# Lie and Conditional Symmetries of a Class of Nonlinear (1 + 2)-Dimensional Boundary Value Problems

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## Abstract

**:**

## 1. Introduction

## 2. Theoretical Background: Definitions and Examples

**Definition 1.**[1] The Lie symmetry X (6) is admitted by the boundary value Problems (4)–(5) if:

- (a)
- $\stackrel{{\displaystyle X}}{{\scriptstyle k}}\left(F\left(x,u,{u}_{x},\dots ,{u}_{x}^{\left(k\right)}\right)-{u}_{t}\right)=0$ when u satisfies (4);
- (b)
- $X\left({s}_{a}(t,x)\right)=0$ when ${s}_{a}(t,x)=0,\phantom{\rule{4pt}{0ex}}a=1,\dots ,p$;
- (c)
- $\stackrel{{\displaystyle X}}{{\scriptstyle {k}_{a}}}\left({B}_{a}\left(t,x,u,{u}_{x},\dots ,{u}_{x}^{\left({k}_{a}\right)}\right)\right)=0$ when ${s}_{a}(t,x)=0$ and ${B}_{a}{|}_{{s}_{a}(t,x)=0}=0$, $a=1,\dots ,p$.

**Remark 1.**Rigorously speaking, one needs to reduce the manifold M by adding the differential consequences of equation $Q\left(u\right)=0$ up to order k, which leads to huge technical problems in the application of the criterion obtained. However, in the case of evolution equations, the resulting symmetries will be still the same provided ${\xi}^{0}(t,x,u)\ne 0$ in Q, because each such differential consequence contains one or more mixed derivatives of the function u w.r.t. the variables t and x, while the evolution equation in question does not involve any such mixed derivatives.

**Definition 2.**BVPs (4)–(5) and (8) are Q-conditionally invariant under operator (9) if:

- (a)
- Criterion (10) is satisfied;
- (b)
- $Q\left({s}_{a}(t,x)\right)=0$ when ${s}_{a}(t,x)=0,$ ${B}_{a}{|}_{{s}_{a}(t,x)=0}=0,$ $a=1,\dots ,p$;
- (c)
- $\stackrel{{\displaystyle Q}}{{\scriptstyle {k}_{a}}}\left({B}_{a}\left(t,x,u,{u}_{x},\dots ,{u}_{x}^{\left({k}_{a}\right)}\right)\right)=0$ when ${s}_{a}(t,x)=0$ and ${B}_{a}{|}_{{s}_{a}(t,x)=0}=0,$ $a=1,\dots ,p$;
- (d)
- there exists a smooth bijective transform (12) mapping $\mathsf{M}$ into ${\mathsf{M}}^{*}$ of the same dimensionality;
- (e)
- ${Q}^{*}\left({\gamma}_{c}^{*}(\tau ,y)\right)=0$ when ${\gamma}_{c}^{*}(\tau ,y)=0$, $c=1,2,\dots ,{p}_{\infty}$;
- (f)
- $\stackrel{{\displaystyle {Q}^{*}}}{{\scriptstyle {k}_{c}^{*}}}\left({\gamma}_{c}^{*}\left(\tau ,y,u,{u}_{y},\dots ,{u}_{y}^{\left({k}_{c}^{*}\right)}\right)\right)=0\phantom{\rule{0.166667em}{0ex}}$ when ${\gamma}_{c}^{*}(\tau ,y)=0$ and ${\gamma}_{c}^{*}{|}_{{\gamma}_{c}^{*}(\tau ,y)=0}=0$, $c=1,\dots ,r$,

**Remark 2.**Because any Q-conditional symmetry operator can be multiplied by an arbitrary function, say ${s}_{a}(t,x)$, Definition 2 implies that the operator Q does not vanish provided ${s}_{a}(t,x)=0$. Rigorously speaking, this restriction is valid also for Definition 1.

**Example 1.**Consider the nonlinear BVP modelling heat transfer in a semi-infinite solid rod, assuming that thermal diffusivity depends on temperature and that the rod is insulated at the left endpoint. Hereafter, we neglect the initial distribution of the temperature in the rod. Thus, the nonlinear BVP reads as:

**Example 2.**Consider the reaction-diffusion-convectionequation:

## 3. Lie Symmetry Classification of the BVPs Class (1)–(3)

- (i)
- The Lie symmetry algebras are the maximal algebras of invariance of the relevant PDEs from the list obtained;
- (ii)
- All PDEs from the list are inequivalent with respect to a set of transformations, which are explicitly (or implicitly) presented and, generally speaking, may not form any group;
- (iii)
- Any other PDE from the class that admits a non-trivial Lie symmetry algebra is reduced by transformations from the set to one of those from the list.

**Lemma 1.**The equivalence group ${E}_{\mathrm{eq}}$ of the PDE class (1) is formed by the transformations:

**Lemma 2.**The equivalence group ${E}_{\mathrm{eq}}^{\mathrm{BVP}}$ of the class of BVPs (1)–(3) is formed by the transformations:

**Theorem 1.**All possible MAIs (up to the equivalent transformations from the group ${E}_{\mathrm{eq}}^{\mathrm{BVP}}$) of Equation (1) for any fixed non-negative function $d\left(u\right)\ne const$ are presented in Table 1. Any other equation of the form (1) is reduced by an equivalence transformation from the group ${E}_{\mathrm{eq}}^{\mathrm{BVP}}$ to one of those given in Table 1.

**Table 1.**Result of group classification of the class of PDEs (1). MAI, maximal algebra of invariance.

Case | $d\left(u\right)$ | Basic Operators of MAI |
---|---|---|

1. | ∀ | $AE(1,2)=\langle T,\phantom{\rule{0.166667em}{0ex}}{X}_{1},\phantom{\rule{0.166667em}{0ex}}{X}_{2},\phantom{\rule{0.166667em}{0ex}}D,\phantom{\rule{0.166667em}{0ex}}{J}_{12}\rangle $ |

2. | ${u}^{k},k\ne 0,-1$ | $AE(1,2),{D}_{k}$ |

3. | ${u}^{-1}$ | $AE(1,2),$ |

$A\left(x\right){\partial}_{{x}_{1}}+B\left(x\right){\partial}_{{x}_{2}}-2{A}_{{x}_{1}}u{\partial}_{u}$ | ||

4. | ${e}^{u}$ | $AE(1,2),{D}_{e}$ |

**Remark 3.**In Table 1, the following designations of the Lie symmetry operators are used:

**Theorem 2.**All possible MAIs (up to equivalent transformations from the group ${E}_{\mathrm{eq}}^{\mathrm{BVP}}$) of the nonlinear BVPs (1)–(3) for any fixed pair $\left(d\right(u),q(t\left)\right)$, where $d\left(u\right)\ne const$, are presented in Table 2. Any other BVP of the form (1)–(3) is reduced by an equivalence transformation from the group ${E}_{\mathrm{eq}}^{\mathrm{BVP}}$ from Lemma 2 to one of those listed in Table 2.

**Table 2.**Result of group classification of the class of BVPs (1)–(3).

Case | $d\left(u\right)$ | $q\left(t\right)$ | Basic Operators of MAI | Relevant Constraints |
---|---|---|---|---|

1. | ∀ | ∀ | ${X}_{1}$ | |

2. | ∀ | ${q}_{0}{t}^{-\frac{1}{2}}$ | ${X}_{1},\phantom{\rule{0.166667em}{0ex}}D$ | |

3. | ∀ | ${q}_{0}$ | ${X}_{1},\phantom{\rule{0.166667em}{0ex}}T$ | |

4. | ∀ | 0 | ${X}_{1},\phantom{\rule{0.166667em}{0ex}}T,\phantom{\rule{0.166667em}{0ex}}D$ | |

5. | ${u}^{k}$ | ${q}_{0}{t}^{p}$ | ${X}_{1},\phantom{\rule{0.166667em}{0ex}}{D}_{kp}$ | $k\ne -2,\phantom{\rule{4pt}{0ex}}p\ne 0$ |

6. | ${u}^{k}$ | ${q}_{0}{e}^{\pm t}$ | ${X}_{1},\phantom{\rule{0.166667em}{0ex}}{D}_{\pm}$ | $k\ne -2$ |

7. | ${u}^{k}$ | ${q}_{0}$ | ${X}_{1},\phantom{\rule{0.166667em}{0ex}}T,\phantom{\rule{0.166667em}{0ex}}{D}_{kp}$ | $p=0$ |

8. | ${u}^{k}$ | 0 | ${X}_{1},\phantom{\rule{0.166667em}{0ex}}T,\phantom{\rule{0.166667em}{0ex}}D,\phantom{\rule{4pt}{0ex}}{D}_{k}$ | $k\ne -1$ |

9. | ${u}^{-2}$ | ∀ | ${X}_{1},\phantom{\rule{0.166667em}{0ex}}{D}_{\pm}$ | $k=-2$ |

10. | ${u}^{-2}$ | ${q}_{0}{t}^{-\frac{1}{2}}$ | ${X}_{1},\phantom{\rule{0.166667em}{0ex}}D,\phantom{\rule{0.166667em}{0ex}}{D}_{k}$ | $k=-2$ |

11. | ${u}^{-1}$ | 0 | ${X}_{1},\phantom{\rule{0.166667em}{0ex}}T,\phantom{\rule{0.166667em}{0ex}}D,\phantom{\rule{0.166667em}{0ex}}{X}^{\infty}$ | $\mathcal{M}$ |

12. | ${e}^{u}$ | ${q}_{0}{t}^{p}$ | ${X}_{1},\phantom{\rule{0.166667em}{0ex}}{D}_{p}$ | $p\ne 0$ |

13. | ${e}^{u}$ | ${q}_{0}{e}^{\pm t}$ | ${X}_{1},\phantom{\rule{0.166667em}{0ex}}{D}_{\pm e}$ | |

14. | ${e}^{u}$ | ${q}_{0}$ | ${X}_{1},\phantom{\rule{0.166667em}{0ex}}T,\phantom{\rule{0.166667em}{0ex}}{D}_{p}$ | $p=0$ |

15. | ${e}^{u}$ | 0 | ${X}_{1},\phantom{\rule{0.166667em}{0ex}}T,\phantom{\rule{0.166667em}{0ex}}D,\phantom{\rule{4pt}{0ex}}{D}_{e}$ |

**Remark 4.**In Table 2 the arbitrary constant ${q}_{0}\ne 0$ and the following designations of the Lie symmetry operators are used:

**Proof.**The proof is based at Definition 2, Lemma 2 and Theorem 1. According to the algorithm described above (see Steps (IV) and (V)), we need to examine the four different cases listed in Table 1. First of all, we should consider Case 1 with the aim to find the principal algebra of invariance, i.e., the invariance algebra, admitting by each BVP of the form (1)–(3). Taking the most general form of the Lie symmetry in this case, one obtains:

- (i)
- If $q\left(t\right)$ is an arbitrary function, then ${\lambda}_{0}={\lambda}_{3}=0$, i.e., $X={X}_{1}$;
- (ii)
- If $q\left(t\right)={q}_{0}/\sqrt{t+{\lambda}_{0}^{*}}$ with ${\lambda}_{0}^{*}={\lambda}_{0}/\left(2{\lambda}_{3}\right)$, then $X={\lambda}_{0}T+{\lambda}_{1}{X}_{1}+{\lambda}_{3}D$ (here, ${\lambda}_{0}$ and ${\lambda}_{3}\ne 0$ are no longer arbitrary);
- (iii)
- If $q\left(t\right)={q}_{0}$, ${q}_{0}$ being a constant, then ${\lambda}_{3}=0$, i.e., $X={\lambda}_{0}{\partial}_{t}+{\lambda}_{2}{\partial}_{{x}_{2}}$.

**Example 3.**The complex function ${z}^{-1}={({x}_{1}+i{x}_{2})}^{-1}$ generates the operator:

## 4. Lie Symmetry Reduction of Some BVPs of the Form (1)–(3)

- (i)
- $k\ne -1,\phantom{\rule{4pt}{0ex}}-2;$
- (ii)
- $k=-2;$
- (iii)
- $k=-1.$

**Remark 5.**The reduced BVPs (75)–(77) were derived under assumption ${x}_{1}>0$. In the case ${x}_{1}<0$, the same problem is obtained, but $z\to +\infty $ should be replaced by $z\to -\infty $.

**Example 4.**Here, we present in Figure 1 a result of numerical simulations of (75)–(77). Note that we take the initial profile ${\mathrm{\varphi}}_{0}\left(z\right)$ to agree the with boundary Conditions (76)–(77), namely $\mathrm{\varphi}(0,z)={\mathrm{\varphi}}_{0}\left(z\right)\equiv -\frac{1}{{q}_{0}(z+{z}_{0})}$, where ${z}_{0}>0,\phantom{\rule{4pt}{0ex}}{q}_{0}<0$ are arbitrary constants. It is appropriate to touch on the large-time behaviour of BVPs (75)–(77). This is best done in polar coordinates. From the symmetry point of view, it means that one uses the ansatz:

## 5. Conditional Symmetry Classification of the BVPs Class (1)–(3)

**Figure 1.**Numerical solution of BVPs (75)–(77) for $\mathrm{\varphi}(0,z)={\mathrm{\varphi}}_{0}\left(z\right)\equiv -\frac{1}{{q}_{0}(z+{z}_{0})}$ and ${z}_{0}=1.0,{q}_{0}=-0.2$.

**Remark 6.**Because each conditional symmetry operator (6) multiplied by an arbitrary smooth function M is again a conditional symmetry, we have examined also the operator $M(t,{x}_{1},{x}_{2},u)Q$ and show that no further results are obtained.

## 6. Some Remarks about the Domain Geometry

## 7. Conclusions

## Acknowledgements

## Author Contributions

## Conflicts of Interest

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Cherniha, R.; King, J.R.
Lie and Conditional Symmetries of a Class of Nonlinear (1 + 2)-Dimensional Boundary Value Problems. *Symmetry* **2015**, *7*, 1410-1435.
https://doi.org/10.3390/sym7031410

**AMA Style**

Cherniha R, King JR.
Lie and Conditional Symmetries of a Class of Nonlinear (1 + 2)-Dimensional Boundary Value Problems. *Symmetry*. 2015; 7(3):1410-1435.
https://doi.org/10.3390/sym7031410

**Chicago/Turabian Style**

Cherniha, Roman, and John R. King.
2015. "Lie and Conditional Symmetries of a Class of Nonlinear (1 + 2)-Dimensional Boundary Value Problems" *Symmetry* 7, no. 3: 1410-1435.
https://doi.org/10.3390/sym7031410