# Fractional Dynamical Systems Solved by a Collocation Method Based on Refinable Spaces

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**—**Theory and Applications)

## Abstract

**:**

## 1. Introduction

## 2. Fractional Dynamical Systems

**Theorem**

**1.**

**Proof.**

## 3. Fractional Cardinal B-Splines and Fractional GP Functions

## 4. Multiresolution Analysis on $\mathbb{R}$ and on $[\mathbf{0},\infty )$

- (i)
- $V}_{j}\subset {V}_{j+1},\phantom{\rule{0.277778em}{0ex}}j\in \mathbb{Z$;
- (ii)
- $\overline{{\bigcup}_{j\in \mathbb{Z}}{V}_{j}}={L}^{2}\left(\mathbb{R}\right)$;
- (iii)
- $\bigcap _{j\in \mathbb{Z}}{V}_{j}=\left\{0\right\}$;
- (iv)
- $f\left(t\right)\in {V}_{j}\leftrightarrow f\left(2t\right)\in {V}_{j+1}$, $j\in \mathbb{Z}$;
- (v)
- There exists an ${L}^{2}\left(\mathbb{R}\right)$-stable basis in ${V}_{0}$.

## 5. The Fractional Derivative of B-Splines

**Theorem**

**2.**

**Proof.**

## 6. The Fractional Collocation Method

## 7. Numerical Results

## 8. Conclusions and Future Work

## Author Contributions

## Funding

## Conflicts of Interest

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**Figure 1.**(

**a**) The graphs of the fractional GP for $\alpha $ = 2:0.25:6, $h=\alpha +3$, classical GP for $\alpha =2,\dots ,6$, and classical B-splines for $h=\alpha $ integer. (

**b**) The graphs of the fractional GP for $\alpha $ = 2:0.25:6, $h=\alpha +0.5$, classical GP for $\alpha =2,\dots ,6$, and classical B-splines for $h=\alpha $ integer.

**Figure 2.**The approximate solutions ${x}_{j},\phantom{\rule{0.166667em}{0ex}}{y}_{j}$ with $j=8$ (red line) obtained with the cubic B-spline ${\phi}_{3,3}\equiv {B}_{3}$ and the exact solutions $x\left(t\right)$, $y\left(t\right)$ (blue dashed line). We consider four different example obtained whit different value for $\gamma $. In (

**a**) we use $\gamma =0.10$, in (

**b**) $\gamma =0.25$, in (

**c**) $\gamma =0.5$ and in (

**d**) $\gamma =0.75$.

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**MDPI and ACS Style**

Pezza, L.; Di Lillo, S.
Fractional Dynamical Systems Solved by a Collocation Method Based on Refinable Spaces. *Axioms* **2023**, *12*, 451.
https://doi.org/10.3390/axioms12050451

**AMA Style**

Pezza L, Di Lillo S.
Fractional Dynamical Systems Solved by a Collocation Method Based on Refinable Spaces. *Axioms*. 2023; 12(5):451.
https://doi.org/10.3390/axioms12050451

**Chicago/Turabian Style**

Pezza, Laura, and Simmaco Di Lillo.
2023. "Fractional Dynamical Systems Solved by a Collocation Method Based on Refinable Spaces" *Axioms* 12, no. 5: 451.
https://doi.org/10.3390/axioms12050451