Convolution Type Research Articles

On the Exact Solution of Nonlocal Euler–Bernoulli Beam Equations via a Direct Approach for Volterra-Fredholm Integro-Differential Equations

First, we develop a direct operator method for solving boundary value problems for a class of nth order linear Volterra–Fredholm integro-differential equations of convolution type. The proposed technique is based on the assumption that the Volterra integro-differential operator is bijective and its inverse is known in closed form. Existence and uniqueness criteria are established and the exact solution is derived. We then apply this method to construct the closed form solution of the fourth order equilibrium equations for the bending of Euler–Bernoulli beams in the context of Eringen’s nonlocal theory of elasticity (two phase integral model) under a transverse distributed load and simply supported boundary conditions. An easy to use algorithm for obtaining the exact solution in a symbolic algebra system is also given.

Inhomogeneous affine Volterra processes

We extend recent results on affine Volterra processes to the inhomogeneous case. This includes moment bounds of solutions of Volterra equations driven by a Brownian motion with an inhomogeneous kernel K(t,s) and inhomogeneous drift and diffusion coefficients b(s,Xs) and σ(s,Xs). In the case of affine b and σσT we show how the conditional Fourier–Laplace functional can be represented by a solution of an inhomogeneous Riccati–Volterra integral equation. For a kernel of convolution type K(t,s)=K¯(t−s) we establish existence of a solution to the stochastic inhomogeneous Volterra equation. If in addition b and σσT are affine, we prove that the conditional Fourier–Laplace functional is exponential–affine in the past path. Finally, we apply these results to an inhomogeneous extension of the rough Heston model used in mathematical finance.

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.

Closed-Form Solution of the Bending Two-Phase Integral Model of Euler-Bernoulli Nanobeams

Recent developments have shown that the widely used simplified differential model of Eringen’s nonlocal elasticity in nanobeam analysis is not equivalent to the corresponding and initially proposed integral models, the pure integral model and the two-phase integral model, in all cases of loading and boundary conditions. This has resolved a paradox with solutions that are not in line with the expected softening effect of the nonlocal theory that appears in all other cases. In addition, it revived interest in the integral model and the two-phase integral model, which were not used due to their complexity in solving the relevant integral and integro-differential equations, respectively. In this article, we use a direct operator method for solving boundary value problems for nth order linear Volterra–Fredholm integro-differential equations of convolution type to construct closed-form solutions to the two-phase integral model of Euler–Bernoulli nanobeams in bending under transverse distributed load and various types of boundary conditions.

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.

$$L^2$$ Estimates of Trilinear Oscillatory Integrals of Convolution Type on $${\mathbb {R}}^2$$

This paper is devoted to $L^2$ estimates for trilinear oscillatory integrals of convolution type on $\mathbb{R}^2$. The phases in the oscillatory factors include smooth functions and polynomials. We shall establish sharp $L^2$ decay estimates of trilinear oscillatory integrals with smooth phases, and then give $L^2$ uniform estimates for these integrals with polynomial phases.

Dualization and Automatic Distributed Parameter Selection of Total Generalized Variation via Bilevel Optimization

Total Generalized Variation (TGV) regularization in image reconstruction relies on an infimal convolution type combination of generalized first- and second-order derivatives. This helps to avoid the staircasing effect of Total Variation (TV) regularization, while still preserving sharp contrasts in images. The associated regularization effect crucially hinges on two parameters whose proper adjustment represents a challenging task. In this work, a bilevel optimization framework with a suitable statistics-based upper level objective is proposed in order to automatically select these parameters. The framework allows for spatially varying parameters, thus enabling better recovery in high-detail image areas. A rigorous dualization framework is established, and for the numerical solution, a Newton type method for the solution of the lower level problem, i.e. the image reconstruction problem, and a bilevel TGV algorithm are introduced. Denoising tests confirm that automatically selected distributed regularization parameters lead in general to improved reconstructions when compared to results for scalar parameters.

Usage of the Fuzzy Laplace Transform Method for Solving One-Dimensional Fuzzy Integral Equations

In this paper, we propose the solution of fuzzy Volterra and Fredholm integral equations with convolution type kernel using fuzzy Laplace transform method (FLTM) under Hukuhara differentiability. It is shown that FLTM is a simple and reliable approach for solving such equations analytically. Finally, the method is illustrated with few examples to show the ability of the proposed method.

Sharp Hardy's type inequality for Laguerre expansions

A method of proving Hardy's type inequality for orthogonal expansions is presented in a rather general setting. Then, sharp multi-dimensional Hardy's inequality associated with the Laguerre functions of convolution type is proved for the type index $\alpha \in [-1/2, \infty)^{d}$. The case of the standard Laguerre functions is also investigated. Moreover, the sharp analogues of Hardy's type inequality involving $L^{1}$ norms in place of $H^{1}$ norms are obtained in both settings.

Maximal singular integral operators acting on noncommutative $$L_p$$-spaces

In this paper, we study the boundedness theory for maximal Calderón–Zygmund operators acting on noncommutative $$L_p$$ -spaces. Our first result is a criterion for the weak type (1, 1) estimate of noncommutative maximal Calderón–Zygmund operators; as an application, we obtain the weak type (1, 1) estimates of operator-valued maximal singular integrals of convolution type under proper regularity conditions. These are the first noncommutative maximal inequalities for families of truly non-positive linear operators. For homogeneous singular integrals, the strong type (p, p) ( $$1<p<\infty $$ ) maximal estimates are shown to be true even for rough kernels. As a byproduct of the criterion, we obtain the noncommutative weak type (1, 1) estimate for Calderón–Zygmund operators with integral regularity condition that is slightly stronger than the Hörmander condition; this provides somewhat an affirmative answer to an open question in the noncommutative Calderón–Zygmund theory.

On the inversion and admissibility for a class of Volterra integro-differential problems

Abstract We study the Volterra integro-differential problems of convolution kernel type from two perspectives: complex inversion formula and the admissibility in the Salamon–Weiss sense (which leads us to consider unbounded control operators). Due to the hybrid Cauchy–Volterra context of these problems, the control operators need to have the admissibility property. First, we show the validity of the inverse formula of the Laplace transform for the resolvent operators associated with some class of Volterra integro-differential problems of convolution type in Banach spaces with the leading operator generating a $C_{0}$-semigroup. This extends and improves the results in Hille &amp; Phillips (1957, 347–348, Theorem 11.6.1), Driouich &amp; El-Mennaoui (1999, 72, Theorem 1), Cioranescu &amp; Lizama (2003, 188–189, Proposition 2), Haase (2008, 84, 81–82, Theorems 4.2 and 4.1) and Fadili &amp; Bounit (2014, 7–8, Proposition 8), including the strong version for this class on UMD spaces. Second, the sufficient or/and necessary conditions for $L^{p}$-admissibility with $p\in (1,+\infty )$ of the control operators are given in terms of the UMD geometric property of its underlying control spaces for a wide class of Volterra integro-differential problems, which provides a generalization of a result known to hold for the standard Cauchy problems shown in Bounit et al. (2010, Proposition 3.2) and the standard Volterra integro-differential problems shown in Fadili &amp; Bounit (2014, 9–10, Theorem 12).

Singular integrals on $$C_{w^*}^{1,\alpha }$$ regular curves in Banach duals

The modern study of singular integral operators on curves in the plane began in the 1970s. Since then, there has been a vast array of work done on the boundedness of singular integral operators defined on lower dimensional sets in Euclidean spaces. In recent years, mathematicians have attempted to push these results into a more general metric setting particularly in the case of singular integral operators defined on curves and graphs in Carnot groups. Suppose $$X = Y^*$$ for a separable Banach space Y. Any separable metric space can be isometrically embedded in such a Banach space via the Kuratowski embedding. The $$w^*$$ -derivative $$\gamma '$$ of a curve $$\gamma :[a,b] \rightarrow X$$ at $$t \in [a,b]$$ satisfies $$\tfrac{\mathrm{d}}{\mathrm{d}s} \langle \gamma (s),y \rangle |_{s=t} = \langle \gamma '(t),y\rangle$$ for any $$y \in Y$$ . Suppose $$\Gamma = \gamma ([a,b])$$ is a curve in X whose $$w^*$$ -derivative is Hölder continuous and bounded away from 0. We prove that any convolution type singular integral operator associated with a 1-dimensional Calderón–Zygmund kernel which is uniformly $$L^2$$ -bounded on lines is $$L^p$$ -bounded along $$\Gamma$$ . We also prove a version of David’s “good lambda” theorem for upper regular measures on doubling metric spaces.

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

Convolution of Decomposition Integrals

Four different types of convolutions of aggregation functions (the upper, the lower, the super-, and the sub-convolution) are examined in the setting of both sub- and super-decomposition integrals defined on a finite space. Examples of the results of the paper are provided. As a by-product, the super-additive transformation of sub-decomposition integrals and the sub-additive transformation of super-decomposition integrals are fully characterized. Possible applications are indicated.

Global well-posedness of a binary–ternary Boltzmann equation

In this paper we show global well-posedness near vacuum for the binary-ternary Boltzmann equation. The binary-ternary Boltzmann equation provides a correction term to the classical Boltzmann equation, taking into account both binary and ternary interactions of particles, and may serve as a more accurate description model for denser gases in non-equilibrium. Well-posedness of the classical Boltzmann equation and, independently, the purely ternary Boltzmann equation follow as special cases. To prove global well-posedness, we use a Kaniel-Shinbrot iteration and related work to approximate the solution of the nonlinear equation by monotone sequences of supersolutions and subsolutions. This analysis required establishing new convolution type estimates to control the contribution of the ternary collisional operator to the model. We show that the ternary operator allows consideration of softer potentials than the one binary operator, consequently our solution to the ternary correction of the Boltzmann equation preserves all the properties of the binary interactions solution. These results are novel for collisional operators of monoatomic gases with either hard or soft potentials that model both binary and ternary interactions.

Multi-directional rain streak removal based on infimal convolution of oscillation TGV

Image deraining is essential in both image processing and computer vision. However it is still challenging to remove rain streaks, due to the fact that they are complex and usually have multiple directions. In this paper, we propose a novel infimal convolution type functional based model to remove multi-directional rain streaks. Specifically, we employ total generalized variation (TGV) to represent the rain-free background and m-fold infimal convolution of oscillation TGV (ICTGVmosci) to capture the rain streaks with m different directions. The existence of solutions to the proposed model is proved. We also design an algorithm using the classical first-order Primal–Dual method for solving the proposed model. Numerical experiments on both synthetic and real images demonstrate that the proposed model outperforms the state-of-the-art methods quantitatively and qualitatively.

About nonlinear integro-differential Volterra and Fredholm equations

Two nonlinear problems in terms of abstract operator equations of the form Bx = f are investigated in this paper. In the first problem the operator B contains a linear differential operator A , the Volterra operator K with kernel of convolution type and the inner product of vectors g ( x )Ф( u ) with nonlinear bounded functionals Φ. The first problem is given by equation Bu ( x ) = Аu ( x ) - Ku ( x ) - g ( x )Ф( u ) = f ( x ) with boundary condition D ( B ) = D ( A ). In the second problem the operator B contains a linear differential operator A and the inner product of vectors g ( x )F( Аu ) with nonlinear bounded on C [ a , b ] functionals F , where F ( Аu ) denotes the nonlinear Fredholm integral. The second problem is presented by equation Bu = Аu - gF ( Au ) = f with boundary condition D ( B ) = D ( A ). A direct method for exact solutions of nonlinear integro-differential Volterra and Fredholm equations is proposed. Specifically, the three theorems about existing exact solutions are proved in this paper. The first theorem is mean that for nonzero constant α0 Volterra integro-differential equation Аu ( x ) - Ku ( x ) = 0 is reducing to Volterra integral equation and has a unique zero solution. During it the operator A - K is closed and continuously invertible. Also, if the functions u ( t ), g ( t ) and f ( t ) are of exponential order α then nonhomogeneous equation Аu ( x ) - Ku ( x ) = f ( x ) for each f ( x ) has a unique solution, shown in this paper. The second theorem is mean that for the first investigated problem with an injective operator A - K , for f ( x ) and g ( x ) from C [ a , b ], the exact solution is given by equation u = ( A - K )- 1 f +( A - K )- 1 g b* for every vector b* = Ф( u ) that solves nonlinear algebraic (transcendental) system of n equations b = Ф(( A - K )- 1 f +( A - K )- 1 g b). And if the last algebraic system has no solution, then investigated problem also has no solution. The third theorem is means that exact solution of the second investigated problem is given by u = A - 1( f+g d*) for every vector d* = F ( Au ) that solves nonlinear algebraic (transcendental) system of n equations d = F ( f + g d). In this case we have same property - if the last algebraic system has no solution, then investigated problem also has no solution. Two particular examples for each considered problem are shown for illustration of exact solutions giving by perform the suggested in this paper methods. In the first example was considered integro-differential Volterra and Fredholm equation and in the second case was considered equation with nonlinear Fredholm integral.

A study on convolutions in weighted lorentz spaces using steklov means

In this study, some estimations of convolution type operators defined with the help of Steklov operator in weighted Lorentz space L_ω^(p,q) (T) are obtained. Also, some basic properties of convolution type operators in these spaces are investigated.

A Multi-Dimensional Graph Convolution Network for EEG Emotion Recognition

Due to the changeable, high-dimensional, non-stationary, and other characteristics of electroencephalography (EEG) signals, the recognition of EEG signals is mostly limited to independent individuals. To deal with these issues, we propose a multi-dimensional graph convolution network (MD-GCN), which integrates EEG signals’ temporal and spatial characteristics and can classify emotions more accurately. First, we use that the asymmetry of neuron activity in the left and right hemispheres is very important for emotion prediction to initialize the adjacency matrix and perform preliminary edge prediction without considering node features. Then, we perform the feature fusion on the Inception network and then input it into the graph convolution network to learn the interrelationship between channels. Finally, we visually analyze the adjacency matrix. To evaluate the performance of the model, we conduct experiments on the SEED dataset and the SEED-IV dataset. The results show that the pre-defined adjacency matrix method can improve the accuracy of emotion recognition and the graph convolution has better performance than the same type of convolution. It also theoretically shows that the emotional state is mainly by the interaction of important brain regions.