相关论文: Approximating semigroups by using pseudospectra

We apply pseudo-spectral methods to integrate functional flow equations with high accuracy, extending earlier work on functional fixed point equations \cite{Borchardt:2015rxa}. The advantages of our method are illustrated with the help of…

The not necessarily unitary evolution operator of a finite dimensional quantum system is studied with the help of a projection operators technique. Applying this approach to the Schr\"odinger equation allows the derivation of an alternative…

We expand our group classification of quasilinear evolution equations (Acta Appl.Math., v.69, 2001) to the case of general evolution equation in one spatial variable. This enables obtaining several new classes of evolution equations with…

The topic of these notes could be easily expanded into a full one-semester course. Nevertheless, we shall try to give some flavour along with theoretical bases of spectral and pseudo-spectral methods. The main focus is made on Fourier-type…

We study diffusion-type equations supported on structures that are randomly varying in time. After settling the issue of well-posedness, we focus on the asymptotic behavior of solutions: our main result gives sufficient conditions for…

We present a new pseudo-spectral code for the simulation of evolution systems that are second order in space. We test this code by evolving a non-linear scalar wave equation. These non-linear waves can be stably evolved using very simple…

With the example of the spherically symmetric scalar wave equation on Minkowski space-time we demonstrate that a fully pseudospectral scheme (i.e. spectral with respect to both spatial and time directions) can be applied for solving…

A novel symmetry method for finding exact solutions to nonlinear PDEs is illustrated by applying it to a semilinear reaction-diffusion equation in multi-dimensions. The method uses a separation ansatz to solve an equivalent first-order…

Pseudospectra and structured pseudospectra are important tools for the analysis of matrices. Their computation, however, can be very demanding for all but small matrices. A new approach to compute approximations of pseudospectra and…

wave solutions to nonlinear partial differential equations. We simplify the so called (G'/G)-expansion method and apply two of those methods to simple physical problems.

In many applications it is important to understand the sensitivity of eigenvalues of a matrix polynomial to perturbations of the polynomial. The sensitivity commonly is described by condition numbers or pseudospectra. However, the…

Here we present exact, stationary, parametric solutions to the Schr\"odinger--Poisson system. We confront two images: on one hand, we draw on the homotopy analysis method which leads us to a nonlinear integral scheme. Indeed, this approach…

There is an abundance of empirical evidence in the numerical relativity literature that the form in which the Einstein evolution equations are written plays a significant role in the lifetime of numerical simulations. This paper attempts to…

We consider the problem of how to compute eigenvalues of a self-adjoint operator when a direct application of the Galerkin (finite-section) method is unreliable. The last two decades have seen the development of the so-called quadratic…

We carry out quantitative studies on the Green operator $ \hat{\mathscr G}$ associated with the Born equation, an integral equation that models electromagnetic scattering, building the strong stability of the evolution semigroup…

In this article we consider the approximation of a variable coefficient (two-sided) fractional diffusion equation (FDE), having unknown $u$. By introducing an intermediate unknown, $q$, the variable coefficient FDE is rewritten as a lower…

We discuss the classical statement of group classification problem and some its extensions in the general case. After that, we carry out the complete extended group classification for a class of (1+1)-dimensional nonlinear…

Motivated by structured parasite populations in aquaculture we consider a class of size-structured population models, where individuals may be recruited into the population with distributed states at birth. The mathematical model which…

Most finite element methods for solving time-harmonic wave-propagation problems lead to a linear system with a non-normal coefficient matrix. The non-normality is due to boundary conditions and losses. One way to solve these systems is to…

In this technical note, we deal with a spectrum approximation problem arising in THREE-like multivariate spectral estimation approaches. The solution to the problem minimizes a suitable divergence index with respect to an a priori spectral…