# A Complex Lie-Symmetry Approach to Calculate First Integrals and Their Numerical Preservation

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## Abstract

**:**

## 1. Introduction

## 2. Symmetries and First Integrals

#### Complex Symmetry Analysis

## 3. Runge–Kutta Methods

#### 3.1. Symplectic Runge–Kutta Methods

#### 3.2. Construction of Symplectic RK Methods

**Gauss, s = 2:**

**Radau I, s = 2:**

**Radau II, s = 2:**

## 4. Construction of First Integrals and Their Numerical Preservation

**Case I**:

**(${\mathit{k}}^{\mathbf{2}}=\mathbf{1}$ and $\mathit{y}$ is real)**

**Case II**:

**(${\mathit{k}}^{\mathbf{2}}=\mathbf{1}$ and $\mathit{y}$ is complex)**

**Case III**:

**($\mathit{k}$ and $\mathit{y}$ are complex)**

## 5. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

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## Share and Cite

**MDPI and ACS Style**

Irshad, W.; Habib, Y.; Farooq, M.U.
A Complex Lie-Symmetry Approach to Calculate First Integrals and Their Numerical Preservation. *Symmetry* **2019**, *11*, 11.
https://doi.org/10.3390/sym11010011

**AMA Style**

Irshad W, Habib Y, Farooq MU.
A Complex Lie-Symmetry Approach to Calculate First Integrals and Their Numerical Preservation. *Symmetry*. 2019; 11(1):11.
https://doi.org/10.3390/sym11010011

**Chicago/Turabian Style**

Irshad, Wajeeha, Yousaf Habib, and Muhammad Umar Farooq.
2019. "A Complex Lie-Symmetry Approach to Calculate First Integrals and Their Numerical Preservation" *Symmetry* 11, no. 1: 11.
https://doi.org/10.3390/sym11010011