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Roman Cherniha

Publications and source records attributed to Roman Cherniha.

47 records · Page 3Linked to original sources
arXiv2012

Lie symmetries and reductions of multi-dimensional boundary value problems of the Stefan type

A new definition of Lie invariance for nonlinear multi-dimensional boundary value problems (BVPs) is proposed by the generalization of known definitions to much wider classes of BVPs. The class of (1+3)-dimensional nonlinear BVPs of the Stefan type, modeling the process of melting and evaporation of metals, is studied in detail. Using the definition proposed, the group classification problem for this class of BVPs is solved and some reductions (with physical meaning) to BVPs of lower dimensionality are made. Examples of how to construct exact solutions of the (1+3)-dimensional nonlinear BVP with the correctly-specified coefficients are presented.

math-ph↗
arXiv2011

New mathematical model for fluid-glucose-albumin transport in peritoneal dialysis

A mathematical model for fluid transport in peritoneal dialysis is constructed. The model is based on a three-component nonlinear system of two-dimensional partial differential equations for fluid, glucose and albumin transport with the relevant boundary and initial conditions. Non-constant steady-state solutions of the model are studied. The restrictions on the parameters arising in the model are established with the aim to obtain exact formulae for the non-constant steady-state solutions. As the result, the exact formulae for the fluid fluxes from blood to tissue and across the tissue were constructed together with two linear autonomous ODEs for glucose and albumin concentrations. The analytical results were checked for their applicability for the description of fluid-glucose-albumin transport during peritoneal dialysis.

math.AP↗
arXiv2010

Conditional symmetries and exact solutions of the diffusive Lotka-Volterra system

Q-conditional symmetries of the classical Lotka-Volterra system in the case of one space variable are completely described and a set of such symmetries in explicit form is constructed. The relevant non-Lie ansätze to reduce the classical Lotka-Volterra systems with correctly-specified coefficients to ODE systems and examples of new exact solutions are found. A possible biological interpretation of some exact solutions is presented.

math-ph↗
arXiv2010

Conditional symmetries for systems of PDEs: new definitions and their application for reaction-diffusion systems

New definitions of $Q$-conditional symmetry for systems of PDEs are presented, which generalize the standard notation of non-classical (conditional) symmetry. It is shown that different types of $Q$-conditional symmetry of a system generate a hierarchy of conditional symmetry operators. A class of two-component nonlinear RD systems is examined to demonstrate applicability of the definitions proposed and it is shown when different definitions of $Q$-conditional symmetry lead to the same operators.

math-ph↗
arXiv2010

Lie Symmetries and Exact Solutions of the Generalized Thin Film Equation

A symmetry group classification for fourth-order reaction-diffusion equations, allowing for both second-order and fourth-order diffusion terms, is carried out. The fourth order equations are treated, firstly, as systems of second-order equations that bear some resemblance to systems of coupled reaction-diffusion equations with cross diffusion, secondly, as systems of a second-order equation and two first-order equations. The paper generalizes the results of Lie symmetry analysis derived earlier for particular cases of these equations. Various exact solutions are constructed using Lie symmetry reductions of the reaction-diffusion systems to ordinary differential equations. The solutions include some unusual structures as well as the familiar types that regularly occur in symmetry reductions, namely self-similar solutions, decelerating and decaying traveling waves, and steady states.

math-ph↗
arXiv2010

The exotic conformal Galilei algebra and nonlinear partial differential equations

The conformal Galilei algebra (CGA) and the exotic conformal Galilei algebra (ECGA) are applied to construct partial differential equations (PDEs) and systems of PDEs, which admit these algebras. We show that there are no single second-order PDEs invariant under the CGA but systems of PDEs can admit this algebra. Moreover, a wide class of nonlinear PDEs exists, which are conditionally invariant under CGA. It is further shown that there are systems of non-linear PDEs admitting ECGA with the realisation obtained very recently in [D. Martelli and Y. Tachikawa, arXiv:0903.5184v2 [hep-th] (2009)]. Moreover, wide classes of non-linear systems, invariant under two different 10-dimensional subalgebras of ECGA are explicitly constructed and an example with possible physical interpretation is presented.

math-ph↗
arXiv2007

New conditional symmetries and exact solutions of nonlinear reaction-diffusion-convection equations. II

In the first part of this paper math-ph/0612078, a complete description of Q-conditional symmetries for two classes of reaction-diffusion-convection equations with power diffusivities is derived. It was shown that all the known results for reaction-diffusion equations with power diffusivities follow as particular cases from those obtained in math-ph/0612078 but not vise versa. In the second part the symmetries obtained in are successfully applied for constructing exact solutions of the relevant equations. In the particular case, new exact solutions of nonlinear reaction-diffusion-convection (RDC) equations arising in application and their natural generalizations are found.

math-ph↗
arXiv2004

On nonlinear partial differential equations with an infinite-dimensional conditional symmetry

The invariance of nonlinear partial differential equations under a certain infinite-dimensional Lie algebra A_N(z) in N spatial dimensions is studied. The special case A_1(2) was introduced in J. Stat. Phys. {\bf 75}, 1023 (1994) and contains the Schrödinger Lie algebra sch_1 as a Lie subalgebra. It is shown that there is no second-order equation which is invariant under the massless realizations of A_N(z). However, a large class of strongly non-linear partial differential equations is found which are conditionally invariant with respect to the massless realization of A_N(z) such that the well-known Monge-Ampere equation is the required additional condition. New exact solutions are found for some representatives of this class.

math-ph↗