Lie symmetries of generalized Burgers equations: application to boundary-value problems

Abstract: There exist several approaches exploiting Lie symmetries in the reduction of boundary-value problems for partial differential equations modelling real-world phenomena to those problems for ordinary differential equations. Using an example of generalized Burgers equations appearing in nonlinear acoustics we show that that the "direct" procedure of solving boundary-value problems using Lie symmetries firstly described by Bluman is more general and straightforward than the method suggested by Moran and Gaggioli in [J. Eng. Math. 3 (1969), 151-162]. After the group classification of a class of generalized Burgers equations with time-dependent viscosity is performed we solve an associated boundary-value problem using the symmetries obtained.

• The paper presents a Lie symmetry analysis that classifies generalized Burgers equations and applies the findings to solve boundary-value problems.
• It employs equivalence transformations and a direct approach to derive invariance conditions and reduce the complex PDEs to simpler ODEs.
• The results demonstrate the efficiency of Lie symmetry methods over traditional techniques, such as those by Moran and Gaggioli.

Lie Symmetries of Generalized Burgers Equations

This paper (1303.3548) presents an analysis of Lie symmetries in generalized Burgers equations, focusing on their application to boundary-value problems (BVPs). The authors demonstrate the efficacy of a "direct" approach, initially proposed by Bluman, for solving BVPs using Lie symmetries, contrasting it with the method outlined by Moran and Gaggioli. The paper includes a group classification of generalized Burgers equations with time-dependent viscosity and solves an associated BVP utilizing the derived symmetries.

Equivalence Transformations of Generalized Burgers Equations

The study investigates equivalence transformations within the class of generalized Burgers equations:

$u_t+a(u^n)_x=g(t)u_{xx}$,

where $a$ is a nonzero constant, $g$ is a smooth nonvanishing function of $t$, and $n \neq 0, 1$. The paper identifies the usual equivalence group $G^{\sim}$ for this class, which involves transformations of the independent and dependent variables, as well as the arbitrary functions $a$ and $g$.

Theorem 1 The usual equivalence group~$G^{\sim}$ of class~\eqref{Eq_GenBurgers} comprises the transformations

$\begin{array}{l} \tilde t=\delta_1t+\delta_2,\quad \tilde x=\delta_3x+\delta_4,\quad \tilde u=\delta_5u, \quad \tilde a=\dfrac{\delta_3}{\delta_1}\delta_5^{1-n}a, \quad \tilde g=\dfrac{\delta_3}^2}{\delta_1} g,\quad \tilde n=n, \end{array}$

where $\delta_j,$ $j=1,\dots,5$, are arbitrary constants with $\delta_1\delta_3\delta_5\not=0$.

A conditional equivalence group is found to exist when $n = 2$, denoted as $\hat G^{\sim}_{2}$, which is broader than $G^{\sim}$.

Theorem 2 The generalized equivalence group~$\hat G<sup>{\sim}_{2}$ of the class, \begin{equation}\label{Eq_GenBurgers_n2} u_t+a(u^2)x=g(t)u{xx}, \end{equation} consists of the transformations

$\begin{array}{l} \tilde t=\dfrac{\alpha t+\beta}{\gamma t+\delta},\quad \tilde x=\dfrac{\kappa x +\mu_1t+\mu_0}{\gamma t+\delta},\quad \tilde u=\dfrac{\sigma}{2a(\alpha\delta-\beta\gamma)}\left(2a\kappa(\gamma t+\delta)u-\kappa\gamma x+\mu_1\delta-\mu_0\gamma\right), \[2ex] \tilde a=\dfrac{a}{\sigma} \quad \mbox{\rm and} \quad \tilde g=\dfrac{\kappa^2}{\alpha\delta-\beta\gamma} g, \end{array}$

where $$\alpha, \beta, \gamma, \delta, \kappa, \mu_1, \mu_0, \sigma$$ are constants defined up to a nonzero multiplier, $\alpha\delta-\beta\gamma\neq0$ and $\kappa\sigma\not=0$.

The authors prove that all admissible point transformations within the class of generalized Burgers equations are exhausted by the transformations in $G^{\sim}$ when $n \neq 2$ and in $\hat G^{\sim}_{2}$ when $n = 2$.

Theorem 3 Let two equations from class~\eqref{Eq_GenBurgers}, $u_t+a(u^n)_x =g(t)u_{xx}$ and\, $$\tilde u_{\tilde t}+\tilde a(\tilde u^{\tilde n})_{\tilde x} =\tilde g(\tilde t)\tilde u_{\tilde x\tilde x}$$, be connected by a point transformation $\mathcal{T}$ in the variables~$t$, $x$ and~$u$. Then the transformation $\mathcal{T}$ is the projection on the space $(t,x,u)$ of a transformation from the group $G^{\sim}$ if $n\not=2$, or from the group $\hat G^{\sim}_{2}$, if $n=2$.

Lie Symmetry Group Classification

The Lie symmetry group classification of the generalized Burgers equation is performed using the classical Lie approach. The classification is conducted up to $G^{\sim}$-equivalence for the general class and up to $\hat G^{\sim}_{2}$-equivalence for the subclass with $n = 2$. The determining equations for the symmetry operators are derived, and the resulting classification is presented in Table 1 of the paper. The kernel of the maximal Lie invariance algebras is found to be one-dimensional for $n \neq 2$ and two-dimensional for $n = 2$.

Theorem 4

The kernel of the maximal Lie invariance algebras of equations from class~\eqref{Eq_GenBurgers} with $n\neq2$ coincides with the one-dimensional algebra $\langle\partial_x\rangle$. All possible $G^\sim$-non-equivalent cases of extension of the maximal Lie invariance algebras are exhausted by the cases 2--4 of Table~1.

Theorem 5

The kernel of the maximal Lie invariance algebras of equations from class~\eqref{Eq_GenBurgers_n2} coincides with the two-dimensional Abelian algebra $$\langle\partial_x,\,2at\partial_x+\partial_u\rangle$$. All possible $\hat G^\sim_2$-non-equivalent cases of extension of the maximal Lie invariance algebras are exhausted by the cases 6--9 of Table~1.

Application to Boundary-Value Problems

The paper addresses the application of Lie symmetries to solve a specific class of BVPs for the generalized Burgers equation. The "direct" approach, as advocated by Bluman, involves first deriving the Lie symmetries of the PDE and then verifying the invariance of the boundary conditions under the action of these symmetries. This approach is compared with the method suggested by Moran and Gaggioli, which is deemed less straightforward.

The authors analyze the invariance of the BVP under the symmetries derived in the group classification. For a specific case where $g(t) = \varepsilon t^{\rho}$, the Lie symmetry $$2t\partial_t+(\rho+1)x\partial_x+\left((\rho-1)/(n-1)\right)u\partial_u$$ is used to reduce the BVP to an ODE. The solution of the reduced ODE is then used to construct a solution for the original BVP. The solution has the form \begin{equation}\label{sol_u} u=t^{-\frac1n}\exp\left[-\frac{1}{2\varepsilon n}x^2 t^{-\frac2n}\right]\left(\gamma^{1-n} +\frac{a(1-n)\sqrt{\pi}{2\varepsilon\sigma}\operatorname{erf}(\sigma xt^{-\frac{1}n})\right)^{\frac1{1-n},\quad \sigma=\sqrt{\frac{n-1}{2n\varepsilon}.\end{equation}

Conclusion

The paper concludes by asserting that the "direct" technique, which utilizes Lie symmetries of PDEs, is more straightforward than other methods, such as the one suggested by Moran and Gaggioli, for solving BVPs. The authors perform a group classification for the class of variable-coefficient generalized Burgers equations and present the results in a compact form using equivalence transformations.