# Nonlocal Symmetries for Time-Dependent Order Differential Equations

## Abstract

**:**

## 1. Introduction

## 2. Time-Dependent Ordinary Differential Equation

## 3. Initial Value Problem and Non-Local Symmetries

## 4. Conclusions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Plot of the solution for the time-dependent order initial problem for the differential equation ${D}^{\alpha \left(t\right)}y=-y$ for $\alpha \in (1,2),t\in (0,1)$. The solution smoothly deforms from exponential decay to trigonometric function, with the increase of $\alpha $.

**Figure 2.**Plot of the solution for the time-dependent order differential equation ${D}^{\alpha \left(t\right)}y=-{t}^{-3/2}$ for $\alpha \in (1,2),t\in (0,1)$. The solution to Equation (8) smoothly deforms from a singular hyperbolic dependence $2/\sqrt{t},{C}_{0}=0$ to a smooth power low $4\sqrt{t}+t,{C}_{0}=0,{C}_{1}=1$ when $\alpha $ increases from 1 to 2. The initial value problem cannot be applied here in the traditional sense, because of the singularity at $t=0$.

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Ludu, A.
Nonlocal Symmetries for Time-Dependent Order Differential Equations. *Symmetry* **2018**, *10*, 771.
https://doi.org/10.3390/sym10120771

**AMA Style**

Ludu A.
Nonlocal Symmetries for Time-Dependent Order Differential Equations. *Symmetry*. 2018; 10(12):771.
https://doi.org/10.3390/sym10120771

**Chicago/Turabian Style**

Ludu, Andrei.
2018. "Nonlocal Symmetries for Time-Dependent Order Differential Equations" *Symmetry* 10, no. 12: 771.
https://doi.org/10.3390/sym10120771