Stable application of Filon–Clenshaw–Curtis rules to singular oscillatory integrals by exponential transformations
- 69 Downloads
Highly oscillatory integrals, having amplitudes with algebraic (or logarithmic) endpoint singularities, are considered. An integral of this kind is first transformed into a regular oscillatory integral over an unbounded interval. After applying the method of finite sections, a composite modified Filon–Clenshaw–Curtis rule, recently developed by the author, is applied on it. By this strategy the original integral can be computed in a more stable manner, while the convergence orders of the composite Filon–Clenshaw–Curtis rule are preserved. By introducing the concept of an oscillation subinterval, we propose algorithms, which employ composite Filon–Clenshaw–Curtis rules on rather small intervals. The integral outside the oscillation subinterval is non-oscillatory, so it can be computed by traditional quadrature rules for regular integrals, e.g. the Gaussian ones. We present several numerical examples, which illustrate the accuracy of the algorithms.
KeywordsFilon–Clenshaw–Curtis rule Highly oscillatory integral Exponential transformation Double exponential transformation Algebraic singularity
Mathematics Subject Classification65D30 65T40
- 9.Majidian, H.: Modified Filon–Clenshaw–Curtis rules for oscillatory integrals with a nonlinear oscillator (2016). arXiv:1604.05074
- 15.Olver, S.: Numerical approximation of highly oscillatory integrals. Ph.D. thesis, University of Cambridge (2008)Google Scholar
- 16.Rizzardi, M.: Detection of the singularities of a complex function by numerical approximations of its Laurent coefficients. Numer. Algoritm. (2017). https://doi.org/10.1007/s11075-017-0349-2