On the Convolution Quadrature Rule for Integral Transforms with Oscillatory Bessel Kernels
Abstract
:1. Introduction
2. Convergence of the Convolution Quadrature Rule
- is analytic in the region
- there exist constants M and such that
- is analytic and without zeros in a neighbourhood of the closed unit disc with the exception of a zero at
- for for some
- for some
- Modified convolution quadrature rule for (1) of the first kind (MCQ1).
3. Application to a Volterra Equation
4. Numerical Results
5. Conclusions
Author Contributions
Funding
Acknowledgments
Conflicts of Interest
Abbreviations
BDF | backward differentiation formula |
CQ | convolution quadrature rule |
FFT | fast Fourier transform |
GMRES | generalized minimal residual method |
HOI | highly oscillatory integral |
HOP | highly oscillatory problem |
ODE | ordinary differential equation |
HOVIE | highly oscillatory Volterra integral equation |
MCQ1 | modified convolution quadrature of the first kind |
MCQ2 | modified convolution quadrature of the second kind |
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Scheme | Initial Basis () | Recurrence for Basis Functions () |
---|---|---|
BDF1 | ||
BDF2 | ||
BDF3 | ||
BDF4 |
20 | 100 | 200 | 400 | 600 | 800 | 1000 | |
---|---|---|---|---|---|---|---|
CQ | |||||||
MCQ1 | |||||||
MCQ2 |
20 | 100 | 200 | 400 | 600 | 800 | 1000 | |
---|---|---|---|---|---|---|---|
CQ | |||||||
MCQ1 | |||||||
MCQ2 |
10 | |||||||
100 | |||||||
200 | |||||||
500 | |||||||
1000 |
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Ma, J.; Liu, H. On the Convolution Quadrature Rule for Integral Transforms with Oscillatory Bessel Kernels. Symmetry 2018, 10, 239. https://doi.org/10.3390/sym10070239
Ma J, Liu H. On the Convolution Quadrature Rule for Integral Transforms with Oscillatory Bessel Kernels. Symmetry. 2018; 10(7):239. https://doi.org/10.3390/sym10070239
Chicago/Turabian StyleMa, Junjie, and Huilan Liu. 2018. "On the Convolution Quadrature Rule for Integral Transforms with Oscillatory Bessel Kernels" Symmetry 10, no. 7: 239. https://doi.org/10.3390/sym10070239