Far field asymptotic of the two-dimensional linearised sloping beach problem
SIAM Journal on Applied Mathematics
A high order, progressive method for the evaluation of irregular oscillatory integrals
Applied Numerical Mathematics
Symmetry reduction of Fourier kernels
Journal of Computational Physics
A method to generate generalized quadrature rule for oscillatory integrals
Applied Numerical Mathematics
Multiple quadrature using highly oscillatory quadrature methods
Journal of Computational and Applied Mathematics - Special issue on proceedings of the international symposium on computational mathematics and applications
Evaluating infinite range oscillatory integrals using generalised quadrature methods
Applied Numerical Mathematics
Numerical quadrature for Bessel transformations
Applied Numerical Mathematics
Numerical Quadrature for Bessel Transformations with High Oscillations
Numerical Analysis and Its Applications
Numerical approximations to integrals with a highly oscillatory Bessel kernel
Applied Numerical Mathematics
Efficient evaluation of oscillatory Bessel Hilbert transforms
Journal of Computational and Applied Mathematics
Journal of Computational and Applied Mathematics
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Generalised quadrature methods rely on generating quadrature rules for given irregular oscillatory weight functions w(x) commonly belonging to the class Cn[a, b], for some usually small n. If these weight functions are known to satisfy Lw = 0 for a differential operator L, then Lagrange's identity gLw - wMg = Z'(w, g) can be used to generate a quadrature rule by forcing exactness for a set of basis functions.Theorems which give conditions under which the computed quadrature rules will yield results correct to a required precision (usually that of the machine being employed) underpin the practical rule, and finite range integrals with weights such as sin(q(x)) and Jn(q(x)) have been successfully integrated, for q(x) ∈ C2[a, b]. Doubly oscillatory weights also become feasible with weights such as Jn(q1(x))Jm(q2(x)).More recent work has considered multiple quadratures and the special problems which arise with the commonly occurring infinite range integrations. In the latter case, the direct approach results in violations of the conditions of the underlying theorem and requires some modification for success.This approach has enabled several diverse practical problems to be attempted including integrals from financial market predictions, from chemical reactor analysis, from coherent optical imaging and from wave analysis on sloping beaches.