Efficient methods for some highly oscillatory integrals and integral equations

2012 ◽  
Vol 42 (7) ◽  
pp. 651-670
Author(s):  
ShuHuang XIANG
Keyword(s):  
Integral Equations ◽  
Oscillatory Integrals ◽  
Highly Oscillatory ◽  
Highly Oscillatory Integrals

scholarly journals Analysis of Generalized Multistep Collocation Solutions for Oscillatory Volterra Integral Equations

Mathematics ◽  
2020 ◽  
Vol 8 (11) ◽  
pp. 2004
Author(s):  
Hao Chen ◽  
Ling Liu ◽  
Junjie Ma
Keyword(s):  
Error Estimate ◽  
Convergence Analysis ◽  
Integral Equations ◽  
Volterra Integral Equations ◽  
Oscillatory Integrals ◽  
Collocation Methods ◽  
Schur Polynomials ◽  
Numerical Tests ◽  
Highly Oscillatory ◽  
Highly Oscillatory Integrals

In this work, we introduce a class of generalized multistep collocation methods for solving oscillatory Volterra integral equations, and study two kinds of convergence analysis. The error estimate with respect to the stepsize is given based on the interpolation remainder, and the nonclassical convergence analysis with respect to oscillation is developed by investigating the asymptotic property of highly oscillatory integrals. Besides, the linear stability is analyzed with the help of generalized Schur polynomials. Several numerical tests are given to show that the numerical results coincide with our theoretical estimates.

scholarly journals Fast Computation of Integrals with Fourier-Type Oscillator Involving Stationary Point

Mathematics ◽  
2019 ◽  
Vol 7 (12) ◽  
pp. 1160 ◽  
Author(s):  
Sakhi Zaman ◽  
Irshad Hussain ◽  
Dhananjay Singh
Keyword(s):  
Collocation Method ◽  
Stationary Point ◽  
Numerical Evaluation ◽  
Oscillatory Integrals ◽  
Test Problems ◽  
Fast Computation ◽  
Splitting Algorithm ◽  
Highly Oscillatory ◽  
Highly Oscillatory Integrals ◽  
Fourier Type

An adaptive splitting algorithm was implemented for numerical evaluation of Fourier-type highly oscillatory integrals involving stationary point. Accordingly, a modified Levin collocation method was coupled with multi-resolution quadratures in order to tackle the stationary point and irregular oscillations of the integrand caused by ω . Some test problems are included to verify the accuracy of the proposed methods.

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