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Weighted averaged finite difference methods for solving fractional diffusion equations are discussed and different formulae of the discretization of the Riemann-Liouville derivative are considered. The stability analysis of the different…
A review of the authors's results is given. Several methods are discussed for solving nonlinear equations $F(u)=f$, where $F$ is a monotone operator in a Hilbert space, and noisy data are given in place of the exact data. A discrepancy…
For solving unsteady hyperbolic conservation laws on cut cell meshes, the so called small cell problem is a big issue: one would like to use a time step that is chosen with respect to the background mesh and use the same time step on the…
We consider the question of determining whether or not a given system of fractional-order differential equations is (asymptotically) stable. In particular, we admit systems where each constituent equation may have its own order, independent…
The time dependent complex Schr\"odinger equation with cubic nonlinearity is solved by constructing differential quadrature algorithm based on sinc functions. Reduction to a coupled system of real equations enables to approach the space…
The article discusses the gradient discretisation method (GDM) for distributed optimal control problems governed by diffusion equation with pure Neumann boundary condition. Using the GDM framework enables to develop an analysis that…
Mathematical modeling of fluid dynamics for computer graphics requires high levels of theoretical rigor to ensure visually plausible and computationally efficient simulations. This paper presents an in-depth theoretical framework analyzing…
The paper describes a novel method for studying the stability of nonautonomous dynamical systems. This method based on the flow and divergence of the vector field with coupling to the method of Lyapunov functions. The necessary and…
The stability and convergence analysis of high-order numerical approximations for the one- and two-dimensional nonlocal wave equations on unbounded spatial domains are considered. We first use the quadrature-based finite difference schemes…
In this article, we introduce the notion of stochastic symmetry of a differential equation. It consists in a stochastic flow that acts over a solution of a differential equation and produces another solution of the same equation. In the…
Maxwell's equations are considered with transparent boundary conditions, for initial conditions and inhomogeneity having support in a bounded, not necessarily convex three-dimensional domain or in a collection of such domains. The numerical…
We propose a quantitative direct method to prove the local stability of a stationary solution for a rough differential equation and its regular discretization scheme. Using Doss-Sussmann technique and stopping time analysis, we provide…
An important problem that arises in many engineering applications is the boundary value problem for ordinary differential equations. There have been many computational methods proposed for dealing with this problem. The convergence of the…
We consider the problem of closeness of solutions of an exact and an averaged difference equations on an infinite interval. Appropriate assertions are derived from one special theorem on the stability under constantly acting perturbations.
We study semilinear elliptic equations \begin{equation*} \begin{cases} -\Delta u = f(u) & \text{in } \Omega, \\ \partial_\nu u = 0 & \text{on } \partial\Omega, \end{cases} \end{equation*} with homogeneous Neumann boundary conditions in…
Eliminating the arbitrary coefficients in the equation of a generic plane curve of order $n$ by computing sufficiently many derivatives, one obtains a differential equation. This is a projective invariant. The first one, corresponding to…
A discrepancy principle for solving nonlinear equations with monotone operators given noisy data is formulated. The existence and uniqueness of the corresponding regularization parameter $a(\delta)$ is proved. Convergence of the solution…
We study a probabilistic numerical method for the solution of both boundary and initial value problems that returns a joint Gaussian process posterior over the solution. Such methods have concrete value in the statistics on Riemannian…
Stochastic differential equations (SDEs) offer powerful and accessible mathematical models for capturing both deterministic and probabilistic aspects of dynamic behavior across a wide range of physical, financial, and social systems.…
This paper gives necessary and sufficient conditions for the convergence of the solution of a weakly damped second order linear differential equation that is subjected to outside forcing, for which solutions of the unforced equation are…