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.

scholarly journals Hermite-Type Collocation Methods to Solve Volterra Integral Equations with Highly Oscillatory Bessel Kernels

Symmetry ◽  
2019 ◽  
Vol 11 (2) ◽  
pp. 168 ◽  
Author(s):  
Chunhua Fang ◽  
Guo He ◽  
Shuhuang Xiang
Keyword(s):  
Integral Equations ◽  
Collocation Method ◽  
Bessel Functions ◽  
Hermite Interpolation ◽  
Volterra Integral Equations ◽  
Collocation Methods ◽  
Fast Computation ◽  
Highly Oscillatory ◽  
Linear Volterra Integral Equations ◽  
Highly Oscillatory Integrals

In this paper, we present two kinds of Hermite-type collocation methods for linear Volterra integral equations of the second kind with highly oscillatory Bessel kernels. One method is direct Hermite collocation method, which used direct two-points Hermite interpolation in the whole interval. The other one is piecewise Hermite collocation method, which used a two-points Hermite interpolation in each subinterval. These two methods can calculate the approximate value of function value and derivative value simultaneously. Both methods are constructed easily and implemented well by the fast computation of highly oscillatory integrals involving Bessel functions. Under some conditions, the asymptotic convergence order with respect to oscillatory factor of these two methods are established, which are higher than the existing results. Some numerical experiments are included to show efficiency of these two methods.

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