Institute of Mathematics of NAS of Ukraine,

3 Tereshchenkivs'ka Str.,

01601 Kyiv,

UKRAINE

and

Department of Mathematics and Statistics,

University of Cyprus,

P. O. Box 20537,

1678 Nicosia,

CYPRUS

E-mail: ivanova@imath.kiev.ua, rop@imath.kiev.ua, christod@ucy.ac.cy

**Enhanced group classification and conservation laws
of variable coefficient diffusion-convection equations**

**Abstract:**

A class of variable coefficient (1+1)-dimensional nonlinear diffusion-convection
equations is investigated. At first, we construct the usual equivalence
group and the extended one including transformations which are nonlocal
with respect to arbitrary elements. The extended equivalence group has
interesting structure since it contains a non-trivial subgroup of gauge
equivalence transformations. For the class under consideration the complete
group classification is performed with respect to extended equivalence
group and with respect to the set of all local transformations. Then, using
the most direct method, we carry out two classifications of local conservation
laws up to equivalence relations generated by both usual and enhanced equivalence
groups. Equivalence with respect to enhanced equivalence group and correct
choice of gauging coefficients of equations play the major role for simple
and clear formulation of the final results.