Related papers: Analytical solutions of the Arrhenius-Semenov prob…
In this article we consider the numerical approximation of the convection-diffusion-reaction equation. One of the main challenges of designing a numerical method for this problem is that boundary layers occurring in the convection-dominated…
A numerical method for approximating weak solutions of an aggregation equation with degenerate diffusion is introduced. The numerical method consists of a stabilized finite element method together with a mass lumping technique and an extra…
The paper deals with a problem of asymptotic soliton like solutions to the Benjamin-Bona-Mahony (BBM) equaion with a small parameter at the highest derivative and variable coefficients depending on the variables $x$, $t$ as well as a small…
In this study, we investigate the dynamics of moving fronts in three-dimensional spaces, which form as a result of in-situ combustion during oil production. This phenomenon is also observed in other contexts, such as various autowave models…
For a homogenization problem associated to a linear elliptic operator, we prove the existence of a distributional corrector and we find an approximation scheme for the homogenized coefficients. We also study the convergence rates in the…
Exact one-dimensional solutions to the equations of fluid dynamics are derived in the large-Pr and small-Pr limits (where Pr is the Prandtl number). The solutions are analogous to the Pr = 3/4 solution discovered by Becker and analytically…
Chernoff approximations to strongly continuous one-parameter semigroups give solutions to a wide class of differential equations. This paper studies the rate of convergence of the Chernoff approximations. We provide simple natural examples…
This paper gives the exact solution in terms of the Karhunen-Lo\`{e}ve expansion to a fractional stochastic partial differential equation on the unit sphere $\mathbb{S}^{2}\subset \mathbb{R}^{3}$ with fractional Brownian motion as driving…
We address the problem of the best uniform approximation by linear combinations of a finite system of functions. If the system is Chebyshev and the problem is unconstrained, then the classical Remez algorithm provides a fast and precise…
This paper develops a two-level fourth-order scheme for solving time-fractional convection-diffusion-reaction equation with variable coefficients subjected to suitable initial and boundary conditions. The basis properties of the new…
We propose a simple algebraic method for constructing exact solutions of equations of two-dimensional hydrodynamics of an incompressible fluid. The problem reduces to consecutively solving three linear partial differential equations for a…
A new exponentially convergent algorithm is proposed for an abstract the first order differential equation with unbounded operator coefficient possessing a variable domain. The algorithm is based on a generalization of the Duhamel integral…
In this paper we consider two different Stefan problems for a semi-infinite material for the non classical heat equation with a source which depends on the heat flux at the fixed face x = 0. One of them (with constant temperature on x = 0)…
This paper concerns the construction and analysis of a numerical scheme for a mixed discrete-continuous fragmentation equation. A finite volume scheme is developed, based on a conservative formulation of a truncated version of the…
The problem of non-perturbative description of stationary flames with arbitrary gas expansion is considered. A general method for deriving equations for the flame front position is developed. On the basis of the Thomson circulation theorem…
We consider the problem of ignition of propagating waves in one-dimensional bistable or excitable systems by an instantaneous spatially extended stimulus. Earlier we proposed a method (Idris and Biktashev, PRL, vol 101, 2008, 244101) for…
Self-similar solution is obtained for propagation of a strong shock, in a flat expanding dusty Friedman universe. Approximate analytic solution was obtained earlier, using relation between self-similar variables, equivalent to the exact…
The non-linear collision-induced breakage equation has significant applications in particulate processes. Two semi-analytical techniques, namely homotopy analysis method (HAM) and accelerated homotopy perturbation method (AHPM) are…
In this chapter we consider different approximations for the one-dimensional one-phase Stefan problem corresponding to the fusion process of a semi-infinite material with a temperature boundary condition at the fixed face and non-linear…
A second-order face-centred finite volume strategy on general meshes is proposed. The method uses a mixed formulation in which a constant approximation of the unknown is computed on the faces of the mesh. Such information is then used to…