47,425 research outputs found
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
Exact Solutions of a Remarkable Fin Equation
A model "remarkable" fin equation is singled out from a class of nonlinear
(1+1)-dimensional fin equations. For this equation a number of exact solutions
are constructed by means of using both classical Lie algorithm and different
modern techniques (functional separation of variables, generalized conditional
symmetries, hidden symmetries etc).Comment: 6 page
Reaction-diffusion systems with constant diffusivities: conditional symmetries and form-preserving transformations
Q-conditional symmetries (nonclassical symmetries) for a general class of
two-component reaction-diffusion systems with constant diffusivities are
studied. Using the recently introduced notion of Q-conditional symmetries of
the first type (R. Cherniha J. Phys. A: Math. Theor., 2010. vol. 43., 405207),
an exhaustive list of reaction-diffusion systems admitting such symmetry is
derived. The form-preserving transformations for this class of systems are
constructed and it is shown that this list contains only non-equivalent
systems. The obtained symmetries permit to reduce the reaction-diffusion
systems under study to two-dimensional systems of ordinary differential
equations and to find exact solutions. As a non-trivial example, multiparameter
families of exact solutions are explicitly constructed for two nonlinear
reaction-diffusion systems. A possible interpretation to a biologically
motivated model is presented
Potential Nonclassical Symmetries and Solutions of Fast Diffusion Equation
The fast diffusion equation is investigated from the
symmetry point of view in development of the paper by Gandarias [Phys. Lett. A
286 (2001) 153-160]. After studying equivalence of nonclassical symmetries with
respect to a transformation group, we completely classify the nonclassical
symmetries of the corresponding potential equation. As a result, new wide
classes of potential nonclassical symmetries of the fast diffusion equation are
obtained. The set of known exact non-Lie solutions are supplemented with the
similar ones. It is shown that all known non-Lie solutions of the fast
diffusion equation are exhausted by ones which can be constructed in a regular
way with the above potential nonclassical symmetries. Connection between
classes of nonclassical and potential nonclassical symmetries of the fast
diffusion equation is found.Comment: 13 pages, section 3 is essentially revise
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
Potential equivalence transformations for nonlinear diffusion-convection equations
Potential equivalence transformations (PETs) are effectively applied to a
class of nonlinear diffusion-convection equations. For this class all possible
potential symmetries are classified and a theorem on connection of them with
point ones via PETs is also proved. It is shown that the known non-local
transformations between equations under consideration are nothing but PETs.
Action of PETs on sets of exact solutions of a fast diffusion equation is
investigated.Comment: 10 page
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