Volterra Integral Operators Research Articles

Synthesis of uniformly distributed optimal control with nonlinear optimization of oscillatory processes

In the article the problem of synthesizing uniformly distributed optimal control for nonlinear optimization of oscillatory processes described by integro-differential partial differential equations with the Volterra integral operator was explored. The study was conducted according to the Bellman-Egorov scheme and an algorithm for constructing a uniformly distributed optimal control in the form of a functional from the state of the controlled process was developed. Sufficient conditions for the solvability of the synthesis problem in nonlinear optimization were established.

Numerical solution for a generalized form of nonlinear cordial Volterra integral equations using quasilinearization and Legendre-collocation methods

In this article, we propose a numerical method for a general form of nonlinear cordial Volterra integral equations. We discuss conditions that under them the problem has solutions. Since the existence of solutions for the problem depends on the solvability of a scalar equation and also a linear form of the problem, then we employ quasilinearization technique in which solving a nonlinear problem is reduced to solve a sequence of linear equations. The existence of solutions of linear equations and their quadratically convergence to the solutions of the nonlinear problem is considered. For the numerical solution of the produced linear equations we apply Legendre-collocation method along with a regularization technique for the quadrature formulas. We discuss the error analysis of the collocation method considering that the cordial Volterra integral operators are noncompact. To test the efficiency and accuracy of the proposed method, the solution of different cases of numerical examples are reported.

Simultaneous space–time Hermite wavelet method for time-fractional nonlinear weakly singular integro-partial differential equations

An innovative simultaneous space–time Hermite wavelet method has been developed to solve weakly singular fractional-order nonlinear integro-partial differential equations in one and two dimensions with a focus whose solutions are intermittent in both space and time. The proposed method is based on multi-dimensional Hermite wavelets and the quasilinearization technique. The simultaneous space–time approach does not fully exploit for time-fractional nonlinear weakly singular integro-partial differential equations. Subsequently, the convergence analysis is challenging when the solution depends on the entire time domain (including past and future time), and the governing equation is combined with Volterra and Fredholm integral operators. Considering these challenges, we use the quasilinearization technique to handle the nonlinearity of the problem and reconstruct it to a linear integro-partial differential equation with second-order accuracy. Then, we apply multi-dimensional Hermite wavelets as attractive candidates on the resulting linearized problems to effectively resolve the initial weak singularity at t=0. In addition, the collocation method is used to determine the tensor-based wavelet coefficients within the decomposition domain. We elaborate on constructing the proposed simultaneous space–time Hermite wavelet method and design comprehensive algorithms for their implementation. Specifically, we emphasize the convergence analysis in the framework of the L2 norm and indicate high accuracy dependent on the regularity of the solution. The stability of the proposed wavelet-based numerical approximation is also discussed in the context of fractional-order nonlinear integro-partial differential equations involving both Volterra and Fredholm operators with weakly singular kernels. The proposed method is compared with existing methods available in the literature. Specifically, we highlighted its high accuracy and compared it with a recently developed hybrid numerical approach and finite difference methods. The efficiency and accuracy of the proposed method are demonstrated by solving several highly intermittent time-fractional nonlinear weakly singular integro-partial differential equations.

On the features of inhomogeneous residual stress identification using the digital speckle interferometry and the hole-drilling method

A phenomenological approach to the actual problem of determining the inhomogeneous residual stress-strain state in the components of high-tech engineering systems at the stages of their design and operation is presented. The approach is based on physical and mechanical methods of measuring displacements. Current physical models describe the physical regularities of the residual states attributed to changes in the structure by the interaction of defects and dislocations in the field of micro- and meso-stresses. At the same time, there are the problems of the transition to the macrolevel, the construction of multilevel models, and the conversion of these models in engineering practice. In the framework of phenomenological approaches, in the general case, the solution of this problem requires the solution of three-dimensional inverse problems of thermoelasticity. A well-known mechanical method for determining a uniform field of residual elastic stresses recommended by ASTM E837 is described. The method proposed earlier by one of the authors for determining an inhomogeneous (in the plane) field of residual elastic stresses is discussed. A method of the three-dimensional inhomogeneous residual elastic stress-strain state determination based on the experimental determination of the displacement vector components by the method of step-by-step point hole-drilling and data of digital speckle interferometry and digital image correlation is developed. The constitutive relations for the components of the displacement vector are written in the form of Volterra integral operators. The basic operator functions are the functions of four variables, i.e., the coordinates of the cylindrical system (r, θ, z) associated with the hole, and the hole depth h. A method for verification of the basic functions is presented. The problem is reduced to the determination of three displacement functions of three variables: hole radius r, h, and z. Numerical simulation of basic functions is carried out. The obtained results are consistent with the known experimental data and calculated values of the deformation on the surface depending on the depth of the hole according to the ASTM E837 Standard.

A fast time domain solver for the equilibrium Dyson equation

We consider the numerical solution of the real-time equilibrium Dyson equation, which is used in calculations of the dynamical properties of quantum many-body systems. We show that this equation can be written as a system of coupled, nonlinear, convolutional Volterra integro-differential equations, for which the kernel depends self-consistently on the solution. As is typical in the numerical solution of Volterra-type equations, the computational bottleneck is the quadratic-scaling cost of history integration. However, the structure of the nonlinear Volterra integral operator precludes the use of standard fast algorithms. We propose a quasilinear-scaling FFT-based algorithm which respects the structure of the nonlinear integral operator. The resulting method can reach large propagation times and is thus well-suited to explore quantum many-body phenomena at low energy scales. We demonstrate the solver with two standard model systems: the Bethe graph and the Sachdev-Ye-Kitaev model.

On the Algebra Generated by Volterra Integral Operators with Homogeneous Kernels and Continuous Coefficients

On the Algebra Generated by Volterra Integral Operators with Homogeneous Kernels and Continuous Coefficients

On Singular Points of Linear Differential-Algebraic Equations with Perturbations in the Form of Integral Operators

The paper consideres linear systems of ordinary differential equations of arbitrary order with a matrix identically degenerate in the domain of definition at the highest derivative of the desired vector function and with loads in the form of Volterra and Fredholm integral operators. The initial value problems are formulated using projections onto admissible sets of initial vectors. Special attention is paid to systems having singular points on the interval of integration. The concept of a singular point is formalized. Their classification in the case of differential equations is given. A number of examples illustrating the theoretical results are presented.

Area functions characterizations of weighted Bergman spaces and area operators

We completely characterize the boundedness and compactness of the area operators on weighted Bergman spaces Ap(ω) over the unit ball induced by Békollé weights. Using the characterization of the boundedness, we obtain some general area formulas in terms of the radial derivative, the complex gradient, and the invariant gradient for Ap(ω). As an application of area operator and the general area formula related to the radial derivative, we characterize the boundedness and compactness of Volterra integral operators.

An FFT method for the numerical differentiation

We consider the numerical differentiation of a function tabulated at equidistant points. The proposed method is based on the Fast Fourier Transform (FFT) and the singular value expansion of a proper Volterra integral operator that reformulates the derivative operator. We provide the convergence analysis of the proposed method and the results of a numerical experiment conducted for comparing the proposed method performance with that of the Neville Algorithm implemented in the NAG library.

Volterra integral operators on a family of Dirichlet-Morrey spaces

A family of Dirichlet-Morrey spaces \(\mathcal{D}_{\lambda,K}\) of functions analytic in the open unit disk \(\mathbb{D}\) are defined in this paper. We completely characterize the boundedness of the Volterra integral operators \(T_g\), \(I_g\) and the multiplication operator \(M_g\) on the space \(\mathcal{D}_{\lambda,K}\). In addition, the compactness and essential norm of the operators \(T_g\) and \(I_g\) on \(\mathcal{D}_{\lambda,K}\) are also investigated.

The Generalized Volterra Integral Operator and Toeplitz Operator on Weighted Bergman Spaces

We study the boundedness and compactness of the generalized Volterra integral operator on weighted Bergman spaces with doubling weights on the unit disc. A generalized Toeplitz operator is defined and the boundedness, compactness and Schatten class membership of this operator are investigated on the Hilbert weighted Bergman space. As an application, Schatten class membership of generalized Volterra integral operators are also characterized. Finally, we also get the characterizations of Schatten class membership of generalized Toeplitz operator and generalized Volterra integral operators on the Hardy space $$H^2$$ .

Dirichlet-Morrey type spaces and Volterra integral operators

Let 0 < p < ∞ and K : [ 0 , ∞ ) → [ 0 , ∞ ) be a nondecreasing and right-continuous function. A new Dirichlet-Morrey type space D p − 1 p , K is introduced in this paper. The boundedness of the identity operator I d from D p − 1 p , K to tent spaces T K p ( μ ) is characterized. As an application, the boundedness, compactness and essential norm of Volterra integral operators on D p − 1 p , K are investigated.

Multistep collocation methods for integral-algebraic equations with non-vanishing delays

In this article, we study the piecewise multistep collocation method for a class of functional integral equations with non-vanishing delays. Based on the notions of the tractability index and the ν-smoothing property of a Volterra integral operator, our numerical analysis and optimal convergence properties are investigated. Here, multistep collocation method which depends on the numerical solution in a fixed number of previous time steps is described by the constructive technique and dividing the definition domain into several subintervals according to the primary discontinuous points associated with the delay function. Numerical experiments confirm the theoretical expectations.

Integral inequalities in amalgam spaces

Weighted norm inequalities for the Volterra integral operator with the kernel are characterized in the amalgam space for certain ranges of indices.

Correct Solvability of Integrodifferential Equations in Spaces of Vector Functions Holomorphic in an Angular Domain

Integrodifferential equations with unbounded operator coefficients in a Hilbert space are studied. The main part of an equation of this kind is an abstract parabolic equation perturbed by a Volterra integral operator. The fundamental difference between this work and the other ones is that integrodifferential equations are considered and studied in this paper for vector functions the arguments of which take values in an angular domain on the complex plane.

Volterra integral operator from weighted Bergman spaces to general function spaces

Volterra integral operator from weighted Bergman spaces to general function spaces

An algebraic study of Volterra integral equations and their operator linearity

The algebraic study of special integral operators led to the notions of Rota-Baxter operators and shuffle products which have found broad applications such as iterated integrals. This paper carries out an algebraic study of general integral operators and equations, and shows that there are rich algebraic structures underlying Volterra integral operators and the corresponding equations. First Volterra integral operators are shown to produce a matching twisted Rota-Baxter algebra satisfying twisted integration-by-parts operator identities. In order to provide a universal space to express general integral equations, free operated algebras are then constructed in terms of bracketed words and rooted trees with decorations on the vertices and edges. Further explicit constructions of the free objects in the category of matching twisted Rota-Baxter algebras are obtained by a twisted and decorated generalization of the shuffle product, providing a universal space for separable Volterra equations. As an application of these algebraic constructions, it is shown that any integral equation with separable Volterra kernels is operator linear in the sense that the equation can be simplified to a linear combination of iterated integrals with the same kernels.

Power set and invariant subspaces of cyclic quasinilpotent operators

For a quasinilpotent operator T on a separable Hilbert space H, Douglas and Yang defined kx=limsup|z|→0log⁡‖(z−T)−1x‖log⁡‖(z−T)−1‖ for x∈H, x≠0, and called Λ(T)={kx:x∈H,x≠0} the power set of T. This paper computes the power set of cyclic quasinilpotent operators and studies the connection with their invariant subspaces. We first prove that the power set of the strictly cyclic quasinilpotent unilateral weighted shift only contains 1. Then we define the strongly strictly cyclic quasinilpotent operator, characterize its invariant subspaces, and show that its power set also only contains 1. Finally, we show that the Volterra integral operator V on L2[0,1], whose power set contains more than one point, is cyclic but not strictly cyclic.

A fast and oblivious matrix compression algorithm for Volterra integral operators

The numerical solution of dynamical systems with memory requires the efficient evaluation of Volterra integral operators in an evolutionary manner. After appropriate discretization, the basic problem can be represented as a matrix-vector product with a lower diagonal but densely populated matrix. For typical applications, like fractional diffusion or large-scale dynamical systems with delay, the memory cost for storing the matrix approximations and complete history of the data then becomes prohibitive for an accurate numerical approximation. For Volterra integral operators of convolution type, the fast and oblivious convolution quadrature method of Schädle, Lopez-Fernandez, and Lubich resolves this issue and allows to compute the discretized evaluation with N time steps in O(N log N) complexity and only requires O(log N) active memory to store a compressed version of the complete history of the data. We will show that this algorithm can be interpreted as an {{mathscr{H}}}-matrix approximation of the underlying integral operator. A further improvement can thus be achieved, in principle, by resorting to {{mathscr{H}}}^{2}-matrix compression techniques. Following this idea, we formulate a variant of the {{mathscr{H}}}^{2}-matrix-vector product for discretized Volterra integral operators that can be performed in an evolutionary and oblivious manner and requires only O(N) operations and O(log N) active memory. In addition to the acceleration, more general asymptotically smooth kernels can be treated and the algorithm does not require a priori knowledge of the number of time steps. The efficiency of the proposed method is demonstrated by application to some typical test problems.

Volterra operators between limits of Bergman-type weighted spaces of analytic functions

We characterize continuity and compactness of the Volterra integral operator $T_g$ with the non-constant analytic symbol $g$ between certain weighted Fréchet or (LB)-spaces of analytic functions on the open unit disc, which arise as projective (resp. inductive) limits of intersections (resp. unions) of Bergman spaces of order $1&amp;lt;p&amp;lt;\infty$ induced by the standard radial weight $(1-|z|^2)^\alpha$ for $0&amp;lt;\alpha&amp;lt;\infty$. Motivated from the earlier results obtained for weighted Bergman spaces of standard weight, we also establish several results concerning the spectrum of the Volterra operators acting on the weighted Bergman Fréchet space $A^p_{\alpha+}$, and acting on the weighted Bergman (LB)-space $A^p_{\alpha-}$.