#
F-Transform Inspired Weak Solution to a Boundary Value Problem^{ †}

^{*}

^{†}

## Abstract

**:**

## 1. Introduction

- functions $p,q$ are bounded and measurable in $(a,b)$,
- function $f\in {L}^{2}(a,b)$,
- $0<{p}_{L}\le p\left(x\right)\le {p}_{R}$, $0\le q\left(x\right)$,

## 2. Prelimanaries

#### 2.1. Basic Notions about ${L}^{2}(a,b)$ and Sobolev Space

#### 2.2. Cut-Off Function

**Definition**

**1.**

**Lemma**

**1.**

#### 2.3. Generalized Uniform Fuzzy Partition

**Definition**

**2.**

**Example**

**1.**

- (i)
- A triangular generating function$${K}^{tr}\left(x\right)=\delta \xb7max(1-|x|,0).$$
- (ii)
- A raised cosine generating function$${K}^{rc}\left(x\right)=\left(\right)open="\{"\; close>\begin{array}{cc}\frac{\delta}{2}(1+cos\left(\pi x\right)),\hfill & -1\le t\le 1,\hfill \\ 0,\hfill & \mathit{otherwise},\hfill \end{array}$$
- (iii)
- A b-spline generating function of degree n$${K}^{bs,n}\left(x\right)=\delta \xb7{\beta}^{n}\left(\right)open="("\; close=")">\frac{(n+1)\xb7x}{2}$$$${\beta}^{0}\left(x\right)=\left(\right)open="\{"\; close>\begin{array}{cc}1,\hfill & -\frac{1}{2}x\frac{1}{2},\hfill \\ \frac{1}{2},\hfill & \left|x\right|=\frac{1}{2},\hfill \\ 0,\hfill & \mathit{otherwise}.\hfill \end{array}$$

**Definition**

**3.**

#### 2.4. Fuzzy Transform of a Higher Degree

**Definition**

**4.**

- (i)
- The direct F${}^{m}$-transform ($m\ge 0$) of f with respect to $\mathcal{A}$ is the set polynomials$${\mathrm{F}}^{m}\left[f\right]=\left(\right)open="\{"\; close="\}">{F}_{k}^{m}\left[f\right]\in {\mathbb{P}}_{m}\left({A}_{k}\right)\mid (f-{F}_{k}^{m}\left[f\right]){\perp}_{{A}_{k}}{\mathbb{P}}_{m}\left({A}_{k}\right),\phantom{\rule{0.166667em}{0ex}}k=M,\dots ,{M}^{\prime}$$${F}_{k}^{m}\left[f\right]$ is called the k-th component of the direct F${}^{m}$-transform.
- (ii)
- The inverse F${}^{m}$-transform of f with respect to $\mathcal{A}$ and the set of the direct F${}^{m}$-transform components ${\mathrm{F}}^{m}\left[f\right]=\left(\right)open="\{"\; close="\}">{F}_{k}^{m}\left[f\right]\mid k=M,\dots ,{M}^{\prime}$ of f, is the function defined as follows:$${\widehat{f}}^{m}\left(x\right)=\sum _{k=M}^{{M}^{\prime}}{F}_{k}^{m}\left[f\right]\left(x\right){A}_{k}\left(x\right),\phantom{\rule{1.em}{0ex}}x\in [a,b].$$

## 3. Test Spaces Constructed with a Generalized Fuzzy Partition

**Remark**

**1.**

**Remark**

**2.**

**Lemma**

**2.**

**Proof.**

**Lemma**

**3.**

**Proof.**

**Theorem**

**1.**

**Proof.**

## 4. Illustration

**Example**

**2**

**Example**

**3**

**Example**

**4**

## 5. Real-Life Application

## 6. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Table 1.**The error estimation and convergence rate of numerical solutions to the analytic solution in Example 2.

# Basis | GFPP | FPP | FEM | |||
---|---|---|---|---|---|---|

Functions | Error | Rate | Error | Rate | Error | Rate |

8 | $5.1\times {10}^{-6}$ | _ | $1.5\times {10}^{-3}$ | _ | $1.3\times {10}^{-2}$ | _ |

16 | $2.9\times {10}^{-8}$ | 7.5 | $1.9\times {10}^{-4}$ | 2.9 | $3.8\times {10}^{-3}$ | 1.8 |

32 | $3.5\times {10}^{-10}$ | 6.4 | $2.3\times {10}^{-5}$ | 3.1 | $9.9\times {10}^{-4}$ | 1.9 |

64 | $6.7\times {10}^{-12}$ | 5.7 | $2.9\times {10}^{-6}$ | 2.9 | $2.6\times {10}^{-4}$ | 1.9 |

128 | $1.7\times {10}^{-13}$ | 5.3 | $3.6\times {10}^{-7}$ | 3.0 | $6.5\times {10}^{-5}$ | 2.2 |

**Table 2.**The error estimation and convergence rate of numerical solutions to the analytic solution in Example 3.

# Basis | GFPP | FPP | FEM | |||
---|---|---|---|---|---|---|

Functions | Error | Rate | Error | Rate | Error | Rate |

8 | $2.8\times {10}^{-4}$ | _ | $7.3\times {10}^{-3}$ | _ | $2.2\times {10}^{-2}$ | _ |

16 | $6.6\times {10}^{-5}$ | 2.1 | $1.6\times {10}^{-3}$ | 2.2 | $5.4\times {10}^{-3}$ | 2.0 |

32 | $2.6\times {10}^{-5}$ | 1.3 | $6.2\times {10}^{-4}$ | 1.4 | $1.7\times {10}^{-3}$ | 1.7 |

64 | $1.2\times {10}^{-5}$ | 1.1 | $3.0\times {10}^{-4}$ | 1.1 | $6.5\times {10}^{-4}$ | 1.4 |

128 | $5.6\times {10}^{-6}$ | 1.1 | $1.5\times {10}^{-4}$ | 1.0 | $2.9\times {10}^{-4}$ | 1.2 |

**Table 3.**The error estimation and convergence rate of numerical solutions to the analytic solution in Example 4.

# Basis | GFPP | FPP | FEM | |||
---|---|---|---|---|---|---|

Functions | Error | Rate | Error | Rate | Error | Rate |

8 | $1.7\times {10}^{-2}$ | _ | $0.9785$ | _ | $1.0055$ | _ |

16 | $1.5\times {10}^{-2}$ | 1.8 | $0.6878$ | 5.1 | $0.8559$ | 2.3 |

32 | $6.0\times {10}^{-3}$ | 1.3 | $0.2447$ | 1.5 | $0.2764$ | 1.6 |

64 | $2.5\times {10}^{-3}$ | 1.3 | $3.9\times {10}^{-2}$ | 2.7 | $7.1\times {10}^{-2}$ | 1.9 |

128 | $2.4\times {10}^{-3}$ | 1.7 | $5.4\times {10}^{-3}$ | 2.9 | $1.8\times {10}^{-2}$ | 2.0 |

# Basis | Linear | # Basis | Quadratic | ||
---|---|---|---|---|---|

Functions | GFPP | DGM | Functions | GFPP | DGM |

16 | $1.14\times {10}^{-2}$ | $4.64\times {10}^{-2}$ | 12 | $1.08\times {10}^{-2}$ | $4.69\times {10}^{-2}$ |

32 | $3.35\times {10}^{-3}$ | $1.69\times {10}^{-2}$ | 24 | $1.99\times {10}^{-3}$ | $5.08\times {10}^{-3}$ |

64 | $9.08\times {10}^{-4}$ | $6.33\times {10}^{-3}$ | 48 | $2.63\times {10}^{-4}$ | $9.08\times {10}^{-4}$ |

128 | $2.49\times {10}^{-4}$ | $2.10\times {10}^{-3}$ | 96 | $2.80\times {10}^{-5}$ | $1.42\times {10}^{-4}$ |

256 | $6.35\times {10}^{-5}$ | $5.22\times {10}^{-4}$ | 192 | $2.21\times {10}^{-5}$ | $2.63\times {10}^{-5}$ |

512 | $2.31\times {10}^{-5}$ | $3.33\times {10}^{-4}$ | 384 | $1.51\times {10}^{-5}$ | $1.67\times {10}^{-5}$ |

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**MDPI and ACS Style**

Nguyen, L.; Perfilieva, I.; Holčapek, M.
F-Transform Inspired Weak Solution to a Boundary Value Problem. *Axioms* **2020**, *9*, 5.
https://doi.org/10.3390/axioms9010005

**AMA Style**

Nguyen L, Perfilieva I, Holčapek M.
F-Transform Inspired Weak Solution to a Boundary Value Problem. *Axioms*. 2020; 9(1):5.
https://doi.org/10.3390/axioms9010005

**Chicago/Turabian Style**

Nguyen, Linh, Irina Perfilieva, and Michal Holčapek.
2020. "F-Transform Inspired Weak Solution to a Boundary Value Problem" *Axioms* 9, no. 1: 5.
https://doi.org/10.3390/axioms9010005