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Methods of numerical integration for rapidly oscillatory integrals

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posted on 09.12.2013, 10:02 by Jonathan R. Webster
This thesis is concerned with the evaluation of rapidly oscillatory integrals, that is integrals in which the integrand has numerous local maxima and minima over the range of integration. Three numerical integration rules are presented. The first is suitable for computing rapidly oscillatory integrals with trigonometric oscillations of the form f(x) exp(irq(x)). The method is demonstrated, empirically, to be convergent and numerically stable as the order of the formula is increased. For other forms of oscillatory behaviour, a second approach based on Lagrange's identity is presented. The technique is suitable for any oscillatory weight function, provided that it satisfies an ordinary linear differential equation of order m :2:: 1. The method is shown to encompass Bessel oscillations, trigonometric oscillations and Fresnel oscillations, and products of these terms. Examples are included which illustrate the efficiency of the method in practical applications. Finally, integrals where the integrand is singular and oscillatory are considered. An extended Clenshaw-Curtis formula is developed for Fourier integrals which exhibit algebraic and logarithmic singularities. An efficient algorithm is presented for the practical implementation of the method.



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© Jonathan Robert Webster

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A Doctoral Thesis. Submitted in partial fulfilment of the requirements for the award of Doctor of Philosophy of Loughborough University.

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