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We discuss some important issues arising from computational efforts in dynamical systems and fluid dynamics. Various individuals have misunderstood these issues since the onset of these problem areas; indeed, they have been routinely…
We consider a non-linear system modelling the dynamics of a linearly elastic body immersed in an incompressible viscous fluid, without damping on the elastic part. We prove local existence of strong solutions and global existence and…
We are concerned with quasilinear symmetrizable partially dissipative hyperbolic systems in the whole space $\mathbb{R}^d$ with $d\geq2$. Following our recent work [10] dedicated to the one-dimensional case, we establish the existence of…
The hierarchy of integrable equations are considered. The dynamical approach to the theory of nonlinear waves is proposed. The special solutions(nonlinear waves) of considered equations are derived. We use powerful methods of computer…
We study the relativistic Euler equations on the Minkowski spacetime background. We make assumptions on the equation of state and the initial data that are relativistic analogs of the well-known physical vacuum boundary condition, which has…
In this paper, we investigate the problem of electromagnetic wave propagation in hyperbolic nonlinear media. To address this problem, we consider the scalar hyperbolic nonlinear Schr\"odinger system and its coupled version, namely…
We are interested in numerical schemes for the simulation of large scale gas networks. Typical models are based on the isentropic Euler equations with realistic gas constant. The numerical scheme is based on transformation of conservative…
In this paper, we propose quasilinearization methods that convert nonlocal fully-nonlinear parabolic systems into the nonlocal quasilinear parabolic systems. The nonlocal parabolic systems serve as important mathematical tools for modelling…
Two effective methods for writing the dynamical equations for non-holonomic systems are illustrated. They are based on the two types of representation of the constraints: by parametric equations or by implicit equations. They can be applied…
We consider a class of doubly nonlinear degenerate hyperbolic-parabolic equations with homogeneous Dirichlet boundary conditions, for which we first establish the existence and uniqueness of entropy solutions. We then turn to the…
We investigate multidimensional model for incompressible non-Newtonian fluids. Using method of energy estimates we prove the property of finite speed of propagations of the solution support for this problem. We find sharp bounds of the…
We discuss solution concepts for linear hyperbolic equations with coefficients of regularity below Lipschitz continuity. Thereby our focus is on theories which are based either on a generalization of the method of characteristics or on…
To further confirm the causality and stability of a second-order hyperbolic system of partial differential equations that models the relativistic dynamics of barotropic fluids with viscosity and heat conduction (H. Freist\"uhler and B.…
In the article we study a hyperbolic-elliptic system of PDE. The system can describe two different physical phenomena: 1st one is the motion of magnetic vortices in the II-type superconductor and 2nd one \ is the collective motion of cells.…
This paper shows nonlinear stability of homogeneous states in second-order hyperbolic systems of partial differential equations that model the dynamics of dissipative relativistic fluids, by checking a dissipativity criterion formulated…
The ultra-relativistic Euler equations for an ideal gas are described in terms of the pressure, the spatial part of the dimensionless four-velocity and the particle density. Radially symmetric solutions of these equations are studied in two…
We show that many important natural science models in their mathematical formulation can be reduced to non-strictly hyperbolic systems of the same kind. This allows the same methods to be applied to them so that some essential results…
A fifth--order nonlinear partial differential equation for the description of nonlinear waves in a liquid with gas bubbles is considered. Special solutions of this equation are studied. Some elliptic and simple periodic traveling waves…
We give an overview of recent advances in analysis of equations of electrodynamics with the aid of biquaternionic technique. We discuss both models with constant and variable coefficients, integral representations of solutions, a numerical…
We introduce a direct method allowing to solve numerically inverse type problems for linear hyperbolic equations. We first consider the reconstruction of the full solution of the wave equation posed in $\Omega\times (0,T)$ - $\Omega$ a…