# A Nyström Method for 2D Linear Fredholm Integral Equations on Curvilinear Domains

^{*}

^{†}

## Abstract

**:**

## 1. Introduction

## 2. Notations and Preliminary Results

#### 2.1. Approximation Tools

#### 2.2. An Algebraic Cubature Formula on Curvilinear Polygons

- $l\cap H$ is connected;
- Every segment q orthogonal to l is such that $q\cap H$ is connected.

**Theorem**

**1**

**.**Let Ω be a compact domain in ${\mathbb{R}}^{2}$, whose boundary (9) is a Jordan curve, and let $\tilde{\mathsf{\Omega}}$ be the domain with the boundary, as in (11). Assume that ${x}^{\prime}$, ${y}^{\prime}$ are at least piecewise Hölder continuous and satisfy (10). The following cubature error estimate holds for Formula (7) with $S=\tilde{P}$

**Theorem**

**2**

**.**Let Ω be as in (9), and $\tilde{P}(t)=(\tilde{x}(t),\tilde{y}(t)),t\in [a,b]$, be the piecewise polynomial approximating curve (10). Assume that the integrand f is Hölder-continuous with constant C and exponent $0<\alpha \le 1$ on the minimal rectangle $\mathcal{R}=[{x}_{1},{x}_{2}]\times [{y}_{1},{y}_{2}]$ containing $\mathsf{\Omega}\cup \tilde{\mathsf{\Gamma}}$, where $\tilde{\mathsf{\Gamma}}=\{\tilde{P}(t),t\in [a,b]\}$. Then, the following estimate holds for the error of the cubature formula (7) with $S=\tilde{P}$:

## 3. Numerical Methods for FIE

#### A Nyström Method

**Theorem**

**3.**

**Remark**

**1.**

**Corollary**

**1.**

**Remark**

**2.**

## 4. Numerical Tests

**Example**

**1.**

**Example**

**2.**

**Example**

**3.**

**Example**

**4.**

#### Comparison with Other Methods

**Example**

**5.**

**Theorem**

**4**

**.**Assume that $ker(I-K)=\{0\}$ in ${C}^{0}({[-1,1]}^{2})$. Denote by ${f}^{\ast}$ the unique solution of (14) in ${C}^{0}({[-1,1]}^{2})$ for a given $g\in {C}^{0}({[-1,1]}^{2})$. If, in addition, for some $r\in \mathbb{N}$,

**Example**

**6.**

**Example**

**7.**

## 5. The Proofs

**Proof**

**of**

**Theorem**

**3.**

- The sequence ${\left\{{K}_{n}\right\}}_{n}$ is collectively compact;
- ${sup}_{n}||{K}_{n}{||}_{{C}^{0}(\mathcal{J})\to {C}^{0}(\mathcal{J})}<+\infty $.

**Proof**

**of**

**Corollary**

**1.**

## 6. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Cubature points on the lune for $ADE=11$ with base-line $x=0.5$: the blue stars are the cubature points and the red line describes the boundary of the domain.

**Figure 2.**Cubaturepoints on the intersection of two disks for $ADE=11$ with base-line $x=0.5$: the blue stars are the cubature points and the red line describes the boundary of the domain.

**Figure 3.**Cubature points on the cardioid for $ADE=11$ with base-line $x=0.25$: the blue stars are the cubature points and the red line describes the boundary of the domain.

**Figure 4.**Cubature points on the deltoid for $ADE=11$ with base-line $y=0$: the blue stars are the cubature points and the red line describes the boundary of the domain.

**Figure 5.**Cubature points on the ellipse for $ADE=11$ with base-line $y=0.5$: the blue stars are the cubature points and the red line describes the boundary of the domain.

**Figure 6.**Cubature points on the unit disk for $ADE=11$ with base-line $x=0$. The blue stars are the cubature points and the red line describes the boundary of the domain.

**Figure 7.**Cubature points on the convex set $\mathsf{\Omega}$ for $ADE=11$ with base-line $y=x$. The blue stars are the cubature points and the red line describes the boundary of the domain.

n | ADE | N | err | cond |
---|---|---|---|---|

6 | 11 | 1254 | $9.325120847790344\times {10}^{-11}$ | $1.361434094160296\times {10}^{2}$ |

11 | 21 | 4059 | $1.476520127656227\times {10}^{-13}$ | $1.404483083850437\times {10}^{2}$ |

16 | 31 | 8464 | $2.233008835035651\times {10}^{-14}$ | $1.414987139794612\times {10}^{2}$ |

n | ADE | N | err | cond |
---|---|---|---|---|

6 | 11 | 1008 | $3.169305573522716\times {10}^{-8}$ | $3.097777365236376\times {10}^{0}$ |

11 | 21 | 3278 | $4.347572560160949\times {10}^{-9}$ | $3.098083282853776\times {10}^{0}$ |

16 | 31 | 6848 | $1.282175247136961\times {10}^{-9}$ | $3.098156965211412\times {10}^{0}$ |

21 | 41 | 11,718 | $5.573608694878142\times {10}^{-10}$ | $3.098185581757940\times {10}^{+00}$ |

n | ADE | N | err | cond |
---|---|---|---|---|

6 | 11 | 1050 | $7.897842369126102\times {10}^{-9}$ | $2.026152320630328\times {10}^{1}$ |

11 | 21 | 3410 | $5.327891709446134\times {10}^{-11}$ | $2.151386616056450\times {10}^{1}$ |

16 | 31 | 7120 | $5.327891709446134\times {10}^{-11}$ | $2.183833727014650\times {10}^{1}$ |

21 | 41 | 12,180 | $1.420771828553943\times {10}^{-12}$ | $2.196345357336691\times {10}^{1}$ |

n | ADE | N | err | cond |
---|---|---|---|---|

6 | 11 | 1050 | $2.338675276913542\times {10}^{-4}$ | $2.516388635642765\times {10}^{4}$ |

11 | 21 | 3410 | $2.888962785136443\times {10}^{-6}$ | $2.586711437154624\times {10}^{4}$ |

16 | 31 | 7120 | $5.314200158048982\times {10}^{-7}$ | $2.600795569124935\times {10}^{4}$ |

21 | 41 | 12,180 | $1.322458726172726\times {10}^{-7}$ | $2.604974962473686\times {10}^{4}$ |

n | ADE | N | err | cond |
---|---|---|---|---|

6 | 11 | 816 | $8.673617379884040\times {10}^{-17}$ | $1.005986730874940\times {10}^{0}$ |

n | ADE | N | err | cond |
---|---|---|---|---|

4 | 7 | 560 | $1.811883976188255\times {10}^{-13}$ | $1.161900962252042\times {10}^{2}$ |

8 | 15 | 2112 | $4.618527782440651\times {10}^{-14}$ | $1.199429863174676\times {10}^{2}$ |

**Table 7.**Numerical results for Example 6: Nyström method based on a tensorial product of two univariate Gaussian formulas.

n | ADE | N | errGauss | condGauss |
---|---|---|---|---|

4 | 7 | 8 | $1.243449787580175\times {10}^{-14}$ | $7.241511696510074\times {10}^{1}$ |

8 | 15 | 64 | $1.421085471520200\times {10}^{-14}$ | $1.081546678145990\times {10}^{2}$ |

n | ADE | N | err | cond |
---|---|---|---|---|

8 | 15 | 144 | $1.088018564132653\times {10}^{-14}$ | $9.799121640776345\times {10}^{1}$ |

16 | 31 | 544 | $2.142730437526552\times {10}^{-14}$ | $9.983873931918762\times {10}^{1}$ |

**Table 9.**Numerical results for Example 7: Nyström method based on a tensorial product of two univariate Gaussian formulas.

n | ADE | N | errGauss | condGauss |
---|---|---|---|---|

8 | 15 | 64 | $2.242650509742816\times {10}^{-14}$ | $9.874466448019098\times {10}^{1}$ |

16 | 31 | 256 | $8.215650382226158\times {10}^{-15}$ | $9.963067955720807\times {10}^{1}$ |

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## Share and Cite

**MDPI and ACS Style**

Laguardia, A.L.; Russo, M.G.
A Nyström Method for 2D Linear Fredholm Integral Equations on Curvilinear Domains. *Mathematics* **2023**, *11*, 4859.
https://doi.org/10.3390/math11234859

**AMA Style**

Laguardia AL, Russo MG.
A Nyström Method for 2D Linear Fredholm Integral Equations on Curvilinear Domains. *Mathematics*. 2023; 11(23):4859.
https://doi.org/10.3390/math11234859

**Chicago/Turabian Style**

Laguardia, Anna Lucia, and Maria Grazia Russo.
2023. "A Nyström Method for 2D Linear Fredholm Integral Equations on Curvilinear Domains" *Mathematics* 11, no. 23: 4859.
https://doi.org/10.3390/math11234859