Calculation of Highly Oscillatory Integrals by Quadrature Method

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1 Calculation of Highly Oscillatory Integrals by Quadrature Methods Krishna Thapa Texas A&M University, College Station, TX May 18, 211

2 Outline Purpose Why Filon Type Quadrature Methods Our Expectations Strategy Questions and/or Suggestions References

3 Purpose In the model of steeply rising potential, λz α, Fulling et al. came across the following oscillatory integral: T (z) = 1 π 3 dρ 1 du 1 u 2 cos(2zρu 2δ(ρu)) (1) Todd Zapattas thesis on WKB approximation for linear potentials also has highly oscillatory integrals that are difficult to calculate numerically. Airy functions with higher frequencies that are much more difficult to approximate numerically arise in both cases. Ψ(x, t) = e i Et ψ(y)u(x, E)u(y, E)ρ(E)dEdy (2)

4 Filon Type Quadrature Method We want to approximate an integral of the type I [f ] = 1 where f and g are smooth functions. f (x)e iωg(x) dx (3) Usually, the numerical integration of these type of integrals is acheived by using Gauss-Christoffel Quadrature methods. The integrand is interpolated at distinct nodes c 1 < c 2 < c 3...c ν by a polynomial p of degree ν 1. I [f ] Q GC = 1 p(x)dx (4) This method fails to give appropriate result when ω 1.

5 Filon Type Quadrature Method An alternative approach to this is to use the method developed by Louis Napoleon George Filon(1928). The function f(x) is interpolated instead of the whole integrand. Q F 1 [f ] = 1 f (x)e iωx dx = ν b k (ω)f (c k ) (5) k=1 where b k (ω) = 1 l k(x)e iωx dx and one needs to know first few moments 1 x m (x)e iωx dx in explicit form. Higher frequency ω ensures smaller error, in the order of O(ω 2 ) as ω

6 Filon Type Quadrature Method Iserles et. al use this Filon-type quadrature method to do the integration. Instead of polynomial expansion, they let the integrating polynomial depend on the derivatives of f. The error further decreases with increased frequency Q F s [f ] I [f ] O(ω s 1 ) (6) They further define a generalized Filon method, Q F s [f ] = 1 f (x)e iωx dx = θ ν j b l,j (ω)f j (c l ) (7) l=1 j= where θ i, θj s and b i,j = 1 α l,j(x)e iωg(x) dx

7 Example Iserles et al. give an example of implementation of this method. with g(x) = x, s = 2, ν = 2, c 1 =, c 2 = 1, and θ 1 = θ 2 =, we get Q A 2 [f ] = eiω f (1) f () iω + eiω f (1) f () ω 2 (8) Q F 2 [f ] = (...)f () + (...)f (1) + (...)f () + (...)f (1) +... (9)

8 Strategy My plan is to apply various methods developed by Iserles et al. to simpler problems whose answer we already know Need to know more about asymptotic behaviour of the functions that I want to integrate Need to know more about stationary points and/or the behaviour at the endpoints Do the required calculation os Power wall problem and WKB approximation problem.

9 Questions Any questions and Suggestions?

10 References A. Iserles and S. Nørsett Efficient quadrature of highly oscillatory integrals using derivatives Proceedings of the Royal Society A (25) Fulling et. al Investigating the Spectral Geometry of a Soft Wall S. Olver Moment-free numerical integration of highly oscillatory functions IMA Journal of Numerical Analysis 26 (26) T. A. Zapata The WKB approximation for linear potential and ceiling Master s Thesis. Texas AM University, 27.

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A Numerical Approach To The Steepest Descent Method K.U. Leuven, Division of Numerical and Applied Mathematics Isaac Newton Institute, Cambridge, Feb 15, 27 Outline 1 One-dimensional oscillatory integrals