Volterra-type Integral Research Articles

Integral and Weighted Composition Operators on Fock-type Spaces

We study various structures of general Volterra-type integral and weighted composition operators acting between two Fock-type spaces mathcal {F}_{varphi }^p and mathcal {F}_{varphi }^q, where varphi is a radial function growing faster than the function zrightarrow |z|^2/2. The main results show that the unboundedness of the Laplacian of varphi provides interesting results on the topological and spectral structures of the operators in contrast to their actions on Fock spaces, where the Laplacian of the weight function is bounded. We further describe the invertible and unitary weighted composition operators. Finally, we show the spaces support no supercyclic weighted composition operator with respect to the pointwise convergence topology and hence with the weak and strong topologies.

Solvability for a nonlocal dispersal model governed by time and space integrals

Abstract This work is to analyze a nonlocal dispersal model governed by a Volterra type integral and two space integrals. A weighted integral is included, and an existence result of solutions for this model is proved. We assume a suitably Hartman-type sign condition and use a sufficiently regular measure of noncompactness. Finally, the degree theory referring to condensing operators is applied.

Regularized Asymptotic Solutions of a Singularly Perturbed Fredholm Equation with a Rapidly Varying Kernel and a Rapidly Oscillating Inhomogeneity

This article investigates an equation with a rapidly oscillating inhomogeneity and with a rapidly decreasing kernel of an integral operator of Fredholm type. Earlier, differential problems of this type were studied in which the integral term was either absent or had the form of a Volterra-type integral. The presence of an integral operator and its type significantly affect the development of an algorithm for asymptotic solutions, in the implementation of which it is necessary to take into account essential singularities generated by the rapidly decreasing kernel of the integral operator. It is shown in tise work that when passing the structure of essentially singular singularities changes from an integral operator of Volterra type to an operator of Fredholm type. If in the case of the Volterra operator they change with a change in the independent variable, then the singularities generated by the kernel of the integral Fredholm-type operators are constant and depend only on a small parameter. All these effects, as well as the effects introduced by the rapidly oscillating inhomogeneity, are necessary to take into account when developing an algorithm for constructing asymptotic solutions to the original problem, which is implemented in this work.

Lerch matrix collocation method for 2D and 3D Volterra type integral and second order partial integro differential equations together with an alternative error analysis and convergence criterion based on residual functions

In this study, second order linear Volterra partial integro-differential equation with two- and three-dimensional are solved by collocation method based on Lerch polynomials. This method is composed of the operational matrix and collocation methods, which are based upon the matrix forms of the Lerch polynomials with the parameter $\lambda$ and Taylor polynomials, and their derivatives and integrals. The approximate solutions of the mentioned equations are investigated in terms of the Lerch polynomials the different values of $\lambda$. Also, to verify the accuracy and efficiency of the present method, an alternative convergence criterion along with error analysis depending on residual function is enhanced. Moreover, the obtained numerical results are compared with other methods and scrutinized by using tables and figures.

Topological Structures of Generalized Volterra-Type Integral Operators

We study the generalized Volterra-type integral and composition operators acting on the classical Fock spaces. We first characterize various properties of the operators in terms of growth and integrability conditions which are simpler to apply than those already known Berezin type characterizations. Then, we apply these conditions to study the compact and Schatten $\mathcal{S}_p$ class difference topological structures of the space of the operators. In particular, we proved that the difference of two Volterra-type integral operators is compact if and only if both are compact.

Volterra Type and Weighted Composition Operators on Weighted Fock Spaces

Bounded and compact product of Volterra type integral and composition operators acting between weighted Fock spaces are described. We also estimate the norms of these operators in terms of Berezin type integral transforms on the complex plan $${\mathbb{C}}$$ . All our results are valid for weighted composition operators acting on the class of Fock spaces considered under appropriate interpretation of the weights.

Product of Volterra Type Integral and Composition Operators on Weighted Fock Spaces

We characterize the bounded, compact, and Schatten class product of Volterra type integral and composition operators acting between weighted Fock spaces. Our results are expressed in terms of certain Berezin type integral transforms on the complex plane ℂ. We also estimate the norms and essential norms of these operators in terms of the integral transforms. All our results are valid for weighted composition operators when acting between the class of weighted Fock spaces considered.

Fractional differential equations through Laguerre expansions in abstract spaces: error estimates

An approximation procedure by the means of expansion with respect to Laguerre orthogonal basis of the space is given. It is applied to solutions of a class of convolution equations, by transforming them to corresponding systems of algebraic equations for the coefficients. Especially, Volterra-type integral and integro-differential equations and system of equations with fractional derivatives modeling viscoelastic rod are solved. Error estimates depending on the order of approximation are derived.

Genetic algorithms as applied to the numerical computation of electromagnetic scattering by weakly nonlinear dielectric cylinders

This paper deals with the application of an optimization procedure based on a genetic algorithm (GA) to the prediction of the electromagnetic fields scattered by weakly nonlinear dielectric objects. Starting by an integral approach and describing the nonlinearities of the constitutive parameters by the Volterra-type integrals, the nonlinear scattering problem is numerically solved by an iterative procedure developed for the minimization of a suitable defined cost function. A GA is applied in order to deal with a large number of unknowns related to the harmonic components of the nonlinear internal electromagnetic field. In a preliminary stage, the behavior of typical parameters of the GA is analyzed; then numerical solutions are carried out and compared with those provided by other methods. Finally, some considerations are made concerning the rate of convergence of the iterative procedure.

Asymptotic properties of Markov operators defined by Volterra type integrals

New sufficient conditions for asymptotic stability of Markov operators are given. These criteria are applied to a class of Volterra type integral operators with advanced argument.

Electromagnetic scattering from nonlinear anisotropic cylinders. I. Fundamental frequency

The solution of the problem of electromagnetic scattering of obliquely incident plane waves by homogeneous, nonlinear anisotropic cylindrical structures is obtained. The medium of the scatterer is characterized by Volterra-type integrals for the electric and magnetic flux density vectors D and B, respectively. The nonlinear problem is solved using the perturbation method. The effects of nonlinearities on the field properties both inside and outside the scatterer, together with the effect on the radar cross section, are investigated for the fundamental frequency components. To demonstrate the validity of the approach, the results obtained by the perturbation method are compared with those obtained using the plane wave representation method of D. Censor (1983), where the iteration method is used to solve the resulting dispersion equation. The results are in very good agreement in both amplitude and phase of the fields for the case of very weak nonlinearity. When the relative magnitude of the nonlinear component of the permittivity is increased, the iteration method shows a faster divergence of the phase from the linear phase. >

Markov operators defined by Volterra type integrals with advanced argument

Markov operators defined by Volterra type integrals with advanced argument

A numerical method for creep computation including heredity

A numerical analysis of structures made of time-dependent materials is discussed, in which the novelty of the formulation is the inclusion of heredity in the constitutive law, thereby permitting continuous computation of steady and unsteady creep, relaxation and recovery. As opposed to the so-called “simple creep theories”, in which the material response depends on its current strain-stress only, hereditary creep theories take into account the whole strain and stress history of the material. Memory of past events is mathematically modelled by means of Volterra-type integrals, which are extremely inconvenient in numerical treatments. An approximate method is presented here, which allows the efficient and effective handling of these integrals. Hereditary effects become significant for materials which exhibit substantial recovery upon unloading, such as plastics, composites and other synthetic materials. Consequently, the approach proposed is aimed mainly at improving the analytical prediction of creep behaviour of structures made of these relatively new and strongly time-dependent materials.

On a multiaxial non-linear hereditary constitutive law for non-ageing materials with fading memory

A hereditary type complementary power potential is postulated which permits definition of the strain rate tensor in terms of the history and current magnitude of a certain combination of stress and stress deviator components. The strain rate tensor is decomposed into elastic and viscous portions, the latter being further separated into reversible and irreversible parts with the reversible portion being represented by Volterra type integrals. For a material obeying Norton's power law, the three-dimensional non-linear constitutive equations are derived in terms of two elastic constants and five viscous material parameters which may be determined from creep tests. These equations are applied to the derivation of general non-linear viscoelastic constitutive relations for a plate element which, in turn, are used to solve the cylindrical time-deflection problem of a long simply supported ice plate subjected to in-plane compressive forces applied along the longitudinal edges. The governing equations are solved numerically, using an incremental approach. An approximate method for the effective numerical treatment of problems involving hereditary type constitutive relations is discussed in detail.

Piecewise Polynomial Collocation for Volterra-type Integral Equations of the Second Kind

The exact solution of a given integral equation of the second kind of Volterra type(with regular or weakly singular kernel) is projected into the space of (continuous) piecewise polynomials of degree m ≧ 1 and with prescribed knots by using collocation techniques. It is shown that a number of discrete methods for the numerical solution of such equations based on product integration techniques or on finite-difference methods are particular discrete versions of collocation methods of the above type. The error behaviour of approximate solutions obtained by collocation (including their discretizations) is discussed.