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In present paper, we establish sufficient conditions for existence and stability of solutions for system of nonlinear implicit fractional differential equations. The main techniques are based on method of successive approximations. Finally,…
For a large class of nonlinear evolution PDEs, and more generally, of nonlinear semigroups, as well as their approximating numerical methods, two rather natural stability type convergence conditions are given, one being necessary, while the…
We consider instabilities of a single mode with finite wavenumber in inversion symmetric spatially one dimensional systems, where the character of the bifurcation changes from sub- to supercritical behaviour. Starting from a general…
The linear Einstein-Boltzmann equations describe the evolution of perturbations in the universe and its numerical solutions play a central role in cosmology. We revisit this system of differential equations and present a detailed…
A number of physical phenomena are described by nonlinear hyperbolic equations. Presence of discontinuous solutions motivates the necessity of development of reliable numerical methods based on the fundamental mathematical properties of…
The linear stability of two exact stationary solutions of the parametrically driven, damped nonlinear Dirac equation is investigated. Stability is ascertained through the resolution of the eigenvalue problem, which stems from the…
The Riccati equation method and an approach of the use of unknown factors is used to establish oscillation, suboscillation and nonoscillation criteria for linear systems of ordinary differential equations. A necessary condition for Lyapunov…
The time-dependent probability density function of a system evolving towards a stationary state exhibits an oscillatory behavior if the eigenvalues of the corresponding evolution operator are complex. The frequencies \omega_n, with which…
We present new criteria for the existence of oscillatory and nonoscillatory solutions of measure delay differential equations with impulses. We deal with the integral forms of the differential equations using the Perron and the…
The effects of nonlinear oscillations in compact stars are attracting considerable current interest. In order to study such phenomena in the framework of fully nonlinear general relativity, highly accurate numerical studies are required. We…
One of the most popular methods of controlling dynamical systems is feedback. It can be used without acquiring detailed knowledge of the underlying system. In this work, we study the stability of fractional-order linear difference equations…
A class of fourth--order neutral type difference equations with quasidifferences and deviating arguments is considered. Our approach is based on studying the considered equation as a system of a four--dimensional difference system. The…
We consider difference equations with several non-monotone deviating arguments and nonnegative coefficients. The deviations (delays and advances) are, generally, unbounded. Sufficient oscillation conditions are obtained in an explicit…
The goal of this work is to determine classes of travelling solitary wave solutions for a differential approximation of a finite difference scheme by means of a hyperbolic ansatz. It is shown that spurious solitary waves can occur in…
Fractional derivatives of Prabhakar type are capturing an increasing interest since their ability to describe anomalous relaxation phenomena (in dielectrics and other fields) showing a simultaneous nonlocal and nonlinear behaviour. In this…
We study a disordered system of interacting bosons described by the Bose-Hubbard Hamiltonian with random tunneling amplitudes. We derive the condition for the stability of the replica-symmetric solution for this model. Following the scheme…
For nonautonomous and nonlinear differential and difference equations depending on a parameter, we formulate sufficient conditions under which they exhibit $C^k$, $k\in \N$ shadowing with respect to a parameter. Our results are applicable…
We present an original study on the numerical stabiliy of explicit schemes solving the incompressible Euler equations on an open domain with slipping boundary conditions. Relying on the skewness property of the non-linear term, we…
Sufficient oscillation conditions involving $\limsup $ and $\liminf $ for first-order differential equations with several non-monotone deviating arguments and nonnegative coefficients are obtained. The results are based on the iterative…
In this paper, we study the qualitative behaviour of approximation schemes for Backward Stochastic Differential Equations (BSDEs) by introducing a new notion of numerical stability. For the Euler scheme, we provide sufficient conditions in…