# Numerical Solution of the Cauchy-Type Singular Integral Equation with a Highly Oscillatory Kernel Function

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## Abstract

**:**

## 1. Introduction

**Theorem**

**1.**

**Proof.**

- 1: The solution $u\left(x\right)$ for $\kappa =1$ is unbounded at both end points $x=\pm 1$:$$u\left(x\right)=af\left(x\right)-\frac{bw\left(x\right)}{\pi}{e}^{-ikx}{{\displaystyle \u2a0d}}_{-1}^{1}\frac{{w}^{*}\left(y\right)f\left(y\right){e}^{iky}}{y-x}dy+Cw\left(x\right),$$$${\int}_{-1}^{1}u\left(y\right){e}^{iky}dy=C.$$Equation (2) gets infinitely many solutions but is unique for the above condition.
- 2: The solution $u\left(x\right)$ is bounded for $\kappa =0$ at $x=\pm 1$ and unbounded at $x=\mp 1$:$$u\left(x\right)=af\left(x\right)-\frac{bw\left(x\right)}{\pi}{e}^{-ikx}{{\displaystyle \u2a0d}}_{-1}^{1}\frac{{w}^{*}\left(y\right)f\left(y\right){e}^{iky}}{y-x}dy,$$Equation (2) gets a unique solution.
- 3: The solution $u\left(x\right)$ is bounded at both end points $x=\pm 1$ for $\kappa =-1$:$$u\left(x\right)=af\left(x\right)-\frac{bw\left(x\right)}{\pi}{e}^{-ikx}{{\displaystyle \u2a0d}}_{-1}^{1}\frac{{w}^{*}\left(y\right)f\left(y\right){e}^{iky}}{y-x}dy.$$Equation (2) has no solution unless it satisfies the following condition:$${\int}_{-1}^{1}\frac{f\left(y\right){e}^{iky}}{w\left(y\right)}dy=0.$$

## 2. Description of the Method

#### Computation of Moments

**Proposition**

**1.**

**Proof.**

## 3. Error Analysis

**Lemma**

**1.**

- (i) if $f\left(x\right)$ is analytic with $\left|f\right(x\left)\right|\le M$ in an ellipse ${\epsilon}_{\rho}$ (Bernstein ellipse) with foci $\pm 1$ and major and minor semiaxis lengths summing to $\rho >1$, then:$$\parallel f-{P}_{N}{\left[f\right]\parallel}_{\infty}\le \frac{4M}{{\rho}^{N}(\rho -1)}.$$
- (ii) if $f\left(x\right)$ has an absolutely continuous $({\kappa}_{0}-1)st$ derivative and a ${\kappa}_{0}th$ derivative ${f}^{\left({\kappa}_{0}\right)}$ of bounded variation ${V}_{{\kappa}_{0}}$ on [−1,1] for some ${\kappa}_{0}\ge 1$, then for $N\ge {\kappa}_{0}+1$:$$\parallel f-{P}_{N}{\left[f\right]}_{\infty}\parallel \le \frac{4{V}_{{\kappa}_{0}}}{{\kappa}_{0}\pi N(N-1)\cdots (N-{\kappa}_{0}+1)}.$$

**Proposition**

**2.**

**Theorem**

**2.**

**Proof.**

## 4. Numerical Examples

**Example**

**1.**

**Example**

**2.**

**Example**

**3.**

## 5. Conclusions

## Supplementary Materials

## Author Contributions

## Funding

## Conflicts of Interest

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k | N = 5 | N = 10 | N = 20 |
---|---|---|---|

50 | 4.6387 × 10^{−9} | 3.9207 × 10^{−14} | 1.1102 × 10^{−16} |

100 | 1.0881 × 10^{−9} | 4.9564 × 10^{−15} | 0 |

1000 | 3.8093 × 10^{−11} | 4.0030 × 10^{−16} | 2.4825 × 10^{−16} |

10,000 | 5.1593 × 10^{−13} | 2.2204 × 10^{−16} | 1.1102 × 10^{−16} |

k | N = 5 | N = 10 | N = 20 |
---|---|---|---|

50 | 1.1156 × 10^{−9} | 9.1854 × 10^{−15} | 1.1102 × 10^{−16} |

100 | 3.2791 × 10^{−10} | 5.6610 × 10^{−16} | 1.1102 × 10^{−16} |

1000 | 1.7225 × 10^{−12} | 2.2204 × 10^{−16} | 2.2204 × 10^{−16} |

10,000 | 7.3056 × 10^{−15} | 3.3307 × 10^{−16} | 3.3307 × 10^{−16} |

x | Error | ||
---|---|---|---|

$\mathit{\kappa}\mathbf{=}\mathbf{-}\mathbf{1}$ | $\mathit{\kappa}\mathbf{=}\mathbf{0}\mathbf{,}\mathit{boundned}\mathit{at}\mathit{x}\mathbf{=}\mathbf{-}\mathbf{1}$ | $\mathit{\kappa}\mathbf{=}\mathbf{0}\mathbf{,}\mathit{boundned}\mathit{at}\mathit{x}\mathbf{=}\mathbf{1}$ | |

−0.6 | 0 | 1.1102 × 10^{−16} | 4.4409 × 10^{−16} |

−0.2 | 3.3307 × 10^{−16} | 2.2204 × 10^{−16} | 4.4409 × 10^{−16} |

0.2 | 2.2204 × 10^{−16} | 4.4409 × 10^{−16} | 0 |

0.6 | 0 | 2.2204 × 10^{−16} | 4.4409 × 10^{−16} |

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**MDPI and ACS Style**

SAIRA; Xiang, S.; Liu, G.
Numerical Solution of the Cauchy-Type Singular Integral Equation with a Highly Oscillatory Kernel Function. *Mathematics* **2019**, *7*, 872.
https://doi.org/10.3390/math7100872

**AMA Style**

SAIRA, Xiang S, Liu G.
Numerical Solution of the Cauchy-Type Singular Integral Equation with a Highly Oscillatory Kernel Function. *Mathematics*. 2019; 7(10):872.
https://doi.org/10.3390/math7100872

**Chicago/Turabian Style**

SAIRA, Shuhuang Xiang, and Guidong Liu.
2019. "Numerical Solution of the Cauchy-Type Singular Integral Equation with a Highly Oscillatory Kernel Function" *Mathematics* 7, no. 10: 872.
https://doi.org/10.3390/math7100872