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Symmetry 2019, 11(2), 168; https://doi.org/10.3390/sym11020168

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

1
College of Mathematics, Hunan Institute of Science and Technology, Yueyang 414006, China
2
College of Economics, Jinan University, Guangzhou 510632, China
3
School of Mathematics and Statistics, Central South University, Changsha 410083, China
*
Authors to whom correspondence should be addressed.
Received: 20 December 2018 / Revised: 21 January 2019 / Accepted: 29 January 2019 / Published: 1 February 2019
(This article belongs to the Special Issue Integral Transforms and Operational Calculus)
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Abstract

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. View Full-Text
Keywords: Volterra integral equations; highly oscillatory Bessel kernel; Hermite interpolation; direct Hermite collocation method; piecewise Hermite collocation method Volterra integral equations; highly oscillatory Bessel kernel; Hermite interpolation; direct Hermite collocation method; piecewise Hermite collocation method
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Fang, C.; He, G.; Xiang, S. Hermite-Type Collocation Methods to Solve Volterra Integral Equations with Highly Oscillatory Bessel Kernels. Symmetry 2019, 11, 168.

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