Nonlinear Systems of Volterra Equations with Piecewise Smooth Kernels: Numerical Solution and Application for Power Systems Operation

The evolutionary integral dynamical models of storage systems are addressed. Such models are based on systems of weakly regular nonlinear Volterra integral equations with piecewise smooth kernels. These equations can have non-unique solutions that depend on free parameters. The objective of this paper was two-fold. First, the iterative numerical method based on the modified Newton–Kantorovich iterative process is proposed for a solution of the nonlinear systems of such weakly regular Volterra equations. Second, the proposed numerical method was tested both on synthetic examples and real world problems related to the dynamic analysis of microgrids with energy storage systems.

Weighted boundedness of multilinear singular integral operators with non-smooth kernels for the extreme cases

The weighted boundedness properties of multilinear operators associated to singular integral operators with non-smooth kernels for extreme cases are obtained.

Study of the algebra of smooth integro-differential operators with applications

We study the algebra of integro-differential operators with smooth coefficients and kernels on a subspace of [Formula: see text]. We find a normal form for elements of this algebra and determine its unit group. The formulation of inverses gives explicit solutions of inhomogeneous linear Volterra integro-differential equations and Volterra integral equations of first kind with smooth kernels.

From non-local Eringen’s model to fractional elasticity

Eringen’s model is one of the most popular theories in non-local elasticity. It has been applied to many practical situations with the objective of removing anomalous stress concentrations around geometric shape singularities, which appear when local modelling is used. Despite the great popularity of Eringen’s model within the mechanical engineering community, even the most basic questions such as the existence and uniqueness of solutions have been rarely considered in research literature for this model. In this work we focus on precisely these questions, proving that the model is in general ill-posed in the case of smooth kernels, the case which appears rather often in numerical studies. We also consider the case of singular, non-smooth kernels and for the paradigmatic case of Riesz potential we establish the well-posedness of the model in fractional Sobolev spaces. For such a kernel, in dimension one the model reduces to the well-known fractional Laplacian. Finally, we discuss possible extensions of Eringen’s model to spatially heterogeneous material distributions.