# On the Method of Transformations: Obtaining Solutions of Nonlinear Differential Equations by Means of the Solutions of Simpler Linear or Nonlinear Differential Equations

^{1}

^{2}

## Abstract

**:**

## 1. Introduction

## 2. Transformations of Linear Equations and Exact Solutions to the Corresponding Nonlinear Equations

#### 2.1. Transformations for the Wave Equation

**Proposition**

**1.**

**Proof.**

**Proposition**

**2.**

**Proof.**

**Proposition**

**3.**

**Proof.**

**Proposition**

**4.**

**Proof.**

**Proposition**

**5.**

**Proof.**

**Proposition**

**6.**

**Proof.**

#### 2.2. Transformations for the Heat Equation

**Proposition**

**7.**

**Proof.**

**Proposition**

**8.**

**Proof.**

**Proposition**

**9.**

**Proof.**

**Proposition**

**10.**

**Proof.**

**Proposition**

**11.**

**Proof.**

**Proposition**

**12.**

**Proof.**

#### 2.3. Transformation for the Laplace Equation

**Proposition**

**13.**

**Proof.**

**Proposition**

**14.**

**Proof.**

**Proposition**

**15.**

## 3. Transformations of Linear and Nonlinear Ordinary Differential Equations

**Proposition**

**16.**

**Proof.**

**Proposition**

**17.**

**Proof.**

**Proposition**

**18.**

**Proof.**

**Proposition**

**19.**

**Proof.**

**Proposition**

**20.**

**Proof.**

**Proposition**

**21.**

**Proof.**

**Proposition**

**22.**

**Proof.**

**Proposition**

**23.**

**Proof.**

**Proposition**

**24.**

## 4. Transformations of Nonlinear Partial Differential Equations

**Proposition**

**25.**

**Proof.**

**Proposition**

**26.**

**Proof.**

**Proposition**

**27.**

**Proof.**

## 5. Concluding Remarks

## Funding

## Conflicts of Interest

## Appendix A. Linear Differential Equations and Their Solutions Used in the Main Text

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**Figure 8.**Illustration of the soliton properties of the waves connected to the bisoliton solution of (127). The parameters of the solution are $c=3.2$, $\alpha =1/2$.

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**MDPI and ACS Style**

Vitanov, N.K.
On the Method of Transformations: Obtaining Solutions of Nonlinear Differential Equations by Means of the Solutions of Simpler Linear or Nonlinear Differential Equations. *Axioms* **2023**, *12*, 1106.
https://doi.org/10.3390/axioms12121106

**AMA Style**

Vitanov NK.
On the Method of Transformations: Obtaining Solutions of Nonlinear Differential Equations by Means of the Solutions of Simpler Linear or Nonlinear Differential Equations. *Axioms*. 2023; 12(12):1106.
https://doi.org/10.3390/axioms12121106

**Chicago/Turabian Style**

Vitanov, Nikolay K.
2023. "On the Method of Transformations: Obtaining Solutions of Nonlinear Differential Equations by Means of the Solutions of Simpler Linear or Nonlinear Differential Equations" *Axioms* 12, no. 12: 1106.
https://doi.org/10.3390/axioms12121106