We study the power spectrum of a space-time dependent neural field which describes the average membrane potential of neurons in a single layer. This neural field is modelled by a dissipative integrodifferential equation, the so-called Amari equation. By considering a small perturbation with respect to a stationary and uniform configuration of the neural field we derive a linearized equation whi...

We consider the problems of numerical solution Cauchy problem for an evolutionary equation with memory when kernel integral term is a difference one. The computational implementation associated need to work approximate all previous points in time. In this paper, considered nonlocal transformed into local one; loosely coupled system additional ordinary differential equations solved. This approac...

• We discretize the 2D peridynamics equation using a time Crank-Nicolson/spatial asymptotically compatible scheme. The boundary conditions prevent. corner reflections for peridynamics. numerical stability condition is proven. aim of this paper to construct accurate absorbing (ABCs) two-dimensional motion which describes nonlocal phenomena arising in continuum mechanics based on integrodifferent...

In this paper we establish some new nonlinear integro-differential inequalities of Gronwall-Bellman-Pachpatte type for function of one independent variable. The purpose of this paper is to extend certain results which proved by Pachpatte in [On some fundamental integrodifferential and integral inequalities, An. Sti. Univ. Al. I. Cuza, Iasi, Vol.23 (1977), 77-86]. The inequalities obtained here ...

My research focuses on scientific computing and kinetic equations. More specifically I develop and analyze numerical methods for solving the Boltzmann transport equation and similar kinetic models, as well as their application to high performance computing resources. My work is supported by NSF-DMS Grant 1217254 “Accurate high performance computing for nonlinear collisional kinetic theory”, (20...

A numerical method is proposed for computing time-periodic and relative time-periodic solutions in dissipative wave systems. In such solutions, the temporal period, and possibly other additional internal parameters such as the propagation constant, are unknown priori and need to be determined along with the solution itself. The main idea of the method is to first express those unknown parameter...

The homotopy analysis method [1, 2], is developed to search the accurate asymptotic solutions of nonlinear problems. This technique has been successfully applied to many nonlinear problems such as the viscous flows of non-Newtonian fluids [3, 4], the Korteweg-de Vries-type equations [5, 6], nonlinear heat transfer [7, 8], finance problems [9, 10], Riemann problems related to nonlinear shallow w...

A governing equation of stable random walks is developed in one dimension. This Fokker-Planck equation is similar to, and contains as a subset, the second-order advection dispersion equation (ADE) except that the order (a) of the highest derivative is fractional (e.g., the 1.65th derivative). Fundamental solutions are Lévy’s a-stable densities that resemble the Gaussian except that they spread ...

Motivated by a wide range of applications in various fields physics and materials science, we consider generalized approach to the evolution polydisperse ensemble spherical particles metastable media. An integrodifferential system governing equations, consisting kinetic equation for particle-size distribution function (Fokker–Planck type equation) balance temperature (concentration) medium, is ...