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We present a numerical method for studying the normal modes of accretion flows around black holes. In this first paper, we focus on two-dimensional, viscous, hydrodynamic disks, for which the linear modes have been calculated analytically…
The problem of optimal recovering high-order mixed derivatives of bivariate functions with finite smoothness is studied. On the basis of the truncation method, an algorithm for numerical differentiation is constructed, which is…
In this article, we consider a class of bi-stable reaction-diffusion equations in two components on the real line. We assume that the system is singularly perturbed, i.e. that the ratio of the diffusion coefficients is (asymptotically)…
Alternative approach for description of the non-equilibrium phenomena arising in solids at a severe external loading is analyzed. The approach is based on the new form of kinetic equations in terms of the internal and modified free energy.…
In this paper, the exponential B-spline functions are used for the numerical solution of the advection-diffusion equation. Two numerical examples\ related to pure advection in a finitely long channel and the distribution of an initial…
We propose a new approach to the numerical solution of ergodic problems arising in the homogenization of Hamilton-Jacobi (HJ) equations. It is based on a Newton-like method for solving inconsistent systems of nonlinear equations, coming…
We describe a new mechanism that triggers periodic orbits in smooth dynamical systems. To this end, we introduce the concept of hybrid bifurcations: Such bifurcations occur when a line of equilibria with an exchange point of normal…
Using the advective Cahn-Hilliard equation as a model, we illuminate the role of advection in phase-separating binary liquids. The advecting velocity is either prescribed, or is determined by an evolution equation that accounts for the…
Computational fluctuating hydrodynamics aims at understanding the impact of thermal fluctuations on fluid motions at small scales through numerical exploration. These fluctuations are modeled as stochastic flux terms and incorporated into…
A pure and incompressible material is confined between two plates such that it is heated from below and cooled from above. When its melting temperature is comprised between these two imposed temperatures, an interface separating liquid and…
This research focuses on the interesting physical phenomenon of the bead-hoop system. The bifurcation can be observed investigating the equilibrium point of the bead, and nonlinear oscillation also occurs from the bead's motion. This paper…
Turbulent convection is ubiquitous in fluid systems. In particular, multi-physical convection problems involve mass, heat, and particle transfer. When the particles are charged and driven by a high-voltage electric field, both buoyancy and…
We consider the regularized Ericksen model of an elastic bar on an elastic foundation on an interval with Dirichlet boundary conditions as a two-parameter bifurcation problem. We explore, using local bifurcation analysis and continuation…
We propose and study discontinuous Galerkin methods for strongly degenerate convection-diffusion equations perturbed by a fractional diffusion (L\'evy) operator. We prove various stability estimates along with convergence results toward…
An isotropic elastic half space is prestrained so that two of the principal axes of strain lie in the bounding plane, which itself remains free of traction. The material is subject to an isotropic constraint of arbitrary nature. A surface…
We study Hopf bifurcation from traveling-front solutions in the Cahn-Hilliard equation. The primary front is induced by a moving source term. Models of this form have been used to study a variety of physical phenomena, including pattern…
We introduce a new method to investigate linear stability of gaseous detonations that is based on an accurate shock-fitting numerical integration of the linearized reactive Euler equations with a subsequent analysis of the computed solution…
Particles suspended in a fluid flow through a curved duct can focus to specific locations within the duct cross-section. This particle focusing is a result of a balance between two dominant forces acting on the particle: (i) the inertial…
In this paper, we employed linear algebra and functional analysis to determine necessary and sufficient conditions for oscillation-free and stable solutions to linear and nonlinear parabolic partial differential equations. We applied…
Multiple equilibrium states arise in many physical systems, including various types of liquid crystal structures. Having the ability to reliably compute such states enables more accurate physical analysis and understanding of experimental…