# On the Convolution Quadrature Rule for Integral Transforms with Oscillatory Bessel Kernels

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Convergence of the Convolution Quadrature Rule

- $F(s)$ is analytic in the region $|\mathrm{arg}(s-c)|<\pi -\phi ,\phi <\pi /2,c\in \mathbb{R};$
- there exist constants M and $\mu ,$ such that $|F(s)|\le {M|s|}^{-\mu}.$

**Remark**

**1.**

**Theorem**

**1.**

- $\delta (\zeta )$ is analytic and without zeros in a neighbourhood of the closed unit disc $|\zeta |\le 1,$ with the exception of a zero at $\zeta =1;$
- $|arg\delta (\zeta )|\le \pi -\alpha ,$ for $|\zeta |<1,$ for some $\alpha >\phi ;$
- $\frac{\delta ({e}^{-h})}{h}=1+O({h}^{p}),$ for some $p\ge 1.$

**Lemma**

**1.**

**Theorem**

**2.**

**Proof.**

- Modified convolution quadrature rule for (1) of the first kind (MCQ1).$$\begin{array}{cc}\hfill {Q}_{h}^{mcq1}(b)=& \mathcal{F}({s}_{h})(f(x)-f(b)){\mid}_{x=1}+f(b){\int}_{0}^{b}{J}_{m}(\omega t)dt.\hfill \end{array}$$

**Theorem**

**3.**

**Remark**

**2.**

## 3. Application to a Volterra Equation

**Lemma**

**2.**

**Proof.**

**Lemma**

**3.**

**Proof.**

**Theorem**

**4.**

**Proof.**

## 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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**Figure 1.**Asymptotic convergence rates of CQ for ${\int}_{0}^{1}{J}_{0}(\omega t)\frac{1}{1+25{t}^{2}}dt$ (

**left**) and ${\int}_{0}^{2}{J}_{1}(\omega t)cos(t){\mathrm{e}}^{-t}dt$ (

**right**).

**Figure 2.**Asymptotic convergence rates of MCQ1 for ${\int}_{0}^{1}{J}_{0}(\omega t)\frac{1}{1+25{t}^{2}}dt$ (

**left**) and ${\int}_{0}^{2}{J}_{1}(\omega t)cos(t){\mathrm{e}}^{-t}dt$ (

**right**).

**Figure 3.**Asymptotic convergence rates of MCQ2 for ${\int}_{0}^{1}{J}_{0}(\omega t)\frac{1}{1+25{t}^{2}}dt$ (

**left**) and ${\int}_{0}^{2}{J}_{1}(\omega t)cos(t){\mathrm{e}}^{-t}dt$ (

**right**).

**Figure 4.**Comparisons between Filon methods and CQ for solving Volterra equations with $f(x)\phantom{\rule{3.33333pt}{0ex}}=\phantom{\rule{3.33333pt}{0ex}}\frac{x}{1+{x}^{2}}.$

Scheme | Initial Basis (${\mathit{\varphi}}_{-\mathit{n}}=0,\mathit{n}\ge 1$) | Recurrence for Basis Functions ($\mathit{j}\ge 1$) |
---|---|---|

BDF1 | $\varphi}_{0}(t)={e}^{-t$ | $j{\varphi}_{j}(t)-t{\varphi}_{j-1}(t)=0$ |

BDF2 | $\varphi}_{0}(t)={e}^{-3t/2$ | $j{\varphi}_{j}(t)-2t{\varphi}_{j-1}(t)+t{\varphi}_{j-2}(t)=0$ |

BDF3 | $\varphi}_{0}(t)={e}^{-11t/6$ | $j{\varphi}_{j}(t)-3t{\varphi}_{j-1}(t)+3t{\varphi}_{j-2}(t)-t{\varphi}_{j-3}(t)=0$ |

BDF4 | $\varphi}_{0}(t)={e}^{-25t/12$ | $j{\varphi}_{j}(t)-4t{\varphi}_{j-1}(t)+6t{\varphi}_{j-2}(t)-4t{\varphi}_{j-3}(t)+t{\varphi}_{j-4}(t)=0$ |

**Table 2.**Comparisons of quadrature rules for ${\int}_{0}^{1}{J}_{0}(\omega t)\frac{1}{1+25{t}^{2}}dt.$

$\mathit{\omega}$ | 20 | 100 | 200 | 400 | 600 | 800 | 1000 |
---|---|---|---|---|---|---|---|

CQ | $9.7\times {10}^{-4}$ | $3.4\times {10}^{-5}$ | $1.1\times {10}^{-5}$ | $9.4\times {10}^{-7}$ | $1.5\times {10}^{-6}$ | $1.3\times {10}^{-6}$ | $1.7\times {10}^{-7}$ |

MCQ1 | $8.3\times {10}^{-4}$ | $5.0\times {10}^{-6}$ | $6.3\times {10}^{-7}$ | $6.6\times {10}^{-8}$ | $2.1\times {10}^{-8}$ | $1.2\times {10}^{-8}$ | $7.2\times {10}^{-9}$ |

MCQ2 | $9.4\times {10}^{-4}$ | $3.5\times {10}^{-5}$ | $1.1\times {10}^{-5}$ | $9.4\times {10}^{-7}$ | $1.5\times {10}^{-6}$ | $1.3\times {10}^{-6}$ | $1.7\times {10}^{-7}$ |

**Table 3.**Comparisons of quadrature rules for ${\int}_{0}^{2}{J}_{1}(\omega t)cos(t){\mathrm{e}}^{-t}dt.$

$\mathit{\omega}$ | 20 | 100 | 200 | 400 | 600 | 800 | 1000 |
---|---|---|---|---|---|---|---|

CQ | $1.3\times {10}^{-5}$ | $7.8\times {10}^{-6}$ | $1.1\times {10}^{-5}$ | $1.3\times {10}^{-6}$ | $1.4\times {10}^{-6}$ | $1.4\times {10}^{-6}$ | $4.0\times {10}^{-7}$ |

MCQ1 | $6.3\times {10}^{-6}$ | $9.3\times {10}^{-7}$ | $1.6\times {10}^{-7}$ | $2.5\times {10}^{-8}$ | $1.9\times {10}^{-8}$ | $9.2\times {10}^{-9}$ | $4.6\times {10}^{-9}$ |

MCQ2 | $1.3\times {10}^{-5}$ | $7.8\times {10}^{-6}$ | $1.1\times {10}^{-5}$ | $1.3\times {10}^{-6}$ | $1.4\times {10}^{-6}$ | $1.4\times {10}^{-6}$ | $4.0\times {10}^{-7}$ |

$\mathit{\omega}$ | $\mathit{x}=0.1$ | $\mathit{x}=0.4$ | $\mathit{x}=0.8$ | $\mathit{x}=1.2$ | $\mathit{x}=1.6$ | $\mathit{x}=1.8$ | $\mathit{x}=2$ |
---|---|---|---|---|---|---|---|

10 | $3.3\times {10}^{-1}$ | $7.6\times {10}^{-2}$ | $4.7\times {10}^{-2}$ | $2.5\times {10}^{-2}$ | $8.0\times {10}^{-3}$ | $1.4\times {10}^{-2}$ | $3.1\times {10}^{-3}$ |

100 | $9.1\times {10}^{-2}$ | $5.5\times {10}^{-3}$ | $6.3\times {10}^{-4}$ | $7.0\times {10}^{-5}$ | $3.5\times {10}^{-4}$ | $3.6\times {10}^{-5}$ | $2.6\times {10}^{-4}$ |

200 | $5.0\times {10}^{-2}$ | $1.4\times {10}^{-4}$ | $3.6\times {10}^{-4}$ | $1.1\times {10}^{-4}$ | $1.2\times {10}^{-4}$ | $1.1\times {10}^{-4}$ | $1.8\times {10}^{-5}$ |

500 | $1.9\times {10}^{-2}$ | $1.9\times {10}^{-4}$ | $1.2\times {10}^{-5}$ | $4.6\times {10}^{-5}$ | $3.7\times {10}^{-5}$ | $2.2\times {10}^{-5}$ | $6.8\times {10}^{-6}$ |

1000 | $1.0\times {10}^{-2}$ | $5.9\times {10}^{-7}$ | $3.9\times {10}^{-5}$ | $1.2\times {10}^{-5}$ | $8.7\times {10}^{-10}$ | $9.6\times {10}^{-6}$ | $9.2\times {10}^{-6}$ |

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**MDPI and ACS Style**

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

**AMA Style**

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 Style**

Ma, 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