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.
Highlights
Volterra integral equations arise from many mathematical problems in engineering and physics [1,2,3]
The numerical solution of a scalar retarded potential integral equation posted on an infinite flat surface, Z
A(·, t), u(·, t) ∈ L2 (R2 ) for t ∈ (0, T ), by taking CFT, Davies and Duncan [2] reformulated it as the following Volterra integral equation of the first kind with highly oscillatory Bessel kernel, 2π
Summary
Volterra integral equations arise from many mathematical problems in engineering and physics [1,2,3]. A(·, t), u(·, t) ∈ L2 (R2 ) for t ∈ (0, T ), by taking CFT, Davies and Duncan [2] reformulated it as the following Volterra integral equation of the first kind with highly oscillatory Bessel kernel, 2π. When ω 1, the Bessel kernel function becomes highly oscillatory. When resort to numerical solutions of Equation (1), the computation of integrals involved Bessel kernel functions is inevitable. We treat the following Volterra integral equation of the second kind with highly oscillatory Bessel kernel u( x ) −. When ω 1, the Bessel kernel function is highly oscillatory, and this makes solving Equation (2) a challenging problem. It is observed from numerical experiments that these methods have higher accuracy as compared with the Direct Filon method in [14]
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