Sergey Shindin

Pseudospectral Laguerre approximation of transport-fragmentation equations

We develop a Laguerre-type pseudo-spectral scheme for solving transport–fragmentation equations. The method converges rapidly for certain type of fragmentation problems and works under mild restrictions on the growth rate of the coefficients of the equation. Rigorous stability and convergence analyses are provided. Numerical simulations illustrate the performance of the scheme.

Analysis of a Chebyshev-type pseudo-spectral scheme for the nonlinear Schrdinger equation

In this paper, we derive several error estimates that are pertinent to the study of Chebyshev-type spectral approximations on the real line. The results are applied to construct a stable and accurate pseudo-spectral Chebyshev scheme for the nonlinear Schrdinger equation. The new technique has several computational advantages as compared to Fourier and Hermite-type spectral schemes, described in the literature (see e.g., [1][3]. Similar to Hermite-type methods, we do not require domain truncation and/or use of artificial boundary conditions. At the same time, the computational complexity is comparable to the best Fourier-type spectral methods described in the literature.

Numerical simulation of a transport fragmentation coagulation model

In this paper, we deal with numerical analysis of a transport fragmentation coagulation equation with power fragmentation rates and separable coagulation kernels. For numerical simulations, the model is rewritten in a conservative form and semi-discretized in space using a Laguerre pseudo-spectral method. Some error estimates are derived and efficiency of the numerical scheme is considered. The paper is concluded with several computational examples.

Asymptotic Analysis of Structured Population Models

Describing real world phenomena we produce models with ever increasing complexity. While very accurate, such models are very costly and cumbersome to analyse and often require data hard to obtain and tend to yield information which is redundant in specific applications. It is thus important to be able to derive simplified sub‐models which still contain relevant information in a particular context but are more tractable. In biological applications this process is called ‘aggregation’ of variables and is often based on separation of multiple time scales in the model. In this paper we describe how techniques of asymptotic analysis of singularly perturbed problems can be used to obtain in a systematic way a complete system of approximating equations and illustrate this approach on a example of a population equation of McKendrick type with age and space structure.