Integral Representation for the Solutions of Autonomous Linear Neutral Fractional Systems with Distributed Delay

The aim of this work is to obtain an integral representation formula for the solutions of initial value problems for autonomous linear fractional neutral systems with Caputo type derivatives and distributed delays. The results obtained improve and extend the corresponding results in the particular case of fractional systems with constant delays and will be a useful tool for studying different kinds of stability properties. The proposed results coincide with the corresponding ones for first order neutral linear differential systems with integer order derivatives.

Explicit Stencil Computation Schemes Generated by Poisson's Formula for the 2D Wave Equation

An approach to building explicit time-marching stencil computation schemes for the transient 2D acoustic wave equation without using finite-difference approximations is proposed and implemented. It is based on using the integral representation formula (Poisson's formula) that provides the exact solution of the initial-value problem for the transient 2D scalar wave equation at any time point through the initial conditions. For the purpose of constructing a two-step time-marching algorithm, a modified integral representation formula involving three time levels is also employed. It is shown that integrals in the two representation formulas are exactly calculated if the initial conditions and the sought solution at each time level as functions of spatial coordinates are approximated by stencil interpolation polynomials in the neighborhood of any point in a 2D Cartesian grid. As a result, if a uniform time grid is chosen, the proposed time-marching algorithm consists of two numerical procedures: 1) the solution calculation at the first time-step through the initial conditions; 2) the solution calculation at the second and next time-steps using a generated two-step numerical scheme. Three particular explicit stencil schemes (with five, nine and 13 space points) are built using the proposed approach. Their stability regions are presented. The obtained stencil expressions are compared with the corresponding finite-difference schemes available in the literature. Their novelty features are discussed. Simulation results with new and conventional schemes are presented for two benchmark problems that have exact solutions. It is demonstrated that using the new first time-step calculation procedure instead of the conventional one can provide a significant improvement of accuracy even for later time steps.

The Newtonian Potential and the Demagnetizing Factors of the General Ellipsoid

The objective of this paper is to present a modern and concise new derivation for the explicit expression of the interior and exterior Newtonian potential generated by homogeneous ellipsoidal domains in $\mathbb{R}^N$ (with $N \geqslant 3$). The very short argument is essentially based on the application of Reynolds transport theorem in connection with Green-Stokes integral representation formula for smooth functions on bounded domains of$\mathbb{R}^N$, which permits to reduce the N-dimensional problem to a 1-dimensional one. Due to its physical relevance, a separate section is devoted to the derivation of the demagnetizing factors of the general ellipsoid which are one of the most fundamental quantities in ferromagnetism.

The Newtonian potential and the demagnetizing factors of the general ellipsoid

The objective of this paper is to present a modern and concise new derivation for the explicit expression of the interior and exterior Newtonian potential generated by homogeneous ellipsoidal domains in R N (with N ≥3). The very short argument is essentially based on the application of Reynold's transport theorem in connection with the Green–Stokes integral representation formula for smooth functions on bounded domains of R N , which permits to reduce the N -dimensional problem to a one-dimensional one. Owing to its physical relevance, a separate section is devoted to the the derivation of the demagnetizing factors of the general ellipsoid which are one of the most fundamental quantities in ferromagnetism.

An integral representation formula for logarithmic potentials and embeddings of Bessel-potential spaces

We give an integral representation formula for logarithmic Riesz potentials. This plays an essential role in proving the sharpness of the embeddings of Bessel-potential spaces, which have logarithmic exponents both in the smoothness and in the underlying Lorentz—Zygmund spaces. These results are natural extensions of those obtained by Edmunds, Gurka, Opic and Trebels.