Integral Equations Of Convolution Type Research Articles

A numerical study of parabolic problems with nonlinear boundary conditions

In this paper, we consider an initial‐boundary value problem for a parabolic equation with nonlinear boundary conditions. The solution to the problem can be expressed as a convolution integral of a Green's function and two unknown functions. We change the problem to a system of two nonlinear Volterra integral equations of convolution type. By using an explicit procedure on the basis of Sinc‐function properties, the resulting integral equations are replaced by a system of nonlinear algebraic equations, whose solution yields an accurate approximate solution to the parabolic problem. Some examples are considered to illustrate the ability of the proposed method. Copyright © 2012 John Wiley & Sons, Ltd.

RISK ANALYSIS OF A ROBOT–SAFETY DEVICE SYSTEM SUBJECTED TO A PRIORITY RULE

We introduce a robot–safety device system characterized by cold stand-by and by an admissible risky state. The system is attended by a single repairman and the robot has overall (break-in) priority in repair with regard to the safety device. We obtain an explicit formula for the point availability of the robot via an integral equation of the renewal-type. The explicit solution requires the notion of effective repair-versus-virtual repair. In order to decide whether the risky state is admissible, we also introduce a risk criterion. The criterion is always satisfied in the case of fast repair. As an example, we consider the case of Weibull–Gnedenko repair and we display a computer-plotted graph of the point availability obtained by a direct numerical solution of a convolution-type integral equation.

Applications of generalized convolutions associated with the Fourier and Hartley transforms

In this paper we present new generalized convolutions with weight-function associated with the Fourier and Hartley transforms, and consider applications. Namely, using the generalized convolutions, we construct normed rings on the space L(R), provide the sufficient and necessary condition for the solvability of a class of integral equations of convolution type, and receive the explicit solutions of those equations.

On a class of integral equations of Urysohn type with strong non-linearity

We study a class of homogeneous and non-homogeneous integral equations of Urysohn type with strong non-linearity on the positive semi-axis. It is assumed that some non-linear integral operator of Wiener-Hopf-Hammerstein type is a local minorant of the corresponding Urysohn operator. Using special methods of the linear theory of convolution-type integral equations, we construct positive solutions for these classes of Urysohn equations. We also study the asymptotic behaviour of these solutions at infinity. As an auxiliary fact in the course of the proof of these assertions, we construct a one-parameter family of positive solutions for non-linear integral equations of Wiener-Hopf-Hammerstein type whose operator is a minorant for the original Urysohn operator. We give particular examples of non-linear integral equations for which all the hypotheses of the main theorems hold.

Weighted Stepanov-like pseudo almost automorphy and applications

In this paper, we propose a new class of functions called weighted Stepanov-like pseudo almost automorphic functions, which generalize in a natural fashion the concept of almost automorphy and its various extensions. We systematically explore the properties of the weighted Stepanov-like pseudo almost automorphic functions in general Banach space including a composition result. As an application, we establish some sufficient criteria for the existence, uniqueness of the weighted pseudo almost automorphic solution to a class of partial neutral functional differential equations and also to a class of nonlinear Volterra integral equations of convolution type with infinite delay in Banach space. Some interesting examples are presented to illustrate the main findings.

An L p -theory for stochastic integral equations

We investigate the stochastic parabolic integral equation of convolution type $$ u=k_1\ast A_pu +\sum\limits_{k=1}^{\infty} k_2\star g^k+u_0, \; t\geq 0, $$ and develop an Lp-theory, 2 ≤ p < ∞, for this equation. The solution u is a function of t, ω, x with ω in a probability space and \({x\in B}\), a σ-finite measure space with positive measure Λ. The kernels k1(t), k2(t) are powers of t, i.e., multiples of tα–1, tβ–1, with \({\alpha \in (0,2), \beta \in (\frac{1}{2},2)}\), respectively. The mapping Ap is such that −Ap is a nonnegative linear operator of \({\mathcal{D}(A_p)\subset L_p(B)}\) into Lp(B). The convolution integrals \({k_2 \star g^k}\) are stochastic Ito-integrals. By combining an approach due to Krylov with transformation techniques and estimates involving fractional powers of (−Ap), we obtain existence and uniqueness results. In the case where Ap is the Laplacian, with B = Rn, sharp regularity results are obtained.

On solving a system of singular Volterra integral equations of convolution type

This paper presents approximate analytical solutions for a system of singular Volterra integral equations of convolution type by using the fractional differential transform method. The solutions are calculated in the form of convergent series with easily computable terms and also the exact solutions can be achieved by well-known series solutions. Several examples are given to demonstrate reliability and performance of the presented method.

Correcting lateral heat conduction effect in image-based heat flux measurements as an inverse problem

The image deconvolution method is developed, which is coupled with the one-dimensional (1D) analytical inverse method to calculate more accurate heat flux fields by correcting the lateral heat conduction effect in image-based surface temperature measurements. The theoretical foundation is a convolution-type integral equation with a Gaussian filter (kernel) that relates a heat flux field obtained by using the 1D inverse method on a surface to the true heat flux field. The accuracy of this method is evaluated and the standard deviation in the Gaussian filter is determined for different materials through simulations. This method is used to calculate heat flux fields in temperature-sensitive-paint measurements on a 7°-half-angle circular cone at Mach 6 in a short-duration hypersonic wind tunnel. In addition, a simple method is proposed to solve a projection problem associated with image deconvolution for a highly curved developable surface.

Convolutions for the Fourier transforms with geometric variables and applications

AbstractThis paper gives a general formulation of convolutions for arbitrary linear operators from a linear space to a commutative algebra, constructs three convolutions for the Fourier transforms with geometric variables and four generalized convolutions for the Fourier‐cosine, Fourier‐sine transforms. With respect to applications, by using the constructed convolutions normed rings on L1(Rn) are constructed, and explicit solutions of integral equations of convolution type are obtained (© 2010 WILEY‐VCH Verlag GmbH &amp; Co. KGaA, Weinheim)

On integral equations of stationary distributions for biological systems

In this paper, properties of solutions of the convolution-type integral equation $$ \left( {1 + w(x)} \right)P(x) = \left( {m*P} \right)(x) + Cm(x) $$ on the real axis are studied. The main concern is to find conditions for the function w(x) and the kernel m(x) sufficient for the existence of an admissible solution P(x), i.e., a solution which has a nonzero limit at infinity. The main results of the paper are the uniqueness theorem for the admissible solution for rapidly decreasing kernels m and the existence theorem for one-sided compactly supported kernels m.

The solvability and explicit solutions of two integral equations via generalized convolutions

This paper presents the necessary and sufficient conditions for the solvability of two integral equations of convolution type; the first equation generalizes from integral equations with the Gaussian kernel, and the second one contains the Toeplitz plus Hankel kernels. Furthermore, the paper shows that the normed rings on L 1 ( R d ) are constructed by using the obtained convolutions, and an arbitrary Hermite function and appropriate linear combination of those functions are the weight-function of four generalized convolutions associating F and F ˇ . The open question about Hermitian weight-function of generalized convolution is posed at the end of the paper.

Generalized convolutions and the integral equations of the convolution type

This article gives six new generalized convolutions of the integral transforms of Fourier type, and investigates a class of integral equations of convolution type by using the constructed convolutions. Namely, the explicit solutions in L 1(ℝ d ) of a class of integral equations of convolution type are obtained.

The Hermite Functions Related to Infinite Series of Generalized Convolutions and Applications

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.

Pseudo-almost periodic solutions of neutral integral and differential equations with applications

The existence and uniqueness of pseudo-almost periodic solutions to general neutral integral equations with deviations are obtained. For this, pseudo-almost periodic functions in two variables are considered. The results extend the corresponding ones to the convolution type integral equations. They are used to study pseudo-almost periodic solutions of general neutral differential equations and to the so-called scalar neutral logistic equation version.

Dynamic eigenvector problem in thermoelectroelasticity of dissipative ionic crystals

A two-dimensional problem in the Stroh formalism is derived for the continuum theory of thermoelectroelasticity with polarization gradients. Dissipative effects are accounted for, according to a constitutive model outlined in previous works. The eigenvector problem is studied in the frequency domain to obtain a representation of the solution in terms of two classes of modes corresponding to opposite signs of imaginary part of the eigenvalues. Impedance and admittance tensors are exploited to express the energy flux of the thermoelectroelastic transformed field across an interface S . The compatibility conditions at S are also derived. The eigenvector equations are then rewritten in the time domain to obtain two convolution-type integral equations for the Hilbert transforms of the real fields corresponding to each mode.

Regularization of Inverse Problems in Reinforced Concrete Fracture

Reinforced concrete beams with flexural cracks are simulated by the bridged crack model. The weight function method of determining stress intensity factors has been followed to derive a transformation between the crack bridging force (the rebar force) and the crack opening displacements (CODs). The matrix of the transformation is then approximated by its finite difference equivalent within finite dimensional vector spaces. Direct problem of the transformation solves for CODs, which require a known rebar force. Alternatively, the inverse problem works out the rebar force from known CODs. However, the inverse transformations of such convolution type integral equations become ill-posed if input CODs are perturbed. The Tikhonov regularization method is followed in its numerical form to regularize the linear ill-posed inverse problem. Restoration of mathematical stability and consistency are demonstrated by specific examples, where the results of the direct and the corresponding inverse problem are cross checked. Results of the direct problem (i.e., the analytical CODs) are deliberately perturbed by adding machine generated random numbers of a certain width. The inverse problems are solved with these CODs to simulate practical situations, where measured CODs data will inevitably be noisy. Computations reveal that the inverse analysis of CODs satisfactorily determines the rebar force without cross-section information.

A specific inverse problem for the Volterra convolution equation

This article attends to the specific inverse problem of determining the difference kernel k of a second kind linear Volterra integral equation of convolution type from a prescribed quotient μ of resolvent r and kernel k of the equation. A problem of this type arises, for example, in the theory of viscoelasticity and leads to a nonlinear integral equation of the third kind , for which several existence theorems are proved. Moreover, the applicability of Tikhonov's regularization method is studied. Finally, the case of non-convolution Volterra equations is also briefly discussed.

Image restoration using L1 norm penalty function

The process of estimating an original image from a given blurred and noisy image is known as image restoration. It is an ill-posed inverse problem, since one of the ways of solving it requires finding a solution to a Fredholm integral equation of convolution type in two-dimensional space. The focus of the article is to achieve a quality edge preserving image restoration using a less expensive (fast) regularization technique with L1 norm penalty function. L1 norm based approaches do not penalize edges or high frequency contents in the restored image compared to L2 norm based approaches. Total variation (TV) is an established L1 norm regularization approach that performs edge preserving image restoration, but at a high computational cost. TV regularization requires linearization of a highly nonlinear penalty term, which increases the restoration time considerably for large scale images. In order to reduce the computational cost, we extend least absolute shrinkage and selection operator (LASSO), an L1 norm minimization statistical modeling technique to image restoration. The penalty function of LASSO is an identity matrix so it is computationally fast. The metrics, like, residual error, peak signal to noise ratio (PSNR), restoration time, edge map of the restored image, and subjective visual evaluation are used to assess the performances of both methods. Based on our experimental results, we show that LASSO achieves similar quality of edge preserving restoration as TV regularization, and is approximately two times faster in computation compared to TV regularization on the same set of images. We also analyze the impact of the different degree of blurring caused by point spread functions (PSFs) corrupted by different signal to noise ratios (SNRs) on image restoration.

Reconstruction of smeared images by various methods

This paper discusses the problem of reconstruction of smeared images. The problem reduces to solving a set of one-dimensional convolution-type integral Fredholm equations of the first kind. Two methods of solving such equations are compared: the method of Fourier transformation, and the method of quadratures (using the method of Tikhonov regularization in both cases). Numerical results are presented, and it is concluded that the method of quadratures is more efficient than the method of Fourier transformation.

A class of integral equations of convolution type

Conditions (both necessary and sufficient) for the existence of a non-trivial bounded solution of the integral equation are obtained for fixed functions and satisfying the following conditions: The existence of the limits is proved and a relation between these limits, the first-order moment , and the integral norm of is found.Bibliography: 9 titles.