Convolution Integral Operators Research Articles

Time-dependent dynamical energy analysis via convolution quadrature

Dynamical Energy Analysis was introduced in 2009 as a novel method for predicting high-frequency acoustic and vibrational energy distributions in complex engineering structures. In this paper we introduce the first time-dependent Dynamical Energy Analysis method. Time-domain models are important in numerous applications including sound simulation in room acoustics, predicting shock-responses in structural mechanics and modelling electromagnetic scattering from conductors. The first step is to reformulate Dynamical Energy Analysis in the time-domain by means of a convolution integral operator. We are then able to employ the Convolution Quadrature method to provide a link between the previous frequency-domain implementations of Dynamical Energy Analysis and fully time-dependent solutions by means of the Z-transform. By combining a modified multistep Convolution Quadrature approach for the time discretisation, together with Galerkin and Petrov-Galerkin methods for the space and momentum discretisations, respectively, we are able to accurately track the propagation of high-frequency transient signals through phase-space. The implementation here is detailed for finite two-dimensional spatial domains and we demonstrate the versatility of our approach by performing a range of numerical experiments for regular, non-convex and irregular geometries as well as different types of wave source.

Generalized Fractional Integral Operator in a Complex Domain

A new fractional integral operator is used to present a generalized class of analytic functions in a complex domain. The method of definition is based on a Hadamard product of analytic function, which is called convolution product. Then we formulate a convolution integral operator acting on the sub-class of normalized analytic functions. Consequently, we investigate the suggested convolution operator geometrically. Differential subordination inequalities, taking the starlike formula are given. Some consequences of well-known results are illustrated. Keywords: Analytic function, subordination and superordination, univalent function, open unit disk, fractional integral operator, convolution operator, fractional calculus.

Convolution Integral Operators in Variable Bounded Variation Spaces

Working in the frame of variable bounded variation spaces in the sense of Wiener, introduced by Castillo, Merentes, and Rafeiro, we prove convergence in variable variation by means of the classical convolution integral operators. In the proposed approach, a crucial step is the convergence of the variable modulus of smoothness for absolutely continuous functions. Several preliminary properties of the variable p(cdot )-variation are also presented.

Inverse Spectral Problem for Integro-Differential Sturm–Liouville Operators with Discontinuity Conditions

We consider the Sturm–Liouville operator perturbed by a convolution integral operator on a finite interval with Dirichlet boundary conditions and discontinuity conditions in the middle of the interval. We study the inverse problem of recovering the convolution term from the spectrum. The problem is reduced to solving the so-called main nonlinear integral equation with a singularity. To derive and study this equation, we do detailed analysis of kernels of transformation operators for the integro-differential expression considered. We prove global solvability of the main equation, which enables us to prove uniqueness of solution of the inverse problem and to obtain necessary and sufficient conditions for its solvability in terms of asymptotics of the spectrum. The proof is constructive and gives an algorithm for solving the inverse problem.

On the Commutation Properties of Finite Convolution and Differential Operators I: Commutation.

The commutation relation $KL = LK$ between finite convolution integral operator $K$ and differential operator $L$ has implications for spectral properties of $K$. We characterize all operators $K$ admitting this commutation relation. Our analysis places no symmetry constraints on the kernel of $K$ extending the well-known results of Morrison for real self-adjoint finite convolution integral operators.

On the Commutation Properties of Finite Convolution and Differential Operators II: Sesquicommutation

We introduce and fully analyze a new commutation relation $\overline{K} L_1 = L_2 K$ between finite convolution integral operator $K$ and differential operators $L_1$ and $L_{2}$, that has implications for spectral properties of $K$. This work complements our explicit characterization of commuting pairs $KL=LK$ and provides an exhaustive list of kernels admitting commuting or sesquicommuting differential operators.

Inverse Spectral Problem for Integrodifferential Sturm-Liouville Operators with Discontinuity Conditions

We consider the Sturm-Liouville operator perturbed by a convolution integral operator on a finite interval with Dirichlet boundary-value conditions and discontinuity conditions in the middle of the interval. We study the inverse problem of restoration of the convolution term by the spectrum. The problem is reduced to solution of the so-called main nonlinear integral equation with a singularity. To derive and investigate this equations, we do detailed analysis of kernels of transformation operators for the integrodifferential expression under consideration. We prove the global solvability of the main equation, this implies the uniqueness of solution of the inverse problem and leads to necessary and sufficient conditions for its solvability in terms of spectrum asymptotics. The proof is constructive and gives the algorithm of solution of the inverse problem.

Convolution Integral Operators

We establish some algebraic, metric, and spectral properties of convolution integral operators in L2(ℝN).

Application of Improved H-Matrices in Micromagnetic Simulations of Spin Torque Oscillator

This paper examines the applicability of hierarchical-matrices (H-matrices) to a computation of the demagnetizing field, which is the most time-consuming part in the micromagnetic simulation of spin torque oscillators (STO). Given that the kernel function of the convolution integral operator for the demagnetizing field has a second-order singularity, efficient approximation cannot be expected using conventional H-matrices employing adaptive cross approximation as the low-rank approximation. We introduce improved H-matrices to overcome this challenge. Furthermore, matrix sizes appearing in STO simulations are relatively small compared with the usual sizes of H-matrices. Through numerical experiments, we confirm the size conditions for memory usage and the computational time of H-matrix-vector products to demonstrate the advantages of using H-matrices.

A New Concept of Multidimensional Variation in the Sense of Riesz and Applications to Integral Operators

In this paper, we introduce and study a new multidimensional generalization in the sense of Tonelli of the Riesz–Medvedev $$\varphi $$ -variation, proving in particular a multidimensional generalization of the Riesz–Medvedev theorem. We finally discuss an application of such concept of variation to some approximation problems: in particular, we obtain some estimates and convergence results by means of convolution integral operators in the space of functions of bounded $$\varphi $$ -variation.

On the half inverse spectral problem for an integro-differential operator

We consider the Sturm–Liouville differential operator on a finite interval with Dirichlet boundary conditions perturbed with a convolution integral operator. The inverse problem is studied of recovering the convolution kernel from a half spectrum, provided that the kernel function on the first half of the interval as well as the Sturm–Liouville potential on the entire interval are known a priori. Besides the uniqueness of a solution of this half inverse problem, we also obtain necessary and sufficient conditions of its solvability. A constructive procedure for solving the inverse problem is also provided.

Directed electromagnetic wave propagation in 1D metamaterial: Projecting operators method

We consider a boundary problem for 1D electrodynamics modeling of a pulse propagation in a metamaterial medium. We build and apply projecting operators to a Maxwell system in time domain that allows to split the linear propagation problem to directed waves for a material relations with general dispersion. Matrix elements of the projectors act as convolution integral operators. For a weak nonlinearity we generalize the linear results still for arbitrary dispersion and derive the system of interacting right/left waves with combined (hybrid) amplitudes. The result is specified for the popular metamaterial model with Drude formula for both permittivity and permeability coefficients. We also discuss and investigate stationary solutions of the system related to some boundary regimes.

Boundedness and compactness of a class of convolution integral operators of fractional integration type

For a class of convolution integral operators whose kernels may have integrable singularities, boundedness and compactness criteria in weighted Lebesgue spaces are obtained.

A Review on Approximation Results for Integral Operators in the Space of Functions of Bounded Variation

We present a review on recent approximation results in the space of functions of bounded variation for some classes of integral operators in the multidimensional setting. In particular, we present estimates and convergence in variation results for both convolution and Mellin integral operators with respect to the Tonelli variation. Results with respect to a multidimensional concept ofφ-variation in the sense of Tonelli are also presented.

FREDHOLM PROPERTY OF COMPOSITE TWO-DIMENSIONAL INTEGRAL OPERATORS WITH HOMOGENEOUS SINGULAR-TYPE KERNELS IN pL SPACE

The authors have previously studied two - dimensional Fredholm integral operators with homogeneous kernels of fiber - singular type. For this class of operators, the symbolic calculus is built using the theory of biloc al operators by V. Pilidi, and Fredholm criterion is formulated through the inversibility of t wo families: the family of one - dimensional convolution operators, and the family of one - dimensional singular integral operators with continuous coefficients. The aim of this work is to study composite two - dimensional integral operators with homogeneous ker nels of fiber singular type analogous to Simonenko’s continual convolution integral operators. This investigation is a part of a more general study of algebra of operators with homogeneous kernels which layers are singular operat ors with piecewise continuo us coefficients. For the considered operators, the symbolic calculus and the necessary and sufficient Fredholm conditions are obtained.

Approximation Results with Respect to Multidimensional φ-Variation for Nonlinear Integral Operators

In this paper we study approximation problems for functions belonging to BV φ-spaces (spaces of functions of bounded φ-variation) in multidimensional setting. In particular, using a multidimensional concept of φ-variation in the sense of Tonelli introduced in [4], we obtain estimates, convergence results and, by means of suitable Lipschitz classes, results about the order of approximation for a family of nonlinear convolution integral operators.

Average error bounds of trigonometric approximation on periodic Wiener spaces

In this paper, we study the approximation of identity operator and the convolution integral operator Bm by Fourier partial sum operators, Fejer operators, Vallee-Poussin operators, Cesaro operators and Abel mean operators, respectively, on the periodic Wiener space (C1(ℝ),Wℴ) and obtain the average error estimations.

Traveling wave solutions in a plant population model with a seed bank

We propose an integro-difference equation model to predict the spatial spread of a plant population with a seed bank. The formulation of the model consists of a nonmonotone convolution integral operator describing the recruitment and seed dispersal and a linear contraction operator addressing the effect of the seed bank. The recursion operator of the model is noncompact, which poses a challenge to establishing the existence of traveling wave solutions. We show that the model has a spreading speed, and prove that the spreading speed can be characterized as the slowest speed of a class of traveling wave solutions by using an asymptotic fixed point theorem. Our numerical simulations show that the seed bank has the stabilizing effect on the spatial patterns of traveling wave solutions.

Some new classes of analytic functions defined using convolution

In this paper, we introduce and study some new subclasses of analytic functions defined in the open unit disc using the convolution technique. Inclusion results, radius problems and several other properties of these classes are discussed. Key words: Convolution integral operator, functions with positive real part, alpha-starlike, bounded Mocanu variation, univalent.

Erratum to: Convergence in Variation and Rate of Approximation for Nonlinear Integral Operators of Convolution Type

In this paper we obtain estimates, convergence results and rate of approximation for functions belonging to BV–spaces (spaces of functions with bounded variation) by means of nonlinear convolution integral operators. We treat both the periodic and the non-periodic case using, respectively, the classical Jordan variation and the multidimensional variation in the sense of Tonelli.