Integration method for oscillatory integrals
In numerical analysis, Filon quadrature or Filon's method is a technique for numerical integration of oscillatory integrals. It is named after English mathematician Louis Napoleon George Filon, who first described the method in 1934.
Description.
The method is applied to oscillatory definite integrals in the form:
formula_1 
where formula_2 is a relatively slowly-varying function and formula_3 is either sine or cosine or a complex exponential that causes the rapid oscillation of the integrand, particularly for high frequencies. In Filon quadrature, the formula_2 is divided into formula_5 subintervals of length formula_6, which are then interpolated by parabolas. Since each subinterval is now converted into a Fourier integral of quadratic polynomials, these can be evaluated in closed-form by integration by parts. For the case of formula_7, the integration formula is given as:
formula_8
where 
formula_9
formula_10
formula_11
formula_12
formula_13
formula_14
Explicit Filon integration formulas for sine and complex exponential functions can be derived similarly. The formulas above fail for small formula_15 values due to catastrophic cancellation; Taylor series approximations must be in such cases to mitigate numerical errors, with formula_16 being recommended as a possible switchover point for 44-bit mantissa.
Modifications, extensions and generalizations of Filon quadrature have been reported in numerical analysis and applied mathematics literature; these are known as Filon-type integration methods. These include Filon-trapezoidal and Filon–Clenshaw–Curtis methods.
Applications.
Filon quadrature is widely used in physics and engineering for robust computation of Fourier-type integrals. Applications include evaluation of oscillatory Sommerfeld integrals for electromagnetic and seismic problems in layered media and numerical solution to steady incompressible flow problems in fluid mechanics, as well as various different problems in neutron scattering, quantum mechanics and metallurgy.
References.
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