# A Numerical Solution of Fredholm Integral Equations of the Second Kind Based on Tight Framelets Generated by the Oblique Extension Principle

## Abstract

**:**

## 1. Introduction

## 2. Preliminary Results

**Definition**

**1.**

**Definition**

**2.**

**Theorem**

**1**

**.**Let $\varphi \in {L}^{2}(\mathbb{R})$ be the compactly supported refinable function with its finitely supported low pass filter ${h}_{0}$. Let $\left\{{h}_{\ell}[k],\phantom{\rule{4.pt}{0ex}}k\in \mathbb{Z},\ell =1,\dots ,r\right\}$ be a set of finitely supported sequences, then the system

## 3. Quasi-Affine **B**-Spline Tight Framelet Systems

**Definition**

**3**

**.**The B-spline ${B}_{m+1}$ is defined as follows by using the convolution

#### 3.1. Framelets by the UEP and Its Generalization

**Definition**

**4**

**.**Let Ψ be defined as in the UEP. A corresponding quasi-affine system from level J is defined as

#### 3.2. Examples of Quasi-Affine B-Spline Tight Framelets

**Example**

**1**(Quasi-affine HAAR framelet (HAAR framelet))

**.**

**Example**

**2**(B

_{2}-UEP)

**.**

**Example**

**3**(B

_{4}-UEP)

**.**

**Example**

**4**(B

_{2}-OEP)

**.**

**Example**

**5**(B

_{4}-OEP)

**.**

## 4. Solving Fredholm Integral Equation via Tight Framelets

## 5. Error Analysis

**Theorem**

**2.**

**Proof.**

## 6. Numerical Performance and Illustrative Examples

**Example**

**6.**

**Example**

**7.**

## 7. Conclusions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 4.**Piecewise linear B-spline, ${B}_{2}(x)$, and the corresponding quasi-affine tight framelets generated by the oblique extension principle (OEP).

**Figure 6.**Illustration of ${B}_{2}$ and ${B}_{4}$, with its corresponding positive function H, respectively.

**Figure 7.**The graphs of u and ${u}_{n}$ for $n=2,3,4$, respectively, based on the quasi-affine HAAR framelet system of Example 6.

**Figure 8.**The graphs of u and ${u}_{n}$ for $n=5,6$, respectively, based on the quasi-affine HAAR framelet system of Example 6.

**Figure 9.**The graphs of ${E}_{n}(x)$ for $n=2,3,6$, respectively, and based on quasi-affine HAAR framelet system of Example 6.

**Figure 10.**The graphs of u and ${u}_{n}$ for $n=2,3,5$, respectively, based on the quasi-affine ${B}_{2}$-UEP framelet system.

**Figure 11.**The graphs of u and ${u}_{n}$ for $n=2,$ based on the quasi-affine ${B}_{2}$-UEP and ${B}_{4}$-UEP framelet systems, respectively, of Example 7.

**Figure 12.**The convergence rate graph of Examples 6 and 7 given in the log-log scale plot, respectively.

**Table 1.**The errors ${e}_{n}$ of Example 6 for five different quasi-affine tight framelet systems generated by the unitary extension principle (UEP) and OEP, for increasing n.

n | HAAR Framelet | ${\mathit{B}}_{2}$-UEP | ${\mathit{B}}_{4}$-UEP | ${\mathit{B}}_{2}$-OEP | ${\mathit{B}}_{4}$-OEP |
---|---|---|---|---|---|

2 | $6.55\times {10}^{-2}$ | $1.83\times {10}^{-3}$ | $8.89\times {10}^{-6}$ | $1.82\times {10}^{-3}$ | $8.24\times {10}^{-6}$ |

3 | $3.27\times {10}^{-2}$ | $4.58\times {10}^{-4}$ | $9.09\times {10}^{-6}$ | $6.78\times {10}^{-4}$ | $1.43\times {10}^{-7}$ |

4 | $1.64\times {10}^{-2}$ | $1.14\times {10}^{-4}$ | $1.39\times {10}^{-7}$ | $3.25\times {10}^{-4}$ | $9.46\times {10}^{-7}$ |

5 | $8.18\times {10}^{-3}$ | $2.86\times {10}^{-5}$ | $2.45\times {10}^{-8}$ | $1.87\times {10}^{-5}$ | $1.03\times {10}^{-9}$ |

6 | $4.09\times {10}^{-3}$ | $7.15\times {10}^{-5}$ | $1.33\times {10}^{-8}$ | $7.01\times {10}^{-6}$ | $5.71\times {10}^{-10}$ |

7 | $1.77\times {10}^{-4}$ | $9.88\times {10}^{-6}$ | $9.33\times {10}^{-9}$ | $8.79\times {10}^{-6}$ | $9.42\times {10}^{-11}$ |

8 | $5.92\times {10}^{-4}$ | $5.73\times {10}^{-6}$ | $7.40\times {10}^{-10}$ | $3.08\times {10}^{-7}$ | $5.08\times {10}^{-12}$ |

9 | $1.70\times {10}^{-5}$ | $4.03\times {10}^{-7}$ | $4.22\times {10}^{-11}$ | $4.01\times {10}^{-8}$ | $3.32\times {10}^{-13}$ |

10 | $8.65\times {10}^{-5}$ | $1.21\times {10}^{-8}$ | $3.52\times {10}^{-12}$ | $1.32\times {10}^{-9}$ | $2.21\times {10}^{-13}$ |

**Table 2.**Numerical results of the function ${u}_{n}$ of Example 6 using different quasi-affine tight framelets and for a level of $n=2$.

x | Exact | HAAR Framelet | ${\mathit{B}}_{2}$-UEP | ${\mathit{B}}_{4}$-UEP | ${\mathit{B}}_{2}$-OEP | ${\mathit{B}}_{4}$-OEP |
---|---|---|---|---|---|---|

$-0.9$ | $1.9000000$ | $1.9723995$ | $1.8998802$ | $1.8999970$ | $1.8998802$ | $1.9000001$ |

$-0.7$ | $1.5444444$ | $1.5235997$ | $1.5457151$ | $1.5444398$ | $1.5457151$ | $1.5444444$ |

$-0.5$ | $1.2777777$ | $1.2128922$ | $1.2748830$ | $1.2777713$ | $1.2748830$ | $1.2777777$ |

$-0.3$ | $1.1000000$ | $1.1093230$ | $1.1012727$ | $1.0999958$ | $1.1012727$ | $1.1000000$ |

$-0.1$ | $1.0111111$ | $1.0057538$ | $1.0109953$ | $1.0111080$ | $1.0109953$ | $1.0111100$ |

$0.0$ | $1.0000000$ | $1.0001228$ | $0.9971065$ | $0.9999965$ | $0.9971065$ | $1.0000000$ |

$0.1$ | $1.0111111$ | $1.0057538$ | $1.0109953$ | $1.0111078$ | $1.0109953$ | $1.0111110$ |

$0.3$ | $1.1000000$ | $1.1093230$ | $1.1012727$ | $1.0999938$ | $1.1012727$ | $1.1000000$ |

$0.5$ | $1.2777777$ | $1.3509844$ | $1.2748830$ | $1.2777699$ | $1.2748830$ | $1.2777777$ |

$0.7$ | $1.5444444$ | $1.5235997$ | $1.5457151$ | $1.5444346$ | $1.5457151$ | $1.5444442$ |

$0.9$ | $1.9000000$ | $1.9723995$ | $1.8998802$ | $1.8999885$ | $1.8998802$ | $1.9000000$ |

**Table 3.**The errors ${e}_{n}$ of Example 7 for five different quasi-affine tight framelet systems generated by the UEP and OEP, for increasing n.

n | HAAR Framelet | ${\mathit{B}}_{2}$-UEP | ${\mathit{B}}_{4}$-UEP | ${\mathit{B}}_{2}$-OEP | ${\mathit{B}}_{4}$-OEP |
---|---|---|---|---|---|

2 | $6.47\times {10}^{-2}$ | $1.04\times {10}^{-3}$ | $6.07\times {10}^{-7}$ | $1.03\times {10}^{-3}$ | $4.45\times {10}^{-7}$ |

3 | $3.23\times {10}^{-2}$ | $4.73\times {10}^{-4}$ | $8.33\times {10}^{-8}$ | $1.08\times {10}^{-4}$ | $1.78\times {10}^{-7}$ |

4 | $2.68\times {10}^{-2}$ | $8.76\times {10}^{-4}$ | $3.35\times {10}^{-8}$ | $1.25\times {10}^{-4}$ | $4.35\times {10}^{-8}$ |

5 | $1.76\times {10}^{-3}$ | $1.56\times {10}^{-5}$ | $1.25\times {10}^{-8}$ | $9.78\times {10}^{-5}$ | $1.58\times {10}^{-9}$ |

6 | $2.11\times {10}^{-4}$ | $1.05\times {10}^{-5}$ | $4.33\times {10}^{-9}$ | $3.45\times {10}^{-6}$ | $6.92\times {10}^{-11}$ |

7 | $7.93\times {10}^{-5}$ | $6.84\times {10}^{-6}$ | $5.53\times {10}^{-10}$ | $2.39\times {10}^{-6}$ | $5.92\times {10}^{-11}$ |

8 | $9.02\times {10}^{-5}$ | $5.56\times {10}^{-6}$ | $5.51\times {10}^{-11}$ | $5.98\times {10}^{-7}$ | $1.82\times {10}^{-13}$ |

9 | $2.50\times {10}^{-6}$ | $1.98\times {10}^{-7}$ | $1.02\times {10}^{-12}$ | $5.45\times {10}^{-8}$ | $3.11\times {10}^{-14}$ |

10 | $4.05\times {10}^{-7}$ | $2.34\times {10}^{-8}$ | $2.42\times {10}^{-13}$ | $2.12\times {10}^{-9}$ | $4.23\times {10}^{-15}$ |

**Table 4.**Numerical results of the function ${u}_{n}$ of Example 7 using different quasi-affine tight framelets and for a level of $n=2$.

x | Exact | HAAR Framelet | ${\mathit{B}}_{2}$-UEP | ${\mathit{B}}_{4}$-UEP | ${\mathit{B}}_{2}$-OEP | ${\mathit{B}}_{4}$-OEP |
---|---|---|---|---|---|---|

$0.0$ | $1.00000000$ | $1.07061012$ | $0.99866232$ | $0.99999601$ | $0.99863491$ | $0.999999958$ |

$0.1$ | $1.10517092$ | $1.09061851$ | $1.10508652$ | $1.10517150$ | $1.10505007$ | $1.105170028$ |

$0.2$ | $1.22140276$ | $1.21247432$ | $1.22209321$ | $1.22140205$ | $1.22203402$ | $1.221402071$ |

$0.3$ | $1.34985880$ | $1.37321902$ | $1.35065092$ | $1.34985878$ | $1.35060677$ | $1.349858253$ |

$0.4$ | $1.49182470$ | $1.55532117$ | $1.49179223$ | $1.49182450$ | $1.49179044$ | $1.491824418$ |

$0.5$ | $1.64872127$ | $1.76165283$ | $1.64658432$ | $1.64872193$ | $1.64660927$ | $1.648721644$ |

$0.6$ | $1.82211880$ | $1.80165032$ | $1.82197332$ | $1.82211894$ | $1.82198698$ | $1.822118840$ |

$0.7$ | $2.01375270$ | $2.01543677$ | $2.01488843$ | $2.01375233$ | $2.01490121$ | $2.013752248$ |

$0.8$ | $2.22554090$ | $2.26031276$ | $2.22684809$ | $2.22554035$ | $2.22686599$ | $2.225540081$ |

$0.9$ | $2.45960311$ | $2.46041532$ | $2.45955456$ | $2.45960353$ | $2.45956180$ | $2.459603410$ |

$1.0$ | $2.71828180$ | $2.72765982$ | $2.71485007$ | $2.71828155$ | $2.71483561$ | $2.718281710$ |

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**MDPI and ACS Style**

Mohammad, M.
A Numerical Solution of Fredholm Integral Equations of the Second Kind Based on Tight Framelets Generated by the Oblique Extension Principle. *Symmetry* **2019**, *11*, 854.
https://doi.org/10.3390/sym11070854

**AMA Style**

Mohammad M.
A Numerical Solution of Fredholm Integral Equations of the Second Kind Based on Tight Framelets Generated by the Oblique Extension Principle. *Symmetry*. 2019; 11(7):854.
https://doi.org/10.3390/sym11070854

**Chicago/Turabian Style**

Mohammad, Mutaz.
2019. "A Numerical Solution of Fredholm Integral Equations of the Second Kind Based on Tight Framelets Generated by the Oblique Extension Principle" *Symmetry* 11, no. 7: 854.
https://doi.org/10.3390/sym11070854