# New Nonlocal Symmetries of Diffusion-Convection Equations and Their Connection with Generalized Hodograph Transformation

## Abstract

**:**

## 1. Introduction

## 2. Lie Symmetries of Given and Intermediate Equations

## 3. Nonlocal Symmetries Generated by Additional Lie Symmetries of the Intermediate Equations

#### 3.1. The Operator X 7th Case

**Theorem 1.**The characteristic Equation (27) determines nonlocal transformation with additional variables, which connects Equations (4) and (7) taking into account Equation (21) and condition Equation (20). Additional independent variable $v(x,t)$ is determined by Equation (7) and additional dependent variable $b(v,t)$ is an arbitrary solution of the Equation (21).

**Theorem 2.**The Equation (4) is invariant under action of the prolonged operator ${X}_{7}^{\star}$ on the manifold, determined by Equation (7), condition Equation (20) and the Equation (21) with their differential consequences.

**Theorem 3.**The operator Equation (28) allows one to construct the associated transformation determined by the formulas

#### 3.2. The Operator X 4th Case

## 4. Nonlocal Invariance Transformations and Generation of Solutions

**Theorem 4.**Equation (4) is nonlocal-invariant under the transformations

## 5. Conclusions

## Acknowledgments

## Conflicts of Interest

## References

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Tychynin, V.
New Nonlocal Symmetries of Diffusion-Convection Equations and Their Connection with Generalized Hodograph Transformation. *Symmetry* **2015**, *7*, 1751-1767.
https://doi.org/10.3390/sym7041751

**AMA Style**

Tychynin V.
New Nonlocal Symmetries of Diffusion-Convection Equations and Their Connection with Generalized Hodograph Transformation. *Symmetry*. 2015; 7(4):1751-1767.
https://doi.org/10.3390/sym7041751

**Chicago/Turabian Style**

Tychynin, Valentyn.
2015. "New Nonlocal Symmetries of Diffusion-Convection Equations and Their Connection with Generalized Hodograph Transformation" *Symmetry* 7, no. 4: 1751-1767.
https://doi.org/10.3390/sym7041751