Analog simulator of integro-differential equations with classical memristors

An analog computer makes use of continuously changeable quantities of a system, such as its electrical, mechanical, or hydraulic properties, to solve a given problem. While these devices are usually computationally more powerful than their digital counterparts, they suffer from analog noise which does not allow for error control. We will focus on analog computers based on active electrical networks comprised of resistors, capacitors, and operational amplifiers which are capable of simulating any linear ordinary differential equation. However, the class of nonlinear dynamics they can solve is limited. In this work, by adding memristors to the electrical network, we show that the analog computer can simulate a large variety of linear and nonlinear integro-differential equations by carefully choosing the conductance and the dynamics of the memristor state variable. We study the performance of these analog computers by simulating integro-differential models related to fluid dynamics, nonlinear Volterra equations for population growth, and quantum models describing non-Markovian memory effects, among others. Finally, we perform stability tests by considering imperfect analog components, obtaining robust solutions with up to 13% relative error for relevant timescales.

Stability of nonlinear Volterra equations

Using a novel approach, we present some new explicit criteria for global exponential stability of the zero solution of general nonlinear time-varying Volterra difference equations. Furthermore, an explicit stability bound for equations subject to nonlinear time-varying perturbations is given. Finally, the obtained results are used to study uniform attraction of equilibrium of discrete-time bidirectional associative memory (BAM) neural networks. Some illustrative examples are given.

A Novel Analytical Implementation of Nonlinear Volterra Integral Equations

This article aims at preferring a new and viable algorithm, specifically a two-step homotopy perturbation transform algorithm (TSHPTA). This novel technique is a feasible way in finding exact solutions with a small amount of calculations. As a simple but typical example, it demonstrates the strength and the great potential of the two-step homotopy perturbation transform method to solve nonlinear Volterra-type integral equations efficiently. The results reveal that the proposed scheme is suitable for the nonlinear Volterra equations.