# Theory of Functional Connections Applied to Linear ODEs Subject to Integral Constraints and Linear Ordinary Integro-Differential Equations

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## Abstract

**:**

## 1. Introduction

## 2. Theory of Functional Connections Summary

## 3. TFC for ODEs with Integral Constraints

#### 3.1. Definite Integral Constraint

#### 3.2. Integral and Linear Constraints

#### Problem #1

#### 3.3. Mixed Constraints

#### Problem #2

#### 3.4. Discussions

## 4. TFC for Linear Ordinary Integro-Differential Equation

#### 4.1. Problem #1

#### 4.2. Problem #2

#### 4.3. Problem #3

#### 4.4. Discussions

## 5. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Appendix A. Chebyshev and Legendre Orthogonal Polynomials

#### Appendix A.1. Definition

#### Appendix A.2. Orthogonality

#### Appendix A.3. Derivatives

#### Appendix A.4. Integral

- Chebyshev indefinite.$$\begin{array}{cc}\hfill \phantom{\rule{1.em}{0ex}}& {\displaystyle {\int}_{-1}^{x}{T}_{0}\left(z\right)\phantom{\rule{0.277778em}{0ex}}\mathrm{d}z=x+1,\phantom{\rule{2.em}{0ex}}{\int}_{-1}^{x}{T}_{1}\left(z\right)\phantom{\rule{0.277778em}{0ex}}\mathrm{d}z={\displaystyle \frac{1}{2}}\phantom{\rule{0.166667em}{0ex}}\left(\right)open="("\; close=")">{x}^{2}-1,}\hfill \end{array}$$
- Chebyshev full range.$$\int}_{-1}^{+1}{T}_{k}\left(x\right)\phantom{\rule{0.277778em}{0ex}}\mathrm{d}x=\left(\right)open="\{"\; close>\begin{array}{cc}0\hfill & \mathrm{for}\phantom{\rule{0.277778em}{0ex}}k\phantom{\rule{0.277778em}{0ex}}\mathrm{odd}\hfill \\ {\displaystyle \frac{{(-1)}^{k}+1}{1-{k}^{2}}}\hfill & \mathrm{for}\phantom{\rule{0.277778em}{0ex}}k\phantom{\rule{0.277778em}{0ex}}\mathrm{even}\hfill \end{array$$
- Chebyshev internal range ($-1\le a<b\le +1$)$$\int}_{a}^{b}{T}_{k}\left(x\right)\phantom{\rule{0.277778em}{0ex}}\mathrm{d}x={\displaystyle \frac{k\phantom{\rule{0.166667em}{0ex}}\left(\right)open="["\; close="]">{T}_{k+1}\left(b\right)-{T}_{k+1}\left(a\right)}{}{k}^{2}-1}-{\displaystyle \frac{b\phantom{\rule{0.166667em}{0ex}}{T}_{k}\left(b\right)-a\phantom{\rule{0.166667em}{0ex}}{T}_{k}\left(a\right)}{k-1}$$
- Legendre indefinite.$$\begin{array}{cc}\hfill \phantom{\rule{1.em}{0ex}}& {\displaystyle {\int}_{-1}^{x}{L}_{0}\left(z\right)\phantom{\rule{0.277778em}{0ex}}\mathrm{d}z=x+1,\phantom{\rule{2.em}{0ex}}{\int}_{-1}^{x}{L}_{1}\left(z\right)\phantom{\rule{0.277778em}{0ex}}\mathrm{d}z={\displaystyle \frac{1}{2}}\phantom{\rule{0.166667em}{0ex}}\left(\right)open="("\; close=")">{x}^{2}-1,}\hfill \end{array}$$
- Legendre full range.$${\int}_{-1}^{+1}{L}_{0}\left(x\right)\phantom{\rule{0.277778em}{0ex}}\mathrm{d}x=2\phantom{\rule{2.em}{0ex}}\mathrm{and}\phantom{\rule{2.em}{0ex}}{\int}_{-1}^{+1}{L}_{k}\left(x\right)\phantom{\rule{0.277778em}{0ex}}\mathrm{d}x=0,\phantom{\rule{1.em}{0ex}}\forall \phantom{\rule{0.277778em}{0ex}}k\ne 0.$$
- Legendre internal range ($-1\le a<b\le +1$)$${\int}_{a}^{b}{L}_{k}\left(x\right)\phantom{\rule{0.277778em}{0ex}}\mathrm{d}x={\displaystyle \frac{{L}_{k+1}\left(b\right)-{L}_{k+1}\left(a\right)+{L}_{k-1}\left(a\right)-{L}_{k-1}\left(b\right)}{2k+1}}.$$

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$\mathit{t}/\mathit{\pi}$ | TFC | X-TFC |
---|---|---|

0.0 | 0.0 | 0.0 |

0.1 | 0.0 | $2.22\times {10}^{-16}$ |

0.2 | 0.0 | $2.22\times {10}^{-16}$ |

0.3 | 0.0 | $0.0$ |

0.4 | 0.0 | $0.0$ |

0.5 | 0.0 | $0.0$ |

0.6 | $4.44\times {10}^{-16}$ | $4.44\times {10}^{-16}$ |

0.7 | $1.11\times {10}^{-16}$ | $2.22\times {10}^{-16}$ |

0.8 | $6.66\times {10}^{-16}$ | $0.0$ |

0.9 | $1.66\times {10}^{-16}$ | $2.77\times {10}^{-16}$ |

1.0 | $2.22\times {10}^{-16}$ | $6.66\times {10}^{-16}$ |

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**MDPI and ACS Style**

De Florio, M.; Schiassi, E.; D’Ambrosio, A.; Mortari, D.; Furfaro, R.
Theory of Functional Connections Applied to Linear ODEs Subject to Integral Constraints and Linear Ordinary Integro-Differential Equations. *Math. Comput. Appl.* **2021**, *26*, 65.
https://doi.org/10.3390/mca26030065

**AMA Style**

De Florio M, Schiassi E, D’Ambrosio A, Mortari D, Furfaro R.
Theory of Functional Connections Applied to Linear ODEs Subject to Integral Constraints and Linear Ordinary Integro-Differential Equations. *Mathematical and Computational Applications*. 2021; 26(3):65.
https://doi.org/10.3390/mca26030065

**Chicago/Turabian Style**

De Florio, Mario, Enrico Schiassi, Andrea D’Ambrosio, Daniele Mortari, and Roberto Furfaro.
2021. "Theory of Functional Connections Applied to Linear ODEs Subject to Integral Constraints and Linear Ordinary Integro-Differential Equations" *Mathematical and Computational Applications* 26, no. 3: 65.
https://doi.org/10.3390/mca26030065