Integral Equations Of Convolution Type Research Articles

On Solutions of a System of Nonlinear Integral Equations of Convolution Type on the Entire Real Line

We consider a special system of integral equations of convolution type with a monotone convex nonlinearity naturally arising when searching for stationary or limit states in various dynamic models of applied nature, for example, in models of the spread of epidemics, and prove theorems stating the existence or absence of a nontrivial bounded solution with limits at +@ depending on the values of these limits and on the structure of the matrix kernel of the system. We also study the uniqueness of such a solution assuming that it exists. Specific examples of systems whose parameters satisfy the conditions stated in our theorems are given.

Ob asimptotike spektra integral'nogo operatora s logarifmicheskim yadrom spetsial'nogo vida

We study the asymptotic behavior of the spectrum of an integral operator similarto an integral operator with a logarithmic kernel depending on the sum of arguments. Bya simple change of variables, the corresponding equation is reduced to an integral equation ofconvolution type defined on a finite interval (as is well known, such equations in the general casecannot be solved by quadratures). Next, using the Fourier transform, the equation is reduced toa conjugation problem for analytic functions and then to an infinite system of linear algebraicequations, the isolation of the main terms in which allows deriving a relation that determinesthe spectrum of the original problem.

On Solutions of a System of Nonlinear Integral Equations of Convolution Type on the Entire Real Line

On Solutions of a System of Nonlinear Integral Equations of Convolution Type on the Entire Real Line

On nonlinear convolution-type integral equations in the theory of $$p$$-adic strings

On nonlinear convolution-type integral equations in the theory of $$p$$-adic strings

On nonlinear convolution-type integral equations in the theory of $p$-adic strings

Исследуется класс интегральных уравнений типа свертки на всей прямой с монотонной и нечетной нелинейностью. Доказаны конструктивные теоремы существования и отсутствия неотрицательных (нетривиальных) и ограниченных решений. Изучается асимптотическое поведение построенного решения на $\pm\infty$. Доказывается также единственность решения в классе неотрицательных (ненулевых) и ограниченных функций. Приводятся частные примеры указанного класса уравнений, имеющих прикладной характер в различных областях математической физики.

A System of Inhomogeneous Integral Equations of Convolution Type with Power Nonlinearity

A System of Inhomogeneous Integral Equations of Convolution Type with Power Nonlinearity

Fast $$\theta $$-Maruyama scheme for stochastic Volterra integral equations of convolution type: mean-square stability and strong convergence analysis

Fast $$\theta $$-Maruyama scheme for stochastic Volterra integral equations of convolution type: mean-square stability and strong convergence analysis

Fractional Fourier cosine and sine Laplace weighted convolution and its application

In this paper, four types of fractional Fourier cosine and sine Laplace weighted convolutions are defined, and the corresponding fractional Fourier cosine and sine Laplace convolution theorems associated with the fractional cosine transform, fractional sine transform and Laplace transform are derived in detail. Furthermore, the relationship between the fractional Fourier cosine (sine) Laplace weighted convolutions and existing convolutions are studied, and Young's type theorem is also investigated. In addition, as an application for fractional Fourier cosine (sine)-Laplace weighted convolution, the system of convolution-type integral equations is considered, the computational complexity is analysed and the explicit solutions for these equations are obtained.

The attractivity of functional hereditary integral equations

In this paper, the problem of attractivity of solutions for functional hereditary integral equations is initiated. A hereditary integral equation is converted into a second kind Volterra integral equation of convolution type by use of the property of convolution. Sufficient conditions for the existence of attractive solutions are established according to the fixed point theorem. The conclusions presented in this paper extend some published results. The effectiveness of the proposed results is illustrated by three numerical examples.

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.

Solvability of a system of integral equations in two variables in the weighted Sobolev space $W^{1,1}_\omega(a,b)$ using a generalized measure of noncompactness

In this paper, we deal with the existence of solutions for a coupled system of integral equations in the Cartesian product of weighted Sobolev spaces E = Wω1,1 (a,b) x Wω1,1 (a,b). The results were achieved by equipping the space E with the vector-valued norms and using the measure of noncompactness constructed in [F.P. Najafabad, J.J. Nieto, H.A. Kayvanloo, Measure of noncompactness on weighted Sobolev space with an application to some nonlinear convolution type integral equations, J. Fixed Point Theory Appl., 22(3), 75, 2020] to applicate the generalized Darbo’s fixed point theorem [J.R. Graef, J. Henderson, and A. Ouahab, Topological Methods for Differential Equations and Inclusions, CRC Press, Boca Raton, FL, 2018].

Solvability of infinite system of nonlinear convolution type integral equations in the tempered sequence space mβ(φ,p)

In this paper, we first introduce the concept of tempered sequence space [Formula: see text]. Then, we construct a Hausdorff measure of noncompactness on this sequence space. Furthermore, by employing this measure of noncompactness we discuss the existence of solutions for an infinite system of nonlinear convolution differential equations in the tempered sequence space [Formula: see text]. Eventually, we demonstrate an example to show the effectiveness and usefulness of the obtained result.

An Algorithm for the Closed-Form Solution of Certain Classes of Volterra–Fredholm Integral Equations of Convolution Type

In this paper, a direct operator method is presented for the exact closed-form solution of certain classes of linear and nonlinear integral Volterra–Fredholm equations of the second kind. The method is based on the existence of the inverse of the relevant linear Volterra operator. In the case of convolution kernels, the inverse is constructed using the Laplace transform method. For linear integral equations, results for the existence and uniqueness are given. The solution of nonlinear integral equations depends on the existence and type of solutions ofthe corresponding nonlinear algebraic system. A complete algorithm for symbolic computations in a computer algebra system is also provided. The method finds many applications in science and engineering.

On One Class of Multidimensional Integral Equations of Convolution Type with Convex Nonlinearity

On One Class of Multidimensional Integral Equations of Convolution Type with Convex Nonlinearity

Nonlinear Integral Equations with Potential-Type Kernels in the Nonperiodic Case

We find conditions under which a generalized potential-type operator acts continuously from a Lebesgue space with a general weight to its dual space and possesses the positivity property. Based on these conditions, we prove the global existence and uniqueness theorems for various classes of nonlinear integral equations of convolution type in real weighted Lebesgue spaces using the method of monotonic (in the Browder–Minty sense) operators. Also we obtain estimates of the norms of solutions, which imply that the corresponding homogeneous equations have only a trivial solution.

Some approximation results for mild solutions of stochastic fractional order evolution equations driven by Gaussian noise

We investigate the quality of space approximation of a class of stochastic integral equations of convolution type with Gaussian noise. Such equations arise, for example, when considering mild solutions of stochastic fractional order partial differential equations but also when considering mild solutions of classical stochastic partial differential equations. The key requirement for the equations is a smoothing property of the deterministic evolution operator which is typical in parabolic type problems. We show that if one has access to nonsmooth data estimates for the deterministic error operator together with its derivative of a space discretization procedure, then one obtains error estimates in pathwise Hölder norms with rates that can be read off the deterministic error rates. We illustrate the main result by considering a class of stochastic fractional order partial differential equations and space approximations performed by spectral Galerkin methods and finite elements. We also improve an existing result on the stochastic heat equation.

Alternating bounded solutions of a class of nonlinear two-dimensional convolution-type integral equations

This paper is devoted to studying a class of nonlinear two-dimensional convolution-type integral equations on R 2 \mathbb {R}^2 . This class of equations has applications in the theory of p p -adic open-closed strings and in the mathematical theory of the spread of epidemics in space and time. The existence of an alternating bounded solution is proved. The asymptotic behaviour of the constructed solution is also studied in a particular case. At the end of the paper, specific applied examples of these equations are given to illustrate the results. UDK 517.968.4.

On the Generalization of the Random Projection Method for Problems of the Recovery of Object Signal Described by Models of Convolution Type

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.

Time-domain formulation of electromagnetic scattering based on a polarization-mode expansion and the principle of least action

An approach to the full wave analysis of the time evolution of the polarization induced in the electromagnetic scattering from dispersive nonmagnetic particles is presented. It is based on the combination of the Hopfield model for the polarization field, the expansion of the polarization field in terms of static longitudinal and transverse modes of the particle, the expansion of the radiation field in terms of transverse wave modes of free space, and the principle of least action. The polarization field is linearly coupled to the electromagnetic field. The losses of the matter are described thorough a linear coupling of the polarization field to a bath of harmonic oscillators with a continuous range of natural frequencies. The set of linear ordinary differential integral equations of convolution type of the overall system is reduced by eliminating both the radiation degrees of freedom and the bath degrees of freedom, and the reduced system of equations is studied. The role played by the radiation field in the coupling between the longitudinal and transverse mode amplitudes of the polarization is described. The principal characteristics of the temporal evolution of the mode amplitudes are found as the particle size varies, including the impulse response. Results are presented for the analytically solvable spherical particle. The proposed approach leads to a general method for the analysis of the temporal evolution of the polarization field induced in dispersive particles of any shape, as well as for the computation of transients and steady states.

A unified construction of product formulas and convolutions for Sturm–Liouville operators

We establish a positive product formula for the solutions of the Sturm–Liouville equation $$\ell (u) = \lambda u$$ , where $$\ell $$ belongs to a general class which includes singular and degenerate Sturm–Liouville operators. Our technique relies on a positivity theorem for possibly degenerate hyperbolic Cauchy problems and on a regularization method which makes use of the properties of the diffusion semigroup generated by the Sturm–Liouville operator. We show that the product formula gives rise to a probability preserving convolution algebra structure on the space of finite measures which satisfies the basic axioms for developing harmonic analysis on the convolution algebra. Unlike previous works, our framework includes a subfamily of Sturm–Liouville operators for which the support of the convolution of Dirac measures is noncompact. The connection with hypergroup theory is discussed. Convolution-type integral equations on weighted Lebesgue spaces are also discussed, and a solvability condition is established.