Solution of an inverse problem for a multidimensional integral equation of convolution type

A VARIATION-OF-CONSTANTS FORMULA FOR NONLINEAR VOLTERRA INTEGRAL EQUATIONS OF CONVOLUTION TYPE

Existence of solutions of functional integral equations of convolution type using a new construction of a measure of noncompactness on [formula omitted

We propose a method of solving coupled thermodiffusion problems for layered bodies of canonical shape. The method is based on separating the coupled system of equations and boundary conditions into independent boundary-value problems. The solution contains arbitrary functions of time determined from a system of second-order integral equations of convolution type obtained as a result of satisfying the boundary conditions.

Introduction. In technical systems, there is a common situation when transformation input-output is described by the integral equation of convolution type. This situation accurses if the object signal is recovered by the results of remote measurements. For example, in spectrometric tasks, for an image deblurring, etc. Matrices of the discrete representation for the output signal and the kernel of convolution are known. We need to find a matrix of the discrete representation of a signal of the object. The well known approach for solving this problem includes the next steps. First, the kernel matrix has to be represented as the Kroneker product. Second, the input-output transformation has to be presented with the usage of Kroneker product matrices. Third, the matrix of the discrete representation of the object has to be found. The object signal matrix estimation obtained with the help of pseudo inverting of Kroneker decomposition matrices is unstable. The instability of the object signal estimation in the case of usage of Kroneker decomposition matrices is caused by their discrete ill posed matrix properties (condition number is big and the series of the singular numbers smoothly decrease to zero). To find solutions of discrete ill-posed problems we developed methods based on the random projection and the random projection with an averaging by the random matrices. These methods provide a stable solutions with a small computational complexity. We consider the problem of object signals recovering in the systems where an input-output transformation is described by the integral equation of a convolution. To find a solution for these problems we need to build a generalization for two-dimensional signals case of the random projection method. Purpose. To develop a stable method of the recovery of object signal for the case in which an input-output transformation is described by the integral equation of a convolution. Results and conclusions. We developed the method of a stable recovery of object signal for the case in which an input-output transformation is described by the integral equation of a convolution. The stable estimation of the object signal is provided by Kroneker decomposition of the kernel matrix of convolution, computation of random projections for Kroneker factorization matrices, and a selection of the optimal dimension of a projector matrix. The method is illustrated by its application in technical problems.

Fast collocation methods for Volterra integral equations of convolution type

Endemic infectious diseases for which infection confers permanent immunity are described by a system of nonlinear Volterra integral equations of convolution type. These constant-parameter models include vital dynamics (birth and deaths), immunization and distributed infectious period. The models are shown to be well posed, the threshold criteria are determined and the asymptotic behavior is analysed. It is concluded that distributed delays do not change the thresholds and the asymptotic behaviors of the models.

We propose a mathematical model of infection spreading among the adult population of certain region. The model is constructed on the basis of some delay differential equations that are supplemented with integral equations of convolution type and the initial data. The variables included in the integral equations and the delay variables take into account the number of individuals in different groups and the transition rate of individuals between the groups which reflects the stages of the disease. Some properties of the solutions of the model are under study including the existence, uniqueness, and nonnegativity of the solution components on the half-axis, as well as the presence and stability of the equilibrium states. We formulate and solve the problem of eliminating infection during finite time. The time for infection eradication is estimated on using the exponentially decreasing component-by-component estimates of the solution. Also we present the results of computational experiments on estimating the eradication time and evaluating the effectiveness of the process of diagnosis and identification of sick (infected) individuals through the procedure of regular medical examinations.

Laguerre transformation as a tool for the numerical solution of integral equations of convolution type

Let A be the set of all complex-valued locally integrable functions defined on [0, +oo), and let T be the topology for A determined by the seminorms 'r(/)=J"o I/O*)I dx for r-\, 2, • • • , so that A is a topological algebra under pointwise addition, complex scalar multiplication, and Laplace convolution.Then the map /->•/' from each element to its quasi-inverse is a homeomprphism of (A, T) onto itself.For each/, g in A the equation v=f+ g *v has a unique solution in A which depends 7"-continuously on fig, and is the T-Iimit of Picard approximations.The set of all/in A with/' in Ll[Q, +oo) is a set of first category in (A, T) but an open subset of A with the metric \\f-g\\ i.For each series 2™=i/?Bz" converging in some neighborhood of z=0, and each element/in A, the series 2nLi/>»/*" converges in TXo some element p*(f) in A.

Inverse problems for the identification of memory kernels in the linear theory of viscoelasticity with constitutive stress-strain-relation of Boltzmann type are dealt with in the case of weakly singular kernels in the space L p , and of continuous kernels with power singularity at zero. The problems are reduced to nonlinear Volterra integral equations of convolution type for which by the method of contraction with weighted norms global existence, uniqueness, and stability of solutions are proved.

In this paper, we show that arbitrary Hermite function or appropriate linear combination of those functions is a weight-function of four explicit generalized convolutions for the Fourier cosine and sine transforms. With respect to applications, normed rings on \({L^1(\mathbb{R}^d)}\) are constructed, and sufficient and necessary conditions for the solvability and explicit solutions in \({L^1(\mathbb{R}^d)}\) of the integral equations of convolution type are provided by using the constructed convolutions.

Qualitative behaviour and stability of solutions of discretised nonlinear Volterra integral equations of convolution type

Volterra functions were introduced at the beginning of the twentieth century as solutions of some integral equations of convolution type with logarithmic kernel. Since then, few authors have studied this family of functions and faced with the problem of providing a clear understanding of their asymptotic behavior for small and large arguments. This paper reviews some of the most important results on Volterra functions and in particular collects, into a quite general framework, several results on their asymptotic expansions; these results turn out to be useful not only for the full understanding of the behavior of the Volterra functions but also for their numerical computation. The connections with integrals of Ramanujan type, which have several important applications, are also discussed.

Part I: Transmutations, Integral Equations and Special Functions.- Some Recent Developments in the Transmutation Operator Approach.- Transmutation Operators and Their Applications.- Hankel Generalized Convolutions with the Associated Legendre Functions in the Kernel and Their Applications.- Second Type Neumann Series Related to Nicholson's and to Dixon-Ferrar Formula.- On Some Generalizations of the Properties of the Multidimensional Generalized Erdelyi-Kober Operators and Their Applications.- Alternative Approach to Miller-Paris Transformations and Their Extensions.- Transmutation Operators For Ordinary Dunkl-Darboux Operators.- Theorems on Restriction of Fourier-Bessel and Multidimensional Bessel Transforms to Spherical Surfaces.- Necessary Condition for the Existence of an Intertwining Operator and Classification of Transmutations on Its Basis.- Polynomial Quantization on Line Bundles.- Fourier-Bessel Transforms of Measures and Qualitative Properties of Solutions of Singular Differential Equations.- Inversion of Hyperbolic B-Potentials.- One-Dimensional and Multi-Dimensional Integral Transforms of Buschman-Erdelyi Type with Legendre Functions in Kernels.- Distributions, Non-smooth Manifolds, Transmutations and Boundary Value Problems.- Part II: Transmutations in ODEs, Direct and Inverse Problems.- On a Transformation Operator Approach in the Inverse Spectral Theory of Integral and Integro-Differential Operators.- Expansion in Terms of Appropriate Functions and Transmutations.- Transmutation Operators as a Solvability Concept of Abstract Singular Equations.- On the Bessel-Wright Operator and Transmutation with Applications.- On a Method of Solving Integral Equation of Carleman Type on the Pair of Segments.- Transmutation Operators Boundary Value Problems.- Solution of Inverse Problems for Differential Operators with Delay.- Part III: Transmutations for Partial and Fractional Differential Equations.- Transmutations of the Composed Erdelyi-Kober Fractional Operators and Their Applications.- Distributed Order Equations in Banach Spaces with Sectorial Operators.- Transformation Operators for Fractional Order Ordinary Differential Equations and Their applications.- Strong Solutions of Semilinear Equations with Lower Fractional Derivatives.- Mean Value Theorems and Properties of Solutions of Linear Differential Equations.- Transmutations for Multi-Term Fractional Operators.- Fractional Bessel Integrals and Derivatives on Semi-axes.- The Fractional Derivative Expansion Method in Nonlinear Dynamics of Structures: A Memorial Essay.- Boundary Value Problem with Integral Condition for the Mixed Type Equation with a Singular Coefficient.

Biased estimation for nonparametric identification of linear systems