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Roman Cherniha

Publications and source records attributed to Roman Cherniha.

At least 19 recordsLinked to original sources
arXiv2026

Nonlinear systems of PDEs admitting infinite-dimensional Lie algebras and their connection with Ricci flows. II: The two-dimensional space case

The work is a natural continuation of that published in Stud Appl Math. 2024; 153:e12737. All possible two-components evolutions systems of (1+2)-dimensional second-order PDEs admitting an infinite-dimensional Lie algebra are constructed. It is shown that a natural generalisation of this Lie algebra to the higher-dimensional case does not lead to a more general result because the infinite-dimensional symmetry is broken. The recently derived system, which is related to Ricci flows, is identified as a very particular case among the evolution systems obtained. All possible stationary solutions of this system in the radially symmetric case are constructed using the surprisingly rich Lie algebra of the reduced system of ODEs. Moreover, it is proved that this Lie algebra is reducible to the fifteen-dimensional algebra of the simplest system of two second-order ODEs. Several time-dependent exact solutions in the radially symmetric case are constructed as well. It is shown that the solutions obtained are bounded and smooth provided arbitrary parameters are correctly specified.

math-ph↗
arXiv2026

A reaction-diffusion system with nonconstant diffusion coefficients: exact and numerical solutions

A Lotka-Volterra type system with porous diffusion, which can be used as an alternative model to the classical Lotka-Volterra system, is under study. Multiparameter families of exact solutions of the system in question are constructed and their properties are established. It is shown that the solutions obtained can satisfy the zero Neumann conditions, which are typical conditions for mathematical models describing real-world processes. It is proved that the system possesses two stable steady-state points provided its coefficients are correctly-specified. In particular, this occurs when the system models the prey-predator interaction. The exact solutions are used for solving boundary-value problems. The analytical results are compared with numerical solutions of the same boundary-value problems but perturbed initial profiles. It is demonstrated that the numerical solutions coincide with the relevant exact solutions with high exactness in the case of sufficiently small perturbations of the initial profiles.

math.AP↗
arXiv2025

Lie symmetry classification and exact solutions of a diffusive Lotka-Volterra system with convection

A mathematical model for description of the viscous fingering induced by a chemical reaction is under study. This complicated five-component model is reduced to a three-component diffusive Lotka-Volterra system with convection by introducing a stream function. The system obtained is examined by the classical Lie method. A complete Lie symmetry classification is derived via a rigorous algorithm. In particular, it is proved that the widest Lie algebras of invariance occur when the stream function generate a linear velocity field. The most interesting cases (from the symmetry and applicability point of view) are further studied in order to derive exact solutions. A wide range of exact solutions are constructed for radially-symmetric stream functions. These solutions include time-dependent and radially symmetric solutions as well as more complicated solutions expressed in terms of the Weierstrass function. It was shown that some of exact solutions can be used for demonstration of spatiotemporal evolution of concentrations corresponding to two reactants and their product.

math-ph↗
arXiv2025

Comments on the paper `Modelling and nonclassical symmetry analysis of a complex porous media flow in a dilating channel'

The Comments are devoted to the recently published paper 'Modelling and nonclassical symmetry analysis of a complex porous media flow in a dilating channel' (Physica D. 481 (2025) 134834), in which a model describing an unsteady two-dimensional viscous incompressible fluid flow through a porous medium is studied. The main theoretical results of that study consists of finding Lie and nonclassical symmetries of a fourth-order PDE, which was derived by simplification of the given model. Here it is shown that the main theoretical results derived therein are incomplete and misleading.

physics.flu-dyn↗
arXiv2024

Symmetries and exact solutions of a reaction-diffusion system arising in population dynamics

A system of two cubic reaction-diffusion equations for two independent gene frequencies arising in population dynamics is studied. Depending on values of coefficients, all possible Lie and $Q$-conditional (nonclassical) symmetries are identified. A wide range of new exact solutions is constructed, including those expressible in terms of a Lambert function and not obtainable by Lie symmetries. An example of a new real-world application of the system is discussed. A general algorithm for finding Q-conditional symmetries of nonlinear evolution systems of the most general form is presented in a useful form for other researchers.

nlin.SI↗
arXiv2024

An age-structured diffusive model for epidemic modelling: Lie symmetries and exact solutions

A new age-structured diffusive model for the mathematical modelling of epidemics is suggested. The model can be considered as a generalization of two models suggested earlier for the same purposes. The Lie symmetry classification of the model is derived. It is shown that the model admits an infinite-dimensional Lie algebra of invariance. Using the Lie symmetries, exact solutions, in particular those of the travelling wave types and in terms of special functions, are constructed. An example of application of the correctly-specified exact solution for calculation of total numbers of infected individuals during an epidemic is presented.

q-bio.PE↗
arXiv2024

Analysis of a Mathematical Model for Fluid Transport in Poroelastic Materials in 2D Space

A mathematical model for the poroelastic materials (PEM) with the variable volume is developed in multidimensional case. Governing equations of the model are constructed using the continuity equations, which reflect the well-known physical laws. The deformation vector is specified using the Terzaghi effective stress tensor. In the two-dimensional space case, the model is studied by analytical methods. Using the classical Lie method, it is proved that the relevant nonlinear system of the (1+2)-dimensional governing equations admits highly nontrivial Lie symmetries leading to an infinite-dimensional Lie algebra. The radially-symmetric case is studied in details. It is shown how correct boundary conditions in the case of PEM in the form of a ring and an annulus are constructed. As a result, boundary-value problems with a moving boundary describing the ring (annulus) deformation are constructed. The relevant nonlinear boundary-value problems are analytically solved in the stationary case. In particular, the analytical formulae for unknown deformations and an unknown radius of the annulus are presented.

math-ph↗
arXiv2024

Exact Nonclassical Symmetry Solutions of Lotka-Volterra Type Population Systems

New classes of conditionally integrable systems of nonlinear reaction-diffusion equations are introduced. They are obtained by extending a well known nonclassical symmetry of a scalar partial differential equation to a vector equation. New exact solutions of nonlinear predator-prey systems, related to the diffusive Lotka-Volterra system, are constructed. An infinite dimensional class of exact solutions is made available. Unlike in the standard Lotka-Volterra system, in the absence of predators, the prey population has a finite carrying capacity, as in the Fisher equation.

nlin.SI↗
arXiv2024

A Mathematical Model for Two Solutes Transport in a Poroelastic Material and Its Applications

Using well-known mathematical foundations of the elasticity theory, a mathematical model for two solutes transport in a poroelastic material (soft tissue is a typical example) is suggested. It is assumed that molecules of essentially different sizes dissolved in fluid and are transported through pores of different sizes. The stress tensor, the main force leading to the material deformation, is taken not only in the standard linear form but also with an additional nonlinear part. The model is constructed in 1D space and consists of six nonlinear equations. It is shown that the governing equations are integrable in stationary case, therefore all steady-state solutions are constructed. The obtained solutions are used in an example for healthy and tumour tissue, in particular, tissue displacements are calculated and compared for parameters taken from experimental data in cases of the linear and nonlinear stress tensors. Since the governing equations are non-integrable in non-stationary case, the Lie symmetry analysis is used in order to construct time-dependent exact solutions. Depending on parameters arising in the governing equations, several special cases with non-trivial Lie symmetries are identified. As a result, multi-parameter families of exact solutions are constructed including those in terms of special functions(hypergeometric and Bessel functions). A possible application of the solutions obtained is demonstrated.

math-ph↗
arXiv2024

Symmetries and exact solutions of the diffusive Holling-Tanner prey-predator model

We consider the classical Holling-Tanner model extended on 1D space by introducing the diffusion term. Making a reasonable simplification, the diffusive Holling-Tanner system is studied by means of symmetry based methods. Lie and Q-conditional (nonclassical) symmetries are identified. The symmetries obtained are applied for finding a wide range of exact solutions, their properties are studied and a possible biological interpretation is proposed. 3D plots of the most interesting solutions are drown as well.

math-ph↗
arXiv2024

The Shigesada-Kawasaki-Teramoto model: conditional symmetries, exact solutions and their properties

We study a simplification of the well-known Shigesada-Kawasaki-Teramoto model, which consists of two nonlinear reaction-diffusion equations with cross-diffusion. A complete set of Q-conditional (nonclassical) symmetries is derived using an algorithm adopted for the construction of conditional symmetries. The symmetries obtained are applied for finding a wide range of exact solutions, possible biological interpretation of some of which being presented. Moreover, an alternative application of the simplified model related to the polymerisation process is suggested and exact solutions are found in this case as well.

math-ph↗
arXiv2024

Comments on the paper "Derivation of lump solutions to a variety of Boussinesq equations with distinct dimensions."

The Comments are devoted to the paper 'Derivation of lump solutions to a variety of Boussinesq equations with distinct dimensions' (Int J Numer Methods Heat Fluid Flow. 2022;32:3072{3082), in which three new generalizations of the classical Boussinesq equation are suggested that were further investigated in several papers. Here it is shown that the equations derived in the above paper are not presented in their canonical forms. It turns out that all three equations can be essentially simplified by the standard technique widely used for linear and quasi-linear PDEs. As a result, it is proved that the equations suggested are not multidimensional generalizations of the Boussinesq equation.

math-ph↗
arXiv2023

Reaction-diffusion equations in mathematical models arising in epidemiology

The review is devoted to analysis of mathematical models used for describing epidemic processes. A main focus is done on the models that are based on partial differential equations (PDEs), especially those that were developed and used for the COVID-19 pandemic modelling. Our attention is paid preferable to the studies in which not only results of numerical simulations are presented but analytical results as well. In particular, travelling fronts (waves), exact solutions, estimation of key epidemic parameters of the epidemic models with governing PDEs (typically reaction-diffusion equations) are discussed. The review may serve as a valuable source for researchers and practitioners in the field of mathematical modelling in epidemiology.

q-bio.PE↗
arXiv2023

Comments on the paper "Solutions of Multitime Reaction-Diffusion PDE"

The Comments are devoted to the paper ``Solutions of Multitime Reaction-Diffusion PDE'' (Mathematics, vol. 10 (2022), 3623), in which main results are misleading and can be derived in a simple way from those obtained earlier. Moreover, it is shown that the exact solution derived therein are simple generalizations of the known solutions and are easily obtainable by the method of differential constrains. The Comments were submitted to the journal Mathematics.

math.AP↗
arXiv2022

A hunter-gatherer-farmer population model: new conditional symmetries and exact solutions with biological interpretation

New Q-conditional (nonclassical) symmetries and exact solutions of the hunter-gatherer-farmer population model proposed by Aoki, Shida and Shigesada (Theor. Popul. Biol. 1996;50:1-17) are constructed. The main method used for the aforementioned purposes is an extension of the nonclassical method for system of partial differential equations. An analysis of properties of the exact solutions obtained and their biological interpretation are carried out. New results are compared with those derived in recent studies devoted to the same model.

nlin.SI↗
arXiv2022

Construction and application of exact solutions of the diffusive Lotka-Volterra system: a review and new results

This review summarizes all known results (up to this date) about methods of integration of the classical Lotka-Volterra systems with diffusion and presents a wide range of exact solutions, which are the most important from applicability point of view. It is the first attempt in this direction. Because the diffusive Lotka-Volterra systems are used for mathematical modeling enormous variety of processes in ecology, biology, medicine, physics and chemistry, the review should be interesting not only for specialists from Applied Mathematics but also those from other branches of Science. The obtained exact solutions can also be used as test problems for estimating the accuracy of approximate analytical and numerical methods for solving relevant boundary value problems.

math-ph↗
arXiv2020

Exact solutions of a mathematical model describing competition and co-existence of different language speakers

The known three-component reaction-diffusion system modeling competition and co-existence of different language speakers is under study. A modification of this system is proposed, which is examined by Lie symmetry method; furthermore exact solutions in the form of traveling fronts are constructed and their properties are identified. Plots of the traveling fronts are presented and the relevant interpretation describing the language shift occurred in Ukraine during the Soviet times is suggested.

nlin.PS↗