# Theory of Functional Connections Extended to Fractional Operators

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## Abstract

**:**

## 1. Introduction

## 2. Background on Fractional Calculus

#### 2.1. The Gamma Function

- $\mathsf{\Gamma}\left(1\right)=1$;
- $\mathsf{\Gamma}(z+1)=z\phantom{\rule{0.166667em}{0ex}}\mathsf{\Gamma}\left(z\right)$, for $z>0$;
- $\mathsf{\Gamma}\left(z\right)$ is logarithmically convex (or superconvex).

#### 2.2. Riemann–Liouville Fractional Integral

#### 2.3. Riemann–Liouville Fractional Derivative

#### 2.4. Caputo Fractional Derivative

#### 2.5. Grünwald–Letnikov Definitions

## 3. Background on the Theory of Functional Connections

#### A Simple Explanatory Example

## 4. Shifted Chebyshev Polynomials

#### Example

## 5. Numerical Examples

#### 5.1. Single Fractional Constraint

#### 5.2. Three Mixed Constraints

#### 5.3. Two Linear Combinations of Fractional Constraints

## 6. Discussion

## Author Contributions

## Funding

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Abbreviations

TFC | Theory of functional connections |

SCP | Shifted Chebyshev polynomials |

## Appendix A. Some Fractional Integrals and Derivatives with Closed-Form Expressions

## Appendix B. Non-Locality of Fractional Operators (Memory Effect)

**Theorem**

**A1.**

**Theorem**

**A2.**

**Memory effect property**: Be $\alpha \in (0,1]$ and calculate the difference between two valuations ${\mathcal{J}}_{0}^{\alpha}f\left(x\right)$ in ${x}_{1}\phantom{\rule{4.pt}{0ex}}\mathrm{and}\phantom{\rule{4.pt}{0ex}}{x}_{2}$ such that ${x}_{1}<{x}_{2}$:

## Appendix C. Least-Squares Approaches

- The common solution: $\mathit{x}={\left({A}^{T}\phantom{\rule{0.166667em}{0ex}}A\right)}^{-1}\phantom{\rule{0.166667em}{0ex}}{A}^{T}\phantom{\rule{0.166667em}{0ex}}\mathit{b}$;
- The QR decomposition: $A=QR$, then $\mathit{x}={R}^{-1}{Q}^{T}\mathit{b}$, where $Q\in SO\left(n\right)$ and R an upper-triangular matrix;
- The SVD decompositions: $A=U\Sigma {V}^{T}$, then $\mathit{x}={A}^{+}\mathit{b}=V{\Sigma}^{+}{U}^{T}\mathit{b},$ where $U\in SO\left(n\right)$ and $V\in SO\left(n\right)$ and where ${\Sigma}^{+}$ is the pseudo-inverse of $\Sigma $, which is formed by replacing every non-zero diagonal entry by its reciprocal and transposing the resulting matrix;
- The Cholesky decomposition: ${A}^{T}A\mathit{x}={U}^{T}U\mathit{x}={A}^{T}\mathit{b}$, then $\mathit{x}={U}^{-1}\left(\right)open="("\; close=")">{U}^{-T}{A}^{T}\mathit{b}$, where U is an upper-triangular matrix.

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**Figure 1.**Functional $f(x,g(x\left)\right)$ (

**left**plot) and its RL derivative ${\mathcal{D}}_{0}^{\alpha}f(x,g\left(x\right))$ (

**right**plot) for some choices of $g\left(x\right)$. Here, $\alpha =0.8$, ${x}_{0}=1$, ${f}_{0}=1$ and $p=2$.

**Figure 2.**Function $g\left(t\right)$ and errors of truncated approximations ${g}_{m}\left(t\right)$ (

**left**plot). RL derivatives ${\mathcal{D}}_{0}^{\alpha}{g}_{m}\left(x\right)$ for $m=12$ (

**right**plot).

**Figure 3.**Residual of the constraints ${C}_{1}$ (

**left**plot) and ${C}_{2}$ (

**right**plot) when applied to the functional $f(x,g(x\left)\right)$ given by (22).

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**MDPI and ACS Style**

Mortari, D.; Garrappa, R.; Nicolò, L.
Theory of Functional Connections Extended to Fractional Operators. *Mathematics* **2023**, *11*, 1721.
https://doi.org/10.3390/math11071721

**AMA Style**

Mortari D, Garrappa R, Nicolò L.
Theory of Functional Connections Extended to Fractional Operators. *Mathematics*. 2023; 11(7):1721.
https://doi.org/10.3390/math11071721

**Chicago/Turabian Style**

Mortari, Daniele, Roberto Garrappa, and Luigi Nicolò.
2023. "Theory of Functional Connections Extended to Fractional Operators" *Mathematics* 11, no. 7: 1721.
https://doi.org/10.3390/math11071721