Convolution Equations Research Articles

The discrete convolution for fractional cosine-sine series and its application in convolution equations

<abstract><p>The fractional sine series (FRSS) and the fractional cosine series (FRCS) were defined. Three types of discrete convolution operations for FRCS and FRSS were introduced, along with a detailed investigation into their corresponding convolution theorems. The interrelationship between these convolution operations was also discussed. Additionally, as an application of the presented results, two forms of discrete convolution equations based on the proposed convolution theorems were examined, resulting in explicit solutions for these equations. Furthermore, numerical simulations were provided to demonstrate that our proposed solution can be easily implemented with low computational complexity.</p> </abstract>

Однородные уравнения типа π-свертки

Exponential polynomials satisfying a homogeneous convolution type equation are called its elementary solutions. The traditional solution of a homogeneous equation of the convolution type involves the proof of an approximation theorem that asserts the density of elementary solutions in the set of all solutions of the equation. This article considers homogeneous equations of the π-convolution type in the space of analytic functions on a convex domain, which generalize the well-known homogeneous equations of the convolution type. Equations of a particular form have been considered many times before. Such equations include homogeneous convolution equations, homogeneous q-sided convolution equations, homogeneous q-sided convolution type equations, and homogeneous-convolution equations. In the article the properties of operators of π-convolution type are investigated and an approximation theorem for a homogeneous equation of π-convolution type in an arbitrary convex domain of the complex plane is proved.

Homogenized Models with Memory Effect for Heterogeneous Periodic Media

The homogenization of initial boundary value problems for heat conduction equations with asymptotically degenerate rapidly oscillating periodic coefficients are considered. Such problems model thermal processes in heterogeneous periodic media. Homogenized problems (whose solutions determine approximate asymptotics for solutions of the original problems) are presented. Estimates for the accuracy of the asymptotics and relevant convergence theorem are discussed. The homogenized problems have the form of initial boundary value problems for integro-differential equations in convolutions. The presence of convolutions in models for media is called the memory effect. Statements about the solvability and regularity for the problems and the homogenized problems are proved. These results are optimal even in the case of zero convolutions, when the homogenized problems coincide with the classical heat conduction problems.

Mean-field limits for non-linear Hawkes processes with excitation and inhibition

We study a multivariate, non-linear Hawkes process ZN on the complete graph with N nodes. Each vertex is either excitatory (probability p) or inhibitory (probability 1−p). We take the mean-field limit of ZN, leading to a multivariate point process Z̄. If p≠12, we rescale the interaction intensity by N and find that the limit intensity process solves a deterministic convolution equation and all components of Z̄ are independent. In the critical case, p=12, we rescale by N1/2 and obtain a limit intensity, which solves a stochastic convolution equation and all components of Z̄ are conditionally independent.

The explicit solutions for a class of fractional Fourier–Laplace convolution equations

In this paper, two types of fractional Fourier–Laplace convolutions are defined, and the corresponding fractional Fourier–Laplace convolution theorems associated with the fractional cosine transform (FRCT), fractional sine transform (FRST) and Laplace transform (LP) are investigated in detail. The relationship between the fractional Fourier–Laplace convolutions and the existing convolutions is given, and Young's type theorem as well as the weighted convolution inequality are also obtained. As an application for fractional Fourier–Laplace convolution, the filter design and the system of convolution-type integral equations are also considered, the computational complexity for the multiplicative filter is analysed, and explicit solutions for these equations are obtained.

Solving convolution equations in $S^{'}_+$ by numerical method

Solving convolution equations in $S^{'}_+$ by numerical method

Characterization of Viscoelastic Poisson's Ratio of Engineering Elastomers via DIC-Based Creep Testing.

New data of creep and viscoelastic Poisson’s ratio, , of five engineering elastomers (Ethylene Propylene-Diene Monomer, Flouroelastomer (Viton®), nitrile butadiene rubber, silicone rubber and neoprene/chloroprene rubber) at different stress (200, 400 and 600 kPa) and temperature (25, 50 and 80 °C) are presented. The was characterized through an experimental methodological approach based on creep testing (30 min) and strain (axial and transverse) measurements by digital image correlation. Initially, creep behavior in axial and transverse directions was characterized for each elastomer and condition, and then each creep curve was fitted to a four-element creep model to obtain the corresponding functions. The obtained functions were used to estimate for prolonged times (300 h) through a convolution equation. Overall, the characterization was achieved for the five elastomers results exhibiting increasing with temperature and time from about 0.3 (for short-term loading) to reach and stabilize at about 0.48 (for long-term loading).

Convolutional Integro-Differential Equations in Banach Spaces With a Noetherian Operator in the Main Part

An initial-value problem for an integro-differential equation of convolution type with a finite index operator for the higher order derivative in Banach spaces is considered. The equations under consideration model the evolution of the processes with "memory" when the current state of the system is influenced not only by the entire history of observations but also by the factors that have formed it and that remain relevant to the current moment of observation. Solutions are constructed in the class of generalized functions with a left bounded support with the use of the theory of fundamental operator functions of degenerate integro-differential operators in Banach spaces. A fundamental operator function that corresponds to the equation under consideration is constructed. Using this function the generalized solution is restored. The relationship between the generalized solution and the classical solution of the original initial-value problem is studied. Two examples of initial-boundary value problems for the integro-differential equations with partial derivatives are considered

In-Flight Spectral Response Function Retrieval of a Multispectral Radiometer Based on the Functional Data Analysis Technique

The spectral response function (SRF) is a crucial parameter in multispectral radiometers, and it influences the radiometric calibration accuracy and quantitative application capabilities. The in-flight SRF often has errors due to prelaunch contamination or postlaunch degradation. This study proposes an innovative new method to retrieve SRFs of multispectral radiometers based on intercomparisons with hyperspectral sounders via the functional data analysis (FDA) technique. Under the FDA framework, all variables, including the hyperspectral radiance and SRF, are regarded as functions rather than discrete data by expanding in the Fourier functional basis. The forward convolution equation is processed directly into a functional integration model rather than a normally pointwise summation; this ensures that the unknown quantities are transformed from numerous SRF samples to several function parameters, thus avoiding the ill-posed problem. The proposed algorithm is verified with both simulated and real data from multiple thermal infrared bands of the FY-3 IRAS and FY-4 AGRI using collocations with METOP-B IASI. All these results demonstrate our algorithm's qualitative and quantitative effectiveness for infrared SRF retrieval. Although the demonstrations are particularly relevant to infrared spectra, the algorithm is universal and also applicable to other spectral bands.

Transmission–propagation–diffraction operator theory for an arbitrary acoustic two-block model

By applying the transmission–propagation–diffraction​ operator theory (TPDOT; in earlier publications abbreviated as TPOT) we have derived an analytical representation of the wavefield in two 3D inhomogeneous acoustic half-spaces separated by a piecewise-smooth interface. We scrutinize the conventional approach and reformulate the mechanical particle motion forward problem in terms of a novel wave amplitude form of counter-propagating wave events. The new wave motion statement is composed of two systems of integral equations describing propagation and transmission of the seismic wavefields. The first system represents the propagation–diffraction operator equation. We express the propagation–diffraction operator as the feasible Kirchhoff-type surface integral with a feasible fundamental solution in the kernel. The feasible fundamental solution generalizes the conventional free space Green’s function for an arbitrary domain case. The second system represents the transmission (refraction/reflection) operator equation. This convolutional interface equation generalizes the Sommerfeld–Weyl spatial-frequency plane wave decomposition in an infinitesimally thin interface layer. We transform the wave amplitude formulation by Neumann iteration into infinite series of counter-propagating multiply transmitted (reflected/refracted) wave modes, representing an analytical solution to the problem. Since the wave motion statement of the forward problem enables analysis of separate wave modes of the total wavefield, it has an advantage over the particle motion statement. The suggested wavefield representation therefore solves both forward and inverse wave motion problems. Calculation of separate wave modes of TPDOT can be done by the software package of the tip-wave superposition method (TWSM) introduced in our earlier papers (in co-authorship). The proposed TPDOT provides the theoretical foundations for TWSM.

Long-term behavior of multilayered angle-ply plate structures with viscoelastic interlayer by state space method

Layered angle-ply plates bonded by viscoelastic adhesives exhibit time-dependent behavior when subjected to sustained loads, which is a common phenomenon in practical engineering. The present work proposes analytical solutions based on the exact elasticity theory for the long-term behavior of multilayered angle-ply plates considering viscoelastic interlayers. In the analytical model, each angle-ply layer involves nine independent orthotropic constants and is modeled on the basis of the three-dimensional (3D) elasticity theory. The interlayer exhibits viscoelastic property and is described by the standard linear solid model. The state space method is employed in order to solve the stresses and displacements for the plate, and the convolution equation for viscoelasticity is figured out by Laplace transform. The results of numerical examples show that the present solution is convergent rapidly and is consistent with the finite element solution. The effects of the viscoelastic material constants, the layer angle and the orthotropic constants on the long-term stresses and displacements in the plate are studied thoroughly by use of the present solution.

Toward “hereditary epidemiology”: A temporal Boltzmann approach to COVID-19 fatality trends

Countless research contributions reflect two major concepts for modeling the spread of the COVID-19 pandemic: (i) ordinary differential equations for population compartments, such as infected or deceased persons (these approaches often exhibit limited predictive capabilities); and (ii) rules applied to digitally realized agents in the populations (these approaches often lack reliable input data and may become computationally overly expensive). As a remedy, we here introduce and discuss convolutional integrodifferential equations adapted from Boltzmann's hereditary mechanics, so as to predict COVID-19 fatality trends from the evolutions of newly infected persons. Replacing the classical statistical reasoning by deliberations arising from the notion of “virus loads” and the corresponding compliance of the infected population to these loads, model errors with respect to data recorded in 102 countries, territories, or US states can be drastically reduced, namely, up to 98% when compared to the traditional kinetics equation of Kermack and McKendrick. The coefficients of determination between model predictions and recorded data range from 94% to 100%, a precision hitherto unachieved in equation-based epidemic modeling.

A weak solution theory for stochastic Volterra equations of convolution type

We obtain general weak existence and stability results for stochastic convolution equations with jumps under mild regularity assumptions, allowing for non-Lipschitz coefficients and singular kernels. Our approach relies on weak convergence in Lp spaces. The main tools are new a priori estimates on Sobolev–Slobodeckij norms of the solution, as well as a novel martingale problem that is equivalent to the original equation. This leads to generic approximation and stability theorems in the spirit of classical martingale problem theory. We also prove uniqueness and path regularity of solutions under additional hypotheses. To illustrate the applicability of our results, we consider scaling limits of nonlinear Hawkes processes and approximations of stochastic Volterra processes by Markovian semimartingales.

ОБ ОДНОМ ОБОБЩЕНИИ ОПЕРАТОРА СДВИГА

Several approaches are known to generalize the shift operator in the complex domain. Specific generalization procedures lead to significantly different operators of the shift type even in one specific area of complex analysis. In this paper, a certain general operator of the shift type is defined. The properties of this operator are studied and the question of the continuity of this definition is considered. The properties of a certain operator to a certain extent re-peat the properties of the previously studied operators of the shift type. This allows us to speak of the possibility of extending the main results on spectral synthesis in the complex domain to convolutional equations of a more general form.

Power and delay efficient fir filter design using ESSA and VL-CSKA based booth multiplier

FIR filter plays a major role in digital image processing applications. The power and delay performance of any FIR filter depends on the switching activities between the filter coefficients (FCs) and its basic arithmetic operations (i.e., multiplication and addition) performed in the convolution equations. In this paper, a new FIR filter is designed using Enhanced Squirrel Search Algorithm (ESSA) and Variable latency Carry skip adder (VL-CSKA) based booth multiplier. The proposed ESSA algorithm selects an optimal FC by minimizing the switching activities of FC based on the ripple contents, power and Transition width parameter to meet the required specifications of FIR filter in the frequency domain. Also, the VL-CSKA based booth multiplier is proposed to reduce the delay of FIR filter with parallel addition of partial products (PPs). In this design, the VL-CSKA adders utilize variable size and compound gate-based skip logic to deduce the delay with low power. The proposed FIR filter is simulated in Xilinx working platform by developing Verilog coding. The simulation result shows that the proposed FIR filter outperforms the state-of-the-art FIR filters by consuming only 0.142 mW power with delay of 28.175 ns.

Generalized spherical mean value operators on Euclidean space

We consider the Neumann version of the spherical mean value operator and its variants in the space of smooth functions, distributions and compactly supported ones. Surjectivity and range characterization issues are addressed from the viewpoint of convolution equations.

Solvability Criterion for Convolution Equations on a Half-Strip

We obtain the criterion of solvability of homogeneous convolution equation in a half-strip. Proof is based on a new decomposition property of the weighted Hardy space. This result has relations to the spectral analysis-synthesis problem, cyclicity problem, information theory. All data generated or analysed during this study are included in this published article.

Explicit solutions of Volterra integro-differential convolution equations

We find an explicit form of solution of a convolution type integro-differential equation of the first order (1) ∂ Q ( t , z ) ∂ z = α ( t , z ) − ∫ 0 t β ( t − u ) Q ( u , z ) d u , ( t , z ) ∈ ( 0 , T ) × ( 0 , T ) , where T > 0 , and α , β are two given locally integrable functions and Q satisfies a boundary condition Q ( t , 0 ) = γ ( t ) for a locally integrable function γ . To achieve this goal we introduce the notion of a biconvolution algebra of locally integrable functions on R + 2 . We investigate the properties of the biconvolution algebra and study the Volterra integral equations of the second kind associated with the biconvolution operation. Finally, we present an explicit form of solution of integro-differential equation associated with a linear operator (2) Λ n = ∂ n + 1 ∂ z ∂ t n + ∑ i = 0 n − 1 λ i ∂ i ∂ t i , λ i ∈ R , i ≤ n − 1 , n ∈ N .

SIMPLIFIED MODEL OF LINEAR INERTIAL MEASUREMENT SYSTEMS

Problem. To increase the metrological reliability of measuring systems at technical objects, the number of sensors measuring the same process parameter is increased to several units and a model of a multi-channel measuring system is synthesized. This synthesis is usually based on the use of Markov's theory of linear filtering, but the presence of a connection between the input and output signals of the linear inertial system through the convolution integral significantly complicates the process of obtaining the optimal device. Goal. The aim of the article is to develop a method for approximating the integral equation of convolution, which describes a linear inertial system, and to estimate the limits of its application on the example of linear inertial sensors. Methodology. Instead of the output signal in the form of a convolution equation, the output signal, defined as the product of the input signal to an unknown time function is used. This function is represented by the Karhunen-Loev series. The distance in the functional space with a quadratic metric between these output signals is minimized by means of a genetic algorithm and the coefficients of the series and, therefore, the unknown function itself, are determined. Results. In the simulation, the relative difference between the output signals, which were calculated from exact and simplified expressions, was determined. Realizations of stationary signals were used as input signals, and the pulse characteristics of the linear inertial system varied over a wide range. The errors of approximation of the integral convolution equation by a simple model do not exceed a few percent. Origina-lity. The approximation of the convolution equation by a simplified model of the system is original and, although it cannot be applied in a wide range of conditions, it is acceptable for a separate class of stationary signals without restrictions. The accuracy of the approximation of the convolution equation is greatest if the width of the spectrum of the input signals is less than the bandwidth of the measuring system. Practical value. The obtained connection between input and output signals based on a simplified model allows to synthesize multi-channel measuring systems using advanced Markov filtering methods for a separate class of stationary input signals. To expand the application of the method in a wide range of conditions, a set of simplified models that are created in advance can be used.

Smoothness of solutions of a convolution equation of restricted type on the sphere

Abstract Let $\mathbb {S}^{d-1}$ denote the unit sphere in Euclidean space $\mathbb {R}^d$ , $d\geq 2$ , equipped with surface measure $\sigma _{d-1}$ . An instance of our main result concerns the regularity of solutions of the convolution equation $$\begin{align*}a\cdot(f\sigma_{d-1})^{\ast {(q-1)}}\big\vert_{\mathbb{S}^{d-1}}=f,\text{ a.e. on }\mathbb{S}^{d-1}, \end{align*}$$ where $a\in C^\infty (\mathbb {S}^{d-1})$ , $q\geq 2(d+1)/(d-1)$ is an integer, and the only a priori assumption is $f\in L^2(\mathbb {S}^{d-1})$ . We prove that any such solution belongs to the class $C^\infty (\mathbb {S}^{d-1})$ . In particular, we show that all critical points associated with the sharp form of the corresponding adjoint Fourier restriction inequality on $\mathbb {S}^{d-1}$ are $C^\infty $ -smooth. This extends previous work of Christ and Shao [4] to arbitrary dimensions and general even exponents and plays a key role in the companion paper [24].