Integral Equations Of Second Kind Research Articles

Spectral methods for weakly singular Volterra integral equations with pantograph delays

In this paper, the convergence analysis of the Volterra integral equation of second kind with weakly singular kernel and pantograph delays is provided. We use some function transformations and variable transformations to change the equation into a new Volterra integral equation with pantograph delays defined on the interval [−1, 1], so that the Jacobi orthogonal polynomial theory can be applied conveniently. We provide a rigorous error analysis for the proposed method in the L ∞-norm and the weighted L 2-norm. Numerical examples are presented to complement the theoretical convergence results.

Analysis of Mode I Conducting Crack in Piezo-Electro-Magneto-Elastic Layer

Within the theory of linear magnetoelectroelasticity, the fracture analysis of a magneto - electrically dielectric crack embedded in a magnetoelectroelastic layer is investigated. The prescribed displacement, electric potential and magnetic potential boundary conditions on the layer surfaces are adopted. Applying the Hankel transform technique, the boundary - value problem is reduced to solving three coupling Fredholm integral equations of second kind. These equations are solved exactly. The corresponding semi - permeable crack - face magnetoelectric boundary conditions are adopted and the electric displacement and magnetic induction of crack interior are obtained explicitly. This field inside the crack is dependent on the material properties, applied loadings, the dielectric permittivity and magnetic permeability of crack interior, and the ratio of the crack length and the layer thickness. Field intensity factors are obtained as explicit expressions.

Propagación de ondas de Rayleigh en medios con grietas

This work is focused on the finding of numerical results for detection and characterization of sub-surface cracks in solids under the incidence of Rayleigh's elastic waves. The results are obtained from boundary integral equations, which belong to the field of dynamics of elasticity. Once applied the boundary conditions, a system of Fredholm's integral equations of second kind and zero order is obtained, which is solved using Gaussian elimination. The method that is used for the solution of such integral equations is known as the Indirect Boundary Element Method, which can be seen as a derivation of the Somigliana's classic theorem. On the basis of the analysis made in the frequency domain, resonance peaks emerge and allow us to infer the presence of cracks through the spectral ratios. Several models of cracked media were analyzed, where analyses reveal the great utility that displays the use of spectral ratios to identify cracks. We studied the effects of orientation and location of cracks. The results show good agreement with the previously published.

Multilevel correction for collocation solutions of Volterra integral equations with proportional delays

In this paper, we propose a convergence acceleration method for collocation solutions of the linear second-kind Volterra integral equations with proportional delay qt $(0<q<1)$ . This convergence acceleration method called multilevel correction method is based on a kind of hybrid mesh, which can be viewed as a combination between the geometric meshes and the uniform meshes. It will be shown that, when the collocation solutions are continuous piecewise polynomials whose degrees are less than or equal to ${m} (m \leqslant 2)$ , the global accuracy of k level corrected approximation is $O(N^{-(2m(k+1)-\varepsilon)})$ , where N is the number of the nodes, and $\varepsilon$ is an arbitrary small positive number.

Convolution/deconvolution of generalized Gaussian kernels with applications to proton/photon physics and electron capture of charged particles

Scatter processes of photons lead to blurring of images produced by CT (computed tomography) or CBCT (cone beam computed tomography). Multiple scatter is described by, at least, one Gaussian kernel. In various tasks, this approximation is crude; we need two/three Gaussian kernels to account for long-range tails (Landau tails), which appear in Molière scatter of protons, energy straggling and electron capture of charged particles passing through matter and Compton scatter. The ideal image (source function) is subjected to Gaussian convolution to yield a blurred image. The inverse problem is to obtain the source image from a detected image. Deconvolution methods of linear combinations of two/three Gaussian kernels with different parameters s0, s1, s2 can be derived via an inhomogeneous Fredholm integral equation of second kind (IFIE2) and Liouville - Neumann series (LNS) to provide the source function ρ. Scatter functions s0, s1, s2 are best determined by Monte-Carlo. An advantage of LNS is given, if the scatter functions s0, s1, s2 depend on coordinates. The convergence criterion can always be satisfied with regard to the above mentioned cases. A generalization is given by an analysis of the Dirac equation and Fermi-Dirac statistics leading to Landau tails applied to Bethe-Bloch equation (BBE) of charged particles and electron capture.

Accuracy Improvement of the Second-Kind Integral Equations for Generally Shaped Objects

In computational electromagnetics, second-kind integral equations are usually considered less accurate than their first-kind counterparts. The loss of the numerical accuracy is mainly due to the discretization error of the identity operators involved in second-kind IEs. In the previous studies, it was shown that by using the Buffa-Christiansen (BC) functions as testing functions, such a discretization error can be suppressed significantly, and the numerical accuracy of the second-kind IEs in the far-field calculation for spherical objects can be improved dramatically. In this paper, this technique is generalized for generally shaped objects in both perfect electric conductor and dielectric cases by using the BC functions as the testing functions, and by handling the near-singularities in the evaluation of the system matrix elements carefully. The extinction theorem is applied for accurate evaluation of the numerical errors in the calculation of scattering problems for generally shaped objects. Several examples are given to demonstrate the performance of this technique, and several important conclusive remarks are drawn.

A note on the alternate trapezoidal quadrature method for Fredholm integral eigenvalue problems

We consider an approximate method based on the alternate trapezoidal quadrature for the eigenvalue problem given by a periodic singular Fredholm integral equation of second kind. For some convolution-type integral kernels, the eigenvalues of the discrete eigenvalue problem provided by the alternate trapezoidal quadrature method have multiplicity at least two, except up to two eigenvalues of multiplicity one. In general, these eigenvalues exhibit some symmetry properties that are not necessarily observed in the eigenvalues of the continuous problem. For a class of Hilbert-type kernels, we provide error estimates that are valid for a subset of the discrete spectrum. This subset is further enlarged in an improved quadrature method presented herein. The results are illustrated through numerical examples.

Numerical Methods for Solving Fredholm Integral Equations of Second Kind

Integral equation has been one of the essential tools for various areas of applied mathematics. In this paper, we review different numerical methods for solving both linear and nonlinear Fredholm integral equations of second kind. The goal is to categorize the selected methods and assess their accuracy and efficiency. We discuss challenges faced by researchers in this field, and we emphasize the importance of interdisciplinary effort for advancing the study on numerical methods for solving integral equations.

Integral-Algebraic Equations: Theory of Collocation Methods I

Our analysis of collocation solutions for general systems of linear integral-algebraic equations (IAEs) is based on the notions of the tractability index and the $\nu$-smoothing property of a Volterra integral operator. These are used to decouple the given IAE system into the inherent system of regular second-kind Volterra integral equations (VIEs) and a system of first-kind VIEs. This decoupling is then used to derive the optimal convergence properties of piecewise polynomial collocation solutions. Numerical examples illustrate the theoretical results.

Vector estimators of the Monte Carlo method with a finite variance

In this paper we study problems related to the finiteness of the variance of weighted Monte Carlo estimators for solving a system of second-kind integral equations. Modifications of the finiteness criterion for the variance of a weighted scalar estimator are based on the construction of an appropriate system of linear integral equations with ‘majorant’ kernels. A majorant criterion of the finiteness of a vector weighted estimator is given. This criterion uses a comparison of the estimator with a scalar estimator obtained by the method of randomization. It is shown that for the corresponding randomized algorithm the variance of the scalar estimator and the mean simulation time for a single trajectory are bounded if the original solution is bounded. The ability to use branching to construct a vector estimator of the solution to a system of integral equations is also studied.

Inverse Scattering Problem for the Maxwell’s Equations

Inverse scattering problem is discussed for the Maxwell's equations. A reduction of the Maxwell's system to a new Fredholm second-kind integral equation with a scalar weakly singular kernel is given for electromagnetic (EM) wave scattering. This equation allows one to derive a formula for the scattering amplitude in which only a scalar function is present. If this function is small (an assumption that validates a Born-type approximation), then formulas for the solution to the inverse problem are obtained from the scattering data: the complex permittivity � 0 (x) in a bounded region DR 3 is found from the scattering amplitude A(�;�;k) known for a fixed k = ! p �0�0 > 0 and all �;� 2 S 2 , where S 2 is the unit sphere in R 3 , �0 and �0 are constant permittivity and magnetic permeability in the exterior region D 0 = R 3 \D. The novel points in this paper include: i) A reduction of the inverse problem for vector EM waves to a vector integral equation with scalar kernel without any symmetry assumptions on the scatterer, ii) A derivation of the scalar integral equation of the first kind for solving the inverse scattering problem, and iii) Presenting formulas for solving this scalar integral equation. The problem of solving this integral equation is an ill-posed one. A method for a stable solution of this problem is given.

Romberg Algorithm to Solve a System of Nonlinear Fredholm Integral Equations of Second Kind

Romberg Algorithm to Solve a System of Nonlinear Fredholm Integral Equations of Second Kind

Well-conditioned boundary integral equations for two- dimensional sound-hard scattering problems in domains with corners

We present several well-posed, well-conditioned direct and indirect integral equation formulations for the solution of two-dimensional acoustic scattering problems with Neumann boundary conditions in domains with corners. We focus mainly on Direct Regularized Combined Field Integral Equations (DCFIE-R) formulations whose name reflects that (1) they consist of combinations of direct boundary integral equations of the second-kind and first-kind integral equations which are preconditioned on the left by coercive boundary single-layer operators, and (2) their unknowns are physical quantities, i.e the total field on the boundary of the scatterer. The DCFIE-R equations are shown to be uniquely solvable in appropriate function spaces under certain assumptions on the coupling parameter. Using Calderon’s identities and the fact that the unknowns are bounded in the neighborhood of the corners, the integral operators that enter the DCFIE-R formulations are recast in a form that involves integral operators that are expressed by convergent integrals only. The polynomially-graded mesh quadrature introduced by Kress [30] enables the high-order resolution of the weak singularities of the kernels of the integral operators and the singularities in the derivatives of the unknowns in the vicinity of the corners. This approach is shown to lead to an efficient, high-order Nystrom method capable of producing solutions of sound-hard scattering problems in domains with corners which require small numbers of Krylov subspace iterations throughout the frequency spectrum. We present a variety of numerical results that support our claims.

Influence of initial stresses on the stressed state of a composite with a periodic system of parallel coaxial normal tensile cracks

Within the framework of the linearized mechanics of deformable solids, we have investigated the axisymmetric problem of fracture of a prestressed composite material with a periodic system of parallel coaxial normal tensile cracks. Using the representations of the general solutions of linearized equilibrium equations via the harmonic potential functions and the technique of Hankel integral transformations, we have reduced the problem to a system of paired integral equations and then to a resolving second-kind Fredholm integral equation. We have obtained expressions for the stress intensity factors near the crack contours and for the case of a layered composite with isotropic layers. We have also analyzed the dependence of stress intensity factors on the initial stresses, the mechanical characteristics of components of the material, and the geometric parameters of the problem.

Optimal homotopy asymptotic method for solving nonlinear Fredholm integral equations of second kind

The aim of this communication is to present the effectiveness of optimal homotopy asymptotic method (OHAM) for solving nonlinear Fredholm integral equations of second kind. Here, we consider the first, second and third order Fredholm integral equations of second kind. Comparisons are made with Adomian Decomposition Method, Discrete Adomian Decomposition Method, Homotopy Perturbation Method, Quasi Interpolation Method, Triangular Functions Method, Trapezoidal rule and Simpson rule. The results reveal that the OHAM is very effective, simple and has close agreement with exact solutions.

Prediction Method of Long-Term Mechanical Behavior of Largely Deformed Sand Asphalt with Constant Loading Creep Tests

Because of large pronounced deformation in the compressive creep process of soft sand asphalt, it is necessary that both the true stress and true strain are measured. Accordingly, instead of fictitious creep compliance (FCC), true creep compliance (TCC) relating the true strain to the true stress is used to characterize the viscoelastic properties of sand asphalt in this paper. The relationship between TCC and FCC is described by a second-kind Volterra integral equation (VIE). A prediction method of long-term mechanical behavior of largely deformed sand asphalt with constant loading creep tests is proposed. In this method, the FCC master curves are constructed based on the time-temperature superposition principle, and then the linear VIE from the FCC to the TCC is solved with the collocation method, and the nonlinear VIE from the TCC to the FCC is solved with an iterative formula. Unconstrained compressive creep tests for 3,600 s were conducted on sand asphalt mixture samples in various temperature and nominal stress conditions. As an application example, the long-term creep behavior of sand asphalt at the given reference temperature is predicted with the proposed method.

Application of A Second Derivative Multi-Step Method to Numerical Solution of Volterra Integral Equation of Second Kind

Normal 0 false false false EN-US X-NONE X-NONE As is known, many problems of natural science are reduced mainly to the solution of nonlinear Volterra integral equations. The method of quadratures that was first applied by Volterra to solving variable boundary integral equations is popular among numerical methods for the solution of such equations. At present, there are different modifications of the method of quadratures that have bounded accuracies. Here we suggest a second derivative multistep method for constructing more exact methods.

Kernel perturbations for a class of second-kind convolution Volterra equations with non-negative kernels

Abstract Within the class of second-kind Volterra equations, an important subclass is the second-kind convolution equations with non-negative kernels k , with ‖ k ‖ 1 1 , and positive forcing terms. Sharper results about the effect of kernel perturbations are obtained if attention is focussed on this subclass. In the standard perturbation and stability analysis for second-kind Volterra integral equations, it is the effect on the solution of perturbations in the forcing term and the stability of numerical methods for their solution that are examined. In applications, where only approximations for the kernel are available, such as arise in rheology, risk analysis, and renewal theory, the effect on the solution of perturbations in the kernel is equally important. In this paper, estimates are derived for this situation. The resolvent kernel framework is used to establish conditions such that, with respect to relative error perturbations in the kernel, the corresponding relative error perturbations in the solution are bounded. Defect renewal equations form a subset of the convolution Volterra integral equations examined.

DYNAMIC INTERACTION OF A RIGID PLATE WITH TRANSVERSELY ISOTROPIC FULL-SPACE

This paper deals with the dynamic interaction of a massless rigid circular plate and a transversely isotropic full-space in frequency domain. The plate is located in an arbitrary place in the full-space in such a way its normal vector is in the direction of the axis of material symmetry of the domain, and undergoes a vertical displacement with a frequency of . Using potential representation and Henkel integral transform, and satisfying the continuity and regularity conditions, the governing equations are transformed to dual integral equations, which can be changed to a Fredholm integral equation of second kind. The Fredholm integral equation is solved analytically in the static case, where is zero, and is numerically evaluated in the general dynamic case. The results are shown to be identical with the simple case of isotropy in both analytical form and numerical evaluation.

Solution and Error Analysis of Two Dimensional Fredholm-Volterra Integral Equations Using Piecewise Constant Functions

In this paper, the piecewise constant Block-Pulse functions and their operational matrices of integration have directly been used to solve a two-dimensional Fredholm-Volterra integral equation of second kind. This method presents a computational technique through converting this integral equation into a system of linear equations which can be easily solved by the known methods. Also the error analysis of this method will be considered. The efficiency and accuracy of the proposed method are illustrated by some examples.