Smoothing transformation and collocation methods for third-kind linear Volterra integral equations

2020 ◽  
Vol 32 (3) ◽  
pp. 361-375
Author(s):  
Xiaoli Xu ◽  
Yu Xiao ◽  
Hui Ming Song
Keyword(s):  
Integral Equations ◽  
Volterra Integral Equations ◽  
Collocation Methods ◽  
Linear Volterra Integral Equations ◽  
Smoothing Transformation

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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