# An Application of Equivalence Transformations to Reaction Diffusion Equations

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## Abstract

**:**

## 1. Introduction

## 2. On Equivalence Transformations and Their Calculation for the Class (1)

#### 2.1. Elements on Equivalence Transformations

#### 2.2. Calculation of Weak Equivalence Transformations

## 3. Symmetries for a Subclass of Advection Reaction Diffusion Systems

**Theorem 1.**The projection of the infinitesimal weak equivalence generator Y for the system (1) on the space $(x,t,u,v)$:

**Corollary 2.**The projection (29) of infinitesimal weak equivalence generator Y for the system (1) on the space $(x,t,u,v)$ is the infinitesimal symmetry generator corresponding to the principal Lie algebra of the class (28) if and only if ${\eta}^{i}=0,\phantom{\rule{0.166667em}{0ex}}{\mu}^{j}=0,\phantom{\rule{0.166667em}{0ex}}i=1,\phantom{\rule{0.166667em}{0ex}}2,\phantom{\rule{0.166667em}{0ex}}j=1,\phantom{\rule{0.166667em}{0ex}}2,\phantom{\rule{0.166667em}{0ex}}3$.

#### 3.1. On the Extensions of the ${L}_{\mathcal{P}}$

- $f={f}_{0}{u}^{r}$.
- $f={f}_{0}{u}^{s}$ and $s\ne r$.

- $f={f}_{0}{u}^{r}$In this case, from Equation (50), we have ${a}_{1}=0$. Moreover, by differentiating Equation (51) with respect to u, we have:$$\begin{array}{ccc}& & {\beta}^{\u2033}+{\beta}^{\prime}(r{{\Gamma}_{1}}^{\prime}-u{{\Gamma}_{1}}^{\u2033})=0.\hfill \end{array}$$$$\begin{array}{ccc}& & u{{\Gamma}_{1}}^{\u2033}-r{{\Gamma}_{1}}^{\prime}={\gamma}_{0},\hfill \end{array}$$$$\begin{array}{ccc}& & {\beta}^{\u2033}={\gamma}_{0}{\beta}^{\prime}.\hfill \end{array}$$
- (a)
- If $r\ne -1$, from Equation (54), we get:$$\begin{array}{ccc}& & {\Gamma}_{1}\left(u\right)=\frac{{c}_{1}{u}^{1+r}}{1+r}-\frac{{\gamma}_{0}u}{r}+{c}_{2}.\hfill \end{array}$$$$\begin{array}{ccc}& & \lambda (t,v)=-\frac{({c}_{2}+{\Gamma}_{2})(r+1){\beta}^{\prime}}{r{{\Gamma}_{2}}^{\prime}},\hfill \end{array}$$$$\begin{array}{c}\hfill \frac{{\beta}^{\prime}}{r{{\Gamma}_{2}}^{\prime 2}}{J}_{1}=0\end{array}$$$$\begin{array}{c}{J}_{1}\equiv h(1+r)({c}_{2}+{\Gamma}_{2}){{\Gamma}_{2}}^{\u2033}-(h(1+2r)-u{h}_{u}){{\Gamma}_{2}}^{\prime 2}+(1+r)({h}_{v}-{\gamma}_{0})({c}_{2}+{\Gamma}_{2}){{\Gamma}_{2}}^{\prime}.\end{array}$$
- i.
- If ${\gamma}_{0}\ne 0$, as from Equation (55), we have:$$\begin{array}{c}\hfill \beta \left(t\right)={b}_{0}+{b}_{1}{e}^{{\gamma}_{0}t},\end{array}$$$$\begin{array}{c}\hfill {X}_{3}={e}^{{\gamma}_{0}t}{\partial}_{t}-\frac{{\gamma}_{0}{e}^{{\gamma}_{0}t}}{r}u{\partial}_{u}-\frac{({c}_{2}+{\Gamma}_{2})(r+1){\gamma}_{0}{e}^{{\gamma}_{0}t}}{r{{\Gamma}_{2}}^{\prime}}{\partial}_{v}.\end{array}$$
- ii.
- If ${\gamma}_{0}=0$, as from Equation (55), we have:$$\begin{array}{c}\hfill \beta \left(t\right)={b}_{0}+{b}_{1}t,\end{array}$$$$\begin{array}{c}\hfill {X}_{3}=t{\partial}_{t}-\frac{u}{r}{\partial}_{u}-\frac{({c}_{2}+{\Gamma}_{2})(r+1)}{r{{\Gamma}_{2}}^{\prime}}{\partial}_{v}.\end{array}$$

- (b)
- If $r=-1$, from Equation (54), we get:$$\begin{array}{ccc}& & {\Gamma}_{1}\left(u\right)={c}_{1}ln\left(u\right)+{\gamma}_{0}u+{c}_{2}.\hfill \end{array}$$$$\begin{array}{ccc}& & \lambda (t,v)=-\frac{{c}_{1}{\beta}^{\prime}}{{{\Gamma}_{2}}^{\prime}},\hfill \end{array}$$$$\begin{array}{c}\hfill \frac{{\beta}^{\prime}}{{{\Gamma}_{2}}^{\prime 2}}{J}_{2}=0\end{array}$$$$\begin{array}{c}\hfill {J}_{2}\equiv h{c}_{1}{{\Gamma}_{2}}^{\u2033}-(h+u{h}_{u}){{\Gamma}_{2}}^{\prime 2}+{c}_{1}({h}_{v}-{\gamma}_{0}){{\Gamma}_{2}}^{\prime}.\end{array}$$
- i.
- If ${\gamma}_{0}\ne 0$, as from Equation (55), we have:$$\begin{array}{c}\hfill \beta \left(t\right)={b}_{0}+{b}_{1}{e}^{{\gamma}_{0}t},\end{array}$$$$\begin{array}{c}\hfill {X}_{3}={e}^{{\gamma}_{0}t}{\partial}_{t}+{\gamma}_{0}{e}^{{\gamma}_{0}t}u{\partial}_{u}-\frac{{c}_{1}{\gamma}_{0}{e}^{{\gamma}_{0}t}}{{{\Gamma}_{2}}^{\prime}}{\partial}_{v}.\end{array}$$
- ii.
- If ${\gamma}_{0}=0$, as from Equation (55), we have:$$\begin{array}{c}\hfill \beta \left(t\right)={b}_{0}+{b}_{1}t,\end{array}$$$$\begin{array}{c}\hfill {X}_{3}=t{\partial}_{t}-\frac{u}{r}{\partial}_{u}-\frac{{c}_{1}}{{{\Gamma}_{2}}^{\prime}}{\partial}_{v}.\end{array}$$

- $f={f}_{0}{u}^{s}$ and $s\ne r$In this case, from Equation (50), we have:$$\begin{array}{c}\hfill \beta \left(t\right)=\frac{{a}_{1}(2r-s)}{r-s}t+{b}_{0},\end{array}$$$$\begin{array}{c}\hfill {a}_{1}u{{\Gamma}_{1}}^{\prime}-{a}_{1}(1+2r-s)({\Gamma}_{1}+{\Gamma}_{2})-(r-s)\lambda {{\Gamma}_{2}}^{\prime}=0.\end{array}$$$$\begin{array}{c}\hfill {a}_{1}(u{{\Gamma}_{1}}^{\u2033}+(s-2r){{\Gamma}_{1}}^{\prime})=0.\end{array}$$$$\begin{array}{c}\hfill u{{\Gamma}_{1}}^{\u2033}+(s-2r){{\Gamma}_{1}}^{\prime}=0.\end{array}$$
- (a)
- If $s\ne 2r+1$, from Equation (75), we get:$$\begin{array}{c}\hfill {\Gamma}_{1}\left(u\right)={c}_{1}+{c}_{2}{u}^{1+2r-s}.\end{array}$$$$\begin{array}{ccc}& & \lambda (t,v)=\frac{{a}_{1}(s-2r-1)({c}_{1}+{\Gamma}_{2})}{(r-s){{\Gamma}_{2}}^{\prime}},\hfill \end{array}$$$$\begin{array}{c}\hfill \frac{{a}_{1}}{(r-s){{\Gamma}_{2}}^{\prime 2}}{J}_{3}=0\end{array}$$$$\begin{array}{c}\hfill {J}_{3}\equiv (1+2r-s)({c}_{1}+{\Gamma}_{2})(h{{\Gamma}_{2}}^{\u2033}+{h}_{v}{{\Gamma}_{2}}^{\prime})+(u{h}_{u}-h(1+4r-2s)){{\Gamma}_{2}}^{\prime 2}.\end{array}$$$$\begin{array}{c}\hfill {X}_{3}=x{\partial}_{x}+\frac{2r-s}{r-s}t{\partial}_{t}+\frac{1}{s-r}u{\partial}_{u}+\frac{(s-2r-1)({c}_{1}+{\Gamma}_{2})}{(r-s){{\Gamma}_{2}}^{\prime}}{\partial}_{v}.\end{array}$$
- (b)
- If $s=2r+1$, from Equation (75), we get:$$\begin{array}{c}\hfill {\Gamma}_{1}\left(u\right)={c}_{1}ln\left(u\right)+{c}_{2},\end{array}$$$$\begin{array}{c}\hfill \beta \left(t\right)=\frac{{a}_{1}}{r+1}t+{b}_{0}.\end{array}$$$$\begin{array}{ccc}& & \lambda (t,v)=-\frac{{c}_{1}{a}_{1}}{(r+1){{\Gamma}_{2}}^{\prime}},\hfill \end{array}$$$$\begin{array}{c}\hfill \frac{{a}_{1}}{(r+1){{\Gamma}_{2}}^{\prime 2}}{J}_{4}=0\end{array}$$$$\begin{array}{c}\hfill {J}_{4}\equiv h{c}_{1}{{\Gamma}_{2}}^{\u2033}-(h+u{h}_{u}){{\Gamma}_{2}}^{\prime 2}+{c}_{1}{h}_{v}{{\Gamma}_{2}}^{\prime}.\end{array}$$$$\begin{array}{c}\hfill {X}_{3}=x{\partial}_{x}+\frac{1}{r+1}t{\partial}_{t}+\frac{1}{r+1}u{\partial}_{u}-\frac{{c}_{1}}{(r+1){{\Gamma}_{2}}^{\prime}}{\partial}_{v}.\end{array}$$

- $f={f}_{0}{u}^{r}$ with $r\ne -1$, ${\Gamma}_{1}\left(u\right)=\frac{{c}_{1}{u}^{1+r}}{1+r}-\frac{{\gamma}_{0}u}{r}+{c}_{2}$ with ${\gamma}_{0}\ne 0$, the functions h and ${\Gamma}_{2}$ linked from the following relation:$$\begin{array}{c}\hfill h(1+r)({c}_{2}+{\Gamma}_{2}){{\Gamma}_{2}}^{\u2033}-(h(1+2r)-u{h}_{u}){{\Gamma}_{2}}^{\prime 2}+(1+r)({h}_{v}-{\gamma}_{0})({c}_{2}+{\Gamma}_{2}){{\Gamma}_{2}}^{\prime}=0.\end{array}$$
- $f={f}_{0}{u}^{r}$ with $r\ne -1$, ${\Gamma}_{1}\left(u\right)=\frac{{c}_{1}{u}^{1+r}}{1+r}+{c}_{2}$ and the functions h and ${\Gamma}_{2}$ linked from the following relation:$$\begin{array}{c}\hfill h(1+r)({c}_{2}+{\Gamma}_{2}){{\Gamma}_{2}}^{\u2033}-(h(1+2r)-u{h}_{u}){{\Gamma}_{2}}^{\prime 2}+(1+r)\left({h}_{v}\right)({c}_{2}+{\Gamma}_{2}){{\Gamma}_{2}}^{\prime}=0.\end{array}$$
- $f=\frac{{f}_{0}}{u}$, ${\Gamma}_{1}\left(u\right)={c}_{1}ln\left(u\right)+{\gamma}_{0}u+{c}_{2}$ with ${\gamma}_{0}\ne 0$ and the functions h and ${\Gamma}_{2}$ linked from the following relation:$$\begin{array}{c}\hfill h{c}_{1}{{\Gamma}_{2}}^{\u2033}-(h+u{h}_{u}){{\Gamma}_{2}}^{\prime 2}+{c}_{1}({h}_{v}-{\gamma}_{0}){{\Gamma}_{2}}^{\prime}=0.\end{array}$$
- $f=\frac{{f}_{0}}{u}$, ${\Gamma}_{1}\left(u\right)={c}_{1}ln\left(u\right)+{c}_{2}$ and the functions h and ${\Gamma}_{2}$ linked from the following relation:$$\begin{array}{c}\hfill h{c}_{1}{{\Gamma}_{2}}^{\u2033}-(h+u{h}_{u}){{\Gamma}_{2}}^{\prime 2}+{c}_{1}\left({h}_{v}\right){{\Gamma}_{2}}^{\prime}=0.\end{array}$$
- $f={f}_{0}{u}^{s}$ with $s\ne r,\phantom{\rule{0.277778em}{0ex}}2r+1$, ${\Gamma}_{1}\left(u\right)={c}_{1}+{c}_{2}{u}^{1+2r-s}$ and the functions h and ${\Gamma}_{2}$ linked from the following relation:$$\begin{array}{c}\hfill (1+2r-s)({c}_{1}+{\Gamma}_{2})(h{{\Gamma}_{2}}^{\u2033}+{h}_{v}{{\Gamma}_{2}}^{\prime})+(u{h}_{u}-h(1+4r-2s)){{\Gamma}_{2}}^{\prime 2}=0.\end{array}$$
- $f={f}_{0}{u}^{2r+1}$, ${\Gamma}_{1}\left(u\right)={c}_{1}ln\left(u\right)+{c}_{2}$ and the functions h and ${\Gamma}_{2}$ linked from the following relation:$$\begin{array}{c}\hfill h{c}_{1}{{\Gamma}_{2}}^{\u2033}-(h+u{h}_{u}){{\Gamma}_{2}}^{\prime 2}+{c}_{1}{h}_{v}{{\Gamma}_{2}}^{\prime}=0.\end{array}$$

#### 3.2. A Special Case

**Remark 1.**It is a simple matter to ascertain that the system (103) admits as the special solution:

## 4. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## References

- Cardile, V.; Torrisi, M.; Tracinà, R. On a reaction-diffusion system arising in the development of Bacterial Colonies. In Proceedings of the 10th International Conference in Modern Group Analysis, Larnaca, Cyprus, 24–31 October 2004; Volume 32, p. 38.
- Medvedev, G.S.; Kaper, T.J.; Kopell, N. A reaction diffusion system with periodic front Dynamics. SIAM J. Appl. Math.
**2000**, 60, 1601–1638. [Google Scholar] [CrossRef] - Torrisi, M.; Tracinà, R. On a class of reaction diffusion systems: Equivalence transformations and symmetries. In Asymptotic Methods in Nonlinear Wave Phenomena; Ruggeri, T., Sammartino, M., Eds.; World Science Publishing Co. Pte. Ltd.: Singapore, 2007; pp. 207–216. [Google Scholar]
- Torrisi, M.; Tracinà, R. Exact solutions of a reaction-diffusion system for Proteus Mirabilis bacterial colonies. Nonlinear Anal. Real World Appl.
**2011**, 12, 1865–1874. [Google Scholar] [CrossRef] - Cherniha, R.; Serov, M. Nonlinear systems of the burgers-type equations: Lie and Q-conditional symmetries, ansätze and solutions. J. Math. Anal. Appl.
**2003**, 282, 305–328. [Google Scholar] [CrossRef] - Cherniha, R.; Wilhelmsson, H. Symmetry and exact solution of heat-mass transfer equations in thermonuclear plasma. Ukr. Math. J.
**1996**, 48, 1434–1449. [Google Scholar] [CrossRef] - Cherniha, R.; Serov, M. Lie and non-Lie symmetries of nonlinear diffusion equations with convection term. Symmetry Nonlinear Math. Phys.
**1997**, 2, 444–449. [Google Scholar] - Cherniha, R.; Serov, M. Symmetries, ansätze and exact solutions of nonlinear second-order evolution equations with convection terms. Eur. J. Appl. Math.
**1998**, 9, 527–542. [Google Scholar] [CrossRef] - Cherniha, R.; Serov, M. Symmetries, ansätze and exact solutions of nonlinear second-order evolution equations with convection terms, II. Eur. J. Appl. Math.
**2006**, 17, 597–605. [Google Scholar] [CrossRef] - Ibragimov, N.H.; Torrisi, M.; Valenti, A. Preliminary group classification of equation v
_{tt}= f(x, v_{x})v_{xx}+ g(x, v_{x}). J. Math. Phys.**1991**, 32, 2988–2995. [Google Scholar] [CrossRef] - Ovsiannikov, L.V. Group Analysis of Differential Equations; Academic Press: New York, NY, USA, 1982. [Google Scholar]
- Freire, I.L.; Torrisi, M. Weak equivalence transformations for a class of models in biomathematics. Abstr. Appl. Anal.
**2014**. [Google Scholar] [CrossRef] - Freire, I.L.; Torrisi, M. Symmetry methods in mathematical modeling Aedes aegypti dispersal dynamics. Nonlinear Anal. Real World Appl.
**2013**, 14, 1300–1307. [Google Scholar] [CrossRef] - Freire, I.L.; Torrisi, M. Similarity solutions for systems arising from an Aedes aegypti model. Commun. Nonlinear Sci. Numer. Simul.
**2014**, 19, 872–879. [Google Scholar] [CrossRef] - Romano, V.; Torrisi, M. Application of weak equivalence transformations to a group analysis of a drift-diffusion model. J. Phys. A Math. Gen.
**1999**, 32, 7953–7963. [Google Scholar] [CrossRef] - Torrisi, M.; Tracinà, R. Equivalence transformations and symmetries for a heat conduction model. Int. J. Non-Linear Mech.
**1998**, 33, 473–487. [Google Scholar] [CrossRef] - Gazeau, J.P.; Winternitz, P. Symmetries of variable-coefficient Korteweg-de Vries equations. J. Math. Phys.
**1992**, 33, 4087–4102. [Google Scholar] [CrossRef] - Winternitz, P.; Gazeau, J.P. Allowed transformations and symmetry classes of variable coefficient Korteweg-de Vries equations. Phys. Lett. A
**1992**, 167, 246–250. [Google Scholar] [CrossRef] - Akhatov, I.S.H.; Gazizov, R.K.; Ibragimov, N.H. Nonlocal symmetries. Heuristic approach. J. Sov. Math.
**1991**, 55, 1401–1450. [Google Scholar] [CrossRef] - Lisle, I.G. Equivalence Transformation for Classes of Differential Equations. Ph.D. Thesis, University of British Columbia, Vancouver, BC, Canada, 1992. [Google Scholar]
- Khalique, C.M.; Mahomed, F.M.; Ntsime, B.P. Group classification of the generalized Emden-Fowler-type equation. Nonlinear Anal. Real World Appl.
**2009**, 10, 3387–3395. [Google Scholar] [CrossRef] - Ibragimov, N.H. CRC Handbook of Lie Group Analysis of Differential Equations; CRC Press: Boca Raton, FL, USA, 1996. [Google Scholar]
- Molati, M.; Khalique, C.M. Lie group classification of a generalized Lane–Emden Type system in two dimensions. J. Appl. Math.
**2012**. [Google Scholar] [CrossRef] - Torrisi, M.; Tracinà, R. Equivalence transformations for systems of first order quasilinear partial differential equations. In Modern Group Analysis VI: Developments in Theory, Computation and Application; New Age International(P) Ltd.: New Delhi, India, 1996; pp. 115–135. [Google Scholar]
- Torrisi, M.; Tracinà, R.; Valenti, A. Group analysis approach for a non linear differential system arising in diffusion phenomena. J. Math. Phys.
**1996**, 37, 4758–4767. [Google Scholar] [CrossRef] - Gambino, G.; Greco, A.M.; Lombardo, M.C. A group analysis via weak equivalence transformations for a model of tumour encapsulation. J. Phys. A
**2004**, 37, 3835–3846. [Google Scholar] [CrossRef] - Ibragimov, N.H.; Säfström, N. The equivalence group and invariant solutions of a tumour growth model. Commun. Nonlinear Sci. Num. Simul.
**2004**, 9, 61–69. [Google Scholar] [CrossRef] - Ibragimov, N.H.; Torrisi, M. A simple method for group analysis and its application to a model of detonation. J. Math. Phys.
**1992**, 33, 3931–3937. [Google Scholar] [CrossRef] - Maidana, N.A.; Yang, H.M. Describing the geographic spread of dengue disease by traveling waves. Math. Biosci.
**2008**, 215, 64–77. [Google Scholar] [CrossRef] [PubMed] - Takahashi, L.T.; Maidana, N.A.; Ferreira, W.C., Jr.; Pulino, P.; Yang, H.M. Mathematical models for the Aedes aegypti dispersal dynamics: Traveling waves by wing and wind. Bull. Math. Biol.
**2005**, 67, 509–528. [Google Scholar] [CrossRef] [PubMed] - Bacani, F.; Freire, I.L.; Maidana, N.A.; Torrisi, M. Modelagem para a dinâmica populacional do Aedes aegypti via simetrias de Lie. Proc. Ser. Braz. Soc. Appl. Comput. Math.
**2015**, 3. [Google Scholar] [CrossRef]

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Torrisi, M.; Tracinà, R.
An Application of Equivalence Transformations to Reaction Diffusion Equations. *Symmetry* **2015**, *7*, 1929-1944.
https://doi.org/10.3390/sym7041929

**AMA Style**

Torrisi M, Tracinà R.
An Application of Equivalence Transformations to Reaction Diffusion Equations. *Symmetry*. 2015; 7(4):1929-1944.
https://doi.org/10.3390/sym7041929

**Chicago/Turabian Style**

Torrisi, Mariano, and Rita Tracinà.
2015. "An Application of Equivalence Transformations to Reaction Diffusion Equations" *Symmetry* 7, no. 4: 1929-1944.
https://doi.org/10.3390/sym7041929