2,406 research outputs found
Enhanced group analysis and conservation laws of variable coefficient reaction-diffusion equations with power nonlinearities
A class of variable coefficient (1+1)-dimensional nonlinear
reaction-diffusion equations of the general form
is investigated. Different kinds of
equivalence groups are constructed including ones with transformations which
are nonlocal with respect to arbitrary elements. For the class under
consideration the complete group classification is performed with respect to
convenient equivalence groups (generalized extended and conditional ones) and
with respect to the set of all point transformations. Usage of different
equivalences and coefficient gauges plays the major role for simple and clear
formulation of the final results. The corresponding set of admissible
transformations is described exhaustively. Then, using the most direct method,
we classify local conservation laws. Some exact solutions are constructed by
the classical Lie method.Comment: 23 pages, minor misprints are correcte
Group classification of variable coefficient quasilinear reaction-diffusion equations
The group classification of variable coefficient quasilinear
reaction-diffusion equations is carried out exhaustively.
This became possible due to usage of a conditional equivalence group found in
the course of the study of admissible point transformation within the class.Comment: 10 pages, submitted to the Proceedings of the XVII Geometrical
Seminar (September 3-8, 2012, Zlatibor, Serbia
Hierarchy of Conservation Laws of Diffusion--Convection Equations
We introduce notions of equivalence of conservation laws with respect to Lie
symmetry groups for fixed systems of differential equations and with respect to
equivalence groups or sets of admissible transformations for classes of such
systems. We also revise the notion of linear dependence of conservation laws
and define the notion of local dependence of potentials. To construct
conservation laws, we develop and apply the most direct method which is
effective to use in the case of two independent variables. Admitting
possibility of dependence of conserved vectors on a number of potentials, we
generalize the iteration procedure proposed by Bluman and Doran-Wu for finding
nonlocal (potential) conservation laws. As an example, we completely classify
potential conservation laws (including arbitrary order local ones) of
diffusion--convection equations with respect to the equivalence group and
construct an exhaustive list of locally inequivalent potential systems
corresponding to these equations.Comment: 24 page
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
Group Analysis of Variable Coefficient Diffusion-Convection Equations. I. Enhanced Group Classification
We discuss the classical statement of group classification problem and some
its extensions in the general case. After that, we carry out the complete
extended group classification for a class of (1+1)-dimensional nonlinear
diffusion--convection equations with coefficients depending on the space
variable. 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 non-local gauge equivalence transformations.
The complete group classification of the class under consideration is carried
out with respect to the extended equivalence group and with respect to the set
of all point transformations. Usage of extended equivalence and correct choice
of gauges of arbitrary elements play the major role for simple and clear
formulation of the final results. The set of admissible transformations of this
class is preliminary investigated.Comment: 25 page
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