8,959 research outputs found
An Application of Equivalence Transformations to Reaction Diffusion Equations
In this paper, we consider a quite general class of advection reaction diffusion systems. By using an equivalence generator, derived in a previous paper, the authors apply a projection theorem to determine some special forms of the constitutive functions that allow the extension by one of the two-dimensional principal Lie algebra. As an example, a special case is discussed at the end of the paper
Group classification of systems of non-linear reaction-diffusion equations with general diffusion matrix. I. Generalized Landau-Ginzburg equations
Group classification of the generalized complex Ginzburg-Landau equations is
presented. An approach to group classification of systems of reaction-diffusion
equations with general diffusion matrix is developed.Comment: The list of additional equivalence transformations is complete
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
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
Enhanced Group Analysis and Exact Solutions of Variable Coefficient Semilinear Diffusion Equations with a Power Source
A new approach to group classification problems and more general
investigations on transformational properties of classes of differential
equations is proposed. It is based on mappings between classes of differential
equations, generated by families of point transformations. A class of variable
coefficient (1+1)-dimensional semilinear reaction-diffusion equations of the
general form () is studied from the
symmetry point of view in the framework of the approach proposed. The singular
subclass of the equations with is singled out. The group classifications
of the entire class, the singular subclass and their images are performed with
respect to both the corresponding (generalized extended) equivalence groups and
all point transformations. The set of admissible transformations of the imaged
class is exhaustively described in the general case . The procedure of
classification of nonclassical symmetries, which involves mappings between
classes of differential equations, is discussed. Wide families of new exact
solutions are also constructed for equations from the classes under
consideration by the classical method of Lie reductions and by generation of
new solutions from known ones for other equations with point transformations of
different kinds (such as additional equivalence transformations and mappings
between classes of equations).Comment: 40 pages, this is version published in Acta Applicanda Mathematica
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
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