Continuous and discrete transformations of a one-dimensional porous medium equation

We consider the one-dimensional porous medium equation $u_t=\left (u^nu_x \right )_x+\frac{\mu}{x}u^nu_x$. We derive point transformations of a general class that map this equation into itself or into equations of a similar class. In some cases this porous medium equation is connected with well known equations. With the introduction of a new dependent variable this partial differential equation can be equivalently written as a system of two equations. Point transformations are also sought for this auxiliary system. It turns out that in addition to the continuous point transformations that may be derived by Lie's method, a number of discrete transformations are obtained. In some cases the point transformations which are presented here for the single equation and for the auxiliary system form cyclic groups of finite order

On the group classification of variable-coefficient nonlinear diffusion–convection equations

AbstractWe consider the variable coefficient diffusion–convection equation of the form f(x)ut=[g(x)D(u)ux]x+h(x)K(u)ux which has considerable interest in mathematical physics, biology and chemistry. We present a complete group classification for this class of equations. Also we derive equivalence transformations between equations that admit Lie symmetries. Furthermore, we obtain mappings that connect variable and constant coefficient equations. Exact solutions of special forms of this equations are constructed using Lie symmetries and equivalence transformations

Group Analysis of Nonlinear Fin Equations

Group classification of a class of nonlinear fin equations is carried out exhaustively. Additional equivalence transformations and conditional equivalence groups are also found. They allow to simplify results of classification and further applications of them. The derived Lie symmetries are used to construct exact solutions of truly nonlinear equations for the class under consideration. Nonclassical symmetries of the fin equations are discussed. Adduced results amend and essentially generalize recent works on the subject [M. Pakdemirli and A.Z. Sahin, Appl. Math. Lett., 2006, V.19, 378-384; A.H. Bokhari, A.H. Kara and F.D. Zaman, Appl. Math. Lett., 2006, V.19, 1356-1340].Comment: 6 page