part integrals in=volving an algebraic singularity.Apart from a historical introduction, the first chapter containssome physical examples of finite-part integrals and their definitionbased on distribution theory. The second chapter treats the most im=portant properties of finite-part integrals; in particular we studytheir behaviour under the most common rules for ordinary integrals.In chapters three and four we derive a quadrature formula for equispacedstations 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 classof orthogonal polynomials. In the fifth and.last chapter we give aderivative-free error bound for the equispaced quadrature formula. Theerror quantities which are independent of the integrand were computedfor the equispaced quadrature formula and are also given. In the caseof some examples, we compare the computed error bounds with the actualerrors.~esides this theoretical investigation df finite-part integrals,we also computed - for several orders of the algebraic singularitythe coefficients for both of the aforesaid quadrature formulae, inwhich the number of stations ranges from three up to twenty. In thecase of the equispaced quadrature fortnu1a,we give the weights and -for int~ger order of the singularity - the coefficients for a numericalderivative of the integrand function. For the Gauss-type quadrature,we give the stations, the corresponding weights and the coefficients ofthe orthogonal polynomials.These data are being published in a separate report [18] whichalso contains detailed instructions on the use of the tables.

[发布机构] Stellenbosch University