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Abstract: In this paper, a numerical scheme based on a Gauss-like cubature formula from Sammariva and Vianello (2007) [1] is introduced for approximate solution of integral equations over a polygonal domain with a piecewise straight lines boundary in . The proposed technique is a meshless like method with sufficient precision, which does not require any discretization of the polygon domain or any preprocessing such as mesh refinement. The error analysis of the method is provided and some numerical experiments are also presented to evaluate the performance of the proposed algorithm. [Copyright &y& Elsevier]

Abstract: The aim of this paper is to develop an analysis of the distribution of the eigenvalues of the acoustic single-layer potential for various low frequency two-dimensional multiple scattering problems. The obstacles are supposed to be distant (dilute media). In [25], it is shown that an approach based on the Gershgorin disks provides limited spectral information. We therefore introduce an alternative approach by applying the power iteration method to the limit matrix (associated with the zero order spatial modes) which results in satisfactory estimates. All these approximations are built for circular cylinders and formally extended to ellipses and rectangles for linear boundary element methods with non uniform meshes. This study is completed in [26] by spectral estimates for the case of close obstacles. [Copyright &y& Elsevier]

Abstract: We propose a new algorithm to compute the teleseismic P and S receiver function synthetics for a multilayered Cartesian structure with anisotropic flat layers. The algorithm is based on the first-order perturbation theory in which the layered background structure is assumed one-dimensional with isotropic variations in vertical direction. Anisotropic velocity perturbations acting as secondary sources constitute the heterogeneities in the medium. The total wavefield is solved using a convolutional type integral equation along with the Green's function of the one-dimensional reference medium extracted using the reflectivity method. The integral equation involves a five-fold integration in space and wavenumber domains. Four of these integrals are achieved analytically and the fifth integral, which is spatial integral in the vertical direction, is performed numerically for which the Born single scattering approximation greatly suffices. The proposed algorithm is demonstrated on some selected numerical examples adapted from published work in the literature. [Copyright &y& Elsevier]

Highlights: [•] A semi-analytical approach for the indentation of a multilayered solid is presented. [•] The method of solution is based on the Hankel transform and the transfer matrix. [•] The implementation is done with Mathematica under the form of a fast and efficient code. [•] The procedure is valid for a large wide material combinations and arbitrary layer’s thickness. [•] A series of results for flat, conical, spherical and blunted punches is given. [Copyright &y& Elsevier]

Abstract: Crack-like cohesive defect propagation within a plane orthotropic linear elastic layer is considered by assuming that the defect, and its growth under load, can be modeled as the evolving separation along a straight, predetermined nonlinear, nonuniform Needleman-type cohesive interface. The analysis exploits a general form of orthotropy rescaling originally developed for the displacement boundary value problem by Krenk (1979). It is shown that when the material is degenerate orthotropic (i.e., ρ =1, ρ is the orthotropic shear parameter) rescaling enables the determination of solutions from isotropic ones and, when the material is fully orthotropic, rescaling allows for solutions to be obtained from problems with the simpler cubic symmetry. (These are well known attributes of linear static sharp crack analysis, which depend on an alternative form of rescaling the traction boundary value problem (Suo, 1990; Suo et al, 1991).) The procedure is demonstrated by obtaining degenerate orthotropic response from isotropic solutions recently obtained by the authors in an investigation of both solitary as well as multiple cohesive defect interaction problems in layered systems under arbitrary loading (Nguyen and Levy, 2009, 2011). In order to obtain fully orthotropic solutions via rescaling, a novel integral equation formulation is developed based on exact infinitesimal strain elasticity solutions for rectangular domains composed of cubically symmetric media and subject to arbitrary loading. Explicit results are obtained for the simple edge notch bend configuration, chosen so as to shed light on the mechanisms of defect propagation in orthotropic layers. It is demonstrated that increasing the orthotropic stiffness ratio can precipitate a quasi-brittle defect growth response. Furthermore, it is well known that in a number of technically important problem geometries and loadings, static sharp crack solutions are only weakly dependent on shear parameter ρ enabling the estimation of fully orthotropic behavior from isotropic solutions (Suo et al, 1991). This result is shown to be true for nonlinear cohesive fracture analysis of the edge notch bend configuration analyzed in this study. [Copyright &y& Elsevier]

Abstract: In this paper, an investigation of radiative transfer in a rectangular medium with one-dimensional or two-dimensional graded index is presented. The integral equations of intensity moments are derived and then the cases of a cold medium exposed to diffuse irradiation at the left boundary are solved by the Nyström method. The results obtained by solving integral equations are in excellent agreement with those obtained by the Monte Carlo method and the discrete ordinates method. For the case with an increasing refractive index in the direction to the right boundary, the distribution of refractive index enhances the radiation in the direction to the right, and so the half-range flux toward the right boundary increases with the increase of the gradient of refractive index. Besides, the half-range flux toward the right boundary decreases as the position considered approaches the top or bottom boundary. [Copyright &y& Elsevier]

Abstract: The aim of this paper is to derive spectral and condition number estimates of the single-layer operator for low-frequency multiple scattering problems. This work extends to dense media the analysis initiated in Antoine and Thierry (in press) [19]. Estimates are obtained first in the case of circular cylinders by Fourier analysis and are next formally adapted to disks, ellipses and rectangles in the framework of boundary element methods. Numerical simulations validating the approach are also given. [Copyright &y& Elsevier]

Abstract: Drug dissolution process of solid dosages is theoretically described by Noyes–Whitney–Nernst equation. However, the analysis of the process is demonstrated assuming some models. Normally, the model-dependent methods are idealized and require some limitations. In this study, Noyes–Whitney integral equation was proposed and applied to represent the drug dissolution profiles of a solid formulation via the non-linear least squares (NLLS) method. The integral equation is a model-free formula involving the dissolution rate constant as a parameter. In the present study, several solid formulations were prepared via changing the blending time of magnesium stearate (MgSt) with theophylline monohydrate, α-lactose monohydrate, and crystalline cellulose. The formula could excellently represent the dissolution profile, and thereby the rate constant and specific surface area could be obtained by NLLS method. Since the long time blending coated the particle surface with MgSt, it was found that the water permeation was disturbed by its layer dissociating into disintegrant particles. In the end, the solid formulations were not disintegrated; however, the specific surface area gradually increased during the process of dissolution. The X-ray CT observation supported this result and demonstrated that the rough surface was dominant as compared to dissolution, and thus, specific surface area of the solid formulation gradually increased. [Copyright &y& Elsevier]

Abstract: This paper is concerned with the existence of integral solutions to a general nonlinear integral equationWith the help of Krasnoselskii’s fixed point theorem and the theory of measure of weak noncompactness, we establish a new and general existence theorem for the nonlinear functional integral equation. Moreover, an example, which can not be treated by the related theorems in [5,20,22], is given to illustrate the new existence theorem. [Copyright &y& Elsevier]

Abstract: By using difference equations, we give some easily applicable sufficient conditions for the non-existence of nonnegative solutions satisfying some additional conditions, of the integral equationwhere and X is a measurable space with probability measure μ. [Copyright &y& Elsevier]

Abstract: We introduce a cyclic generalized -weakly contractive mappings for a map in a metric space and originate existence and uniqueness results of fixed points for such mappings. Examples are given to support the usability of our results. At the end of the results, an application to the study of existence and uniqueness of solutions for a class of nonlinear integral equations is presented. [Copyright &y& Elsevier]

Abstract: It is a classical fact that the exponential function is a solution of the integral equation . If we slightly modify this equation to with , it seems that no classical techniques apply to yield solutions. In this article, we consider the parameter . We will show the existence of a solution which takes the values of the Thue–Morse sequence on the odd integers. [Copyright &y& Elsevier]

Abstract: In this paper, a novel hybrid method is presented for solving the second kind linear Volterra integral equations. Due to the powerful regression ability of least squares support vector regression (LS-SVR), we approximate the unknown function of integral equations by using LS-SVR in intervals with known numerical solutions. The trapezoid quadrature is used to approximate subsequent integrations in intervals with unknown numerical solutions. The feasibility of the proposed method is examined on some integral equations. Experimental results of comparison with analytic and repeated modified trapezoid quadrature method’s solutions show that the proposed algorithm could reach a very high accuracy. The proposed algorithm could be a good tool for solving the second kind linear Volterra integral equations. [Copyright &y& Elsevier]

Abstract: Recently there have been many theoretical works on statistical inference in quantum systems. In the present Letter, we deal with the estimation of an unknown wave function under the assumption of a general parametric model. We propose the Bayesian estimation of a wave function and show the optimality result when we adopt the Bures distance as a loss function. We see some examples where the quantum state is better estimated by our method than by that based on the maximum-likelihood estimate. [Copyright &y& Elsevier]

Abstract: The Zhdanov–Trubnikov equation describing wrinkled premixed flames is studied, using pole decompositions as starting points. Its one-parameter () nonlinearity generalises the Michelson–Sivashinsky equation () to a stronger Darrieus–Landau instability. The shapes of steady flame crests (or periodic cells) are deduced from Laguerre (or Jacobi) polynomials when , which numerical resolutions confirm. Large wrinkles are analysed via a pole density: adapting results of Dunkl relates their shapes to the generating function of Meixner–Pollaczek polynomials, which numerical results confirm for (reduced stabilisation). Although locally ill-behaved if (over-stabilisation) such analytical solutions can yield accurate flame shapes for . Open problems are invoked. [Copyright &y& Elsevier]

Abstract: In the present work, a non-local boundary value problem with special gluing conditions for a mixed parabolic–hyperbolic equation with parameter is considered. The parabolic part of this equation is a fractional analogue of heat equation and the hyperbolic part is the telegraph equation. The considered problem is reduced, for positive values of the parameter, to an equivalent system of the second kind Volterra integral equations. Due to the influence of the fractional diffusion equation, the looked for solution belongs to a specific class of functions. The method of the Green functions and the properties of integro-differential operators are on the basis of the investigation. [Copyright &y& Elsevier]

Abstract: In this paper, we extend the coupled fixed point theorems for mixed monotone operators obtained by Bhaskar and Lakshmikantham [T.G. Bhaskar, V. Lakshmikantham, Fixed point theorems in partially ordered metric spaces and applications, Nonlinear Anal. 65 (2006) 1379–1393] and Luong and Thuan [N.V. Luong and N.X. Thuan, Coupled fixed points in partially ordered metric spaces and application, Nonlinear Anal. 74 (2011) 983–992], by weakening the involved contractive condition. An example as well as an application to nonlinear Fredholm integral equations is also given in order to illustrate the effectiveness of our generalizations. [Copyright &y& Elsevier]

Abstract: We propose iterated fast multiscale Galerkin methods for the second kind Fredholm integral equations with mildly weakly singular kernel by combining the advantages of fast methods and iteration post-processing methods. To study the super-convergence of these methods, we develop a theoretical framework for iterated fast multiscale schemes, and apply the scheme to integral equations with weakly singular kernels. We show theoretically that even the computational complexity is almost optimal, our schemes improve the accuracy of numerical solutions greatly, and exhibit the global super-convergence. Numerical examples are presented to illustrate the theoretical results and the efficiency of the methods. [Copyright &y& Elsevier]

Abstract: This article deals with the influence of interfacial CPE (constant phase element) behaviour on linear scan voltammograms (LSV) and cyclic voltammograms (CV). Theoretical investigation is carried out using the generalised Mittag–Leffler function which arises naturally in the solution of fractional order integral or differential equations. First, ohmic drop and CPE effects are analysed in the absence of electrochemical reaction through the closed-form expression of current vs. time. Next, the combined effects of ohmic drop, CPE and faradaic processes are modelled using the integral equation approach. Finally, the example of one-step electrochemical reaction, investigated under semi-infinite linear diffusion conditions, is dealt with for illustration of the calculation procedure. [Copyright &y& Elsevier]

Abstract: In this paper the authors develop an algorithm for solving the time-dependent heat conduction equation in an analytical, exact fashion for a two-component domain. The Green’s function approach is used to get the formal solution of the problem. As an intermediate result an integral-equation for the temperature history at the domain interface is formulated which can be solved analytically. The Green’s function approach in conjunction with the integral-equation method is very useful in cases were strong discontinuities or jumps occur. The system parameters and the initial conditions of the investigated problem give rise to two jumps in the temperature field. In conjunction to the analytical solution a pure numerical solution is obtained by using the FEM (finite element method) , and compared with the solution obtained by the integral-equation approach. [Copyright &y& Elsevier]

Abstract: The statements on solvability, solution estimates, and well-posedness of equations with conditionally covering mappings are proved. The results obtained are applied to the investigation of Volterra equations (including integral equations) unsolved for the unknown function. [Copyright &y& Elsevier]

Abstract: In this paper, we prove existence, uniqueness and comparison results for solutions of differential and integral equations in Banach spaces containing Henstock–Kurzweil integrable functions from a compact real interval to an ordered Banach space. [Copyright &y& Elsevier]

Abstract: In this work we study a system of an integral equation of Volterra type coupled with an original renewal equation. This model arises in the context of cell motility (Oelz et al., 2008 ): the integral equation describes the trajectory of a binding site which is connected via transiently remodelling linkages to the substrate and which evolves driven by a given force. The renewal model accounts for the remodelling process of linkages which attach and break with given probabilities. In the present paper we analyze existence and uniqueness issues for the coupled system of interest and provide a rigorous justification of the asymptotic limit of infinitesimally rapid turnover of linkages. The renewal model for the age distribution of linkages differs from more classical ones in that it describes competition between population size and birth and because it admits a new and specific Lyapunov functional. On the other side, using a comparison principle which applies to non-convolution linear Volterra kernels and the peculiar transport properties of the linkages, one establishes a convergence result when the turnover parameter ε tends to zero. [Copyright &y& Elsevier]

Abstract: The published traditional crack problem solutions usually consider cracks located in the planes, parallel to the plane of isotropy, which is usually denoted as z = 0. We consider here case of a crack located in the plane x = 0 and subjected to arbitrary normal or tangential loading. The case of elliptic crack is considered in detail. Complete solution for the fields of displacements and stresses is presented single contour integrals of elementary integrands. Stress intensity factors are computed explicitly. [Copyright &y& Elsevier]

Abstract: The plane elasticity problem of a functionally graded semi-infinite plane, containing periodic imbedded or edge cracks perpendicular to the free surface is considered. Cracks are subjected to mode one mechanical or thermal loadings, which are represented by crack surface tractions. Young’s modulus, conduction coefficient, coefficient of thermal expansion are taken as exponentially varying functions of the depth coordinate where as Poisson ratio and thermal diffusivity are assumed to be constant. Fourier integrals and Fourier series are used in the formulation which lead to a Cauchy type singular integral equation. The unknown function which is the derivative of crack surface displacement is numerically solved and used in the calculation of stress intensity factors. Limited finite element calculations are done for verification of the results which demonstrate the strong dependence of stress intensity factors on geometric and material parameters. [Copyright &y& Elsevier]

Abstract: An exact analysis of the mechanics of interface failure is presented for a trilayer composite system consisting of geometrically and materially distinct linear elastic layers separated by straight nonlinear, uniform and nonuniform decohesive interfaces. The technical significance of this system stems from its utility in representing two slabs joined together by a third adhesive layer whose thickness cannot be neglected. The formulation, based on exact infinitesimal strain elasticity solutions for rectangular domains, employs a methodology recently developed by the authors to investigate both solitary defect as well as multiple defect interaction problems in layered systems under arbitrary loading. Interfacial integral equations, governing the normal and tangential displacement jump components at the interfaces, are solved for the uniformly loaded trilayer system. Interfacial defects, taken in the form of interface perturbations and nonbonded portions of interface, are modeled by coordinate dependent interface strengths. They are examined in a variety of configurations chosen so as to shed light on the various interfacial failure mechanisms active in layered systems. [Copyright &y& Elsevier]

Abstract: Using the setting of cone metric space, a fixed point theorem is proved for two maps, and several corollaries are obtained. In these cases, the cone does not need to be normal. These results generalize several well known compatible recent and classical results in the literature. As an application, the existence of solution of an integral equation is presented. [Copyright &y& Elsevier]

Abstract: We consider an American put option under the CEV process. This corresponds to a free boundary problem for a PDE. We show that this free boundary satisfies a nonlinear integral equation, and analyze it in the limit of small , where is the interest rate and is the volatility. We use perturbation methods to find that the free boundary behaves differently for five ranges of time to expiry. [Copyright &y& Elsevier]

Abstract: The interfacial stress arising between thin structures bonded to a 2D elastic substrate has been investigated in the present work. Chebyshev polynomials have been adopted to approximate the shear stress occurring across the contact region. Perfect adhesion has been assumed among the coating structures and the underlying substrate, leading to a singular integral equation which has been reduced to an algebraic system. Thin bonded structures having several geometric configurations under different load conditions have been considered. In particular, the stress concentration in the neighborhood of the coating edges and around the points of load application has been evaluated in detail. [Copyright &y& Elsevier]

Abstract: The eigenfunctions and eigenvalues of the Karhunen–Loeve integral equation are found for an exponential covariance function in a spherical system. The solution is given in terms of a set of subsidiary integral equations, the kernels of which are the spherical harmonic moments of the covariance function. These are relatively simple and can be solved numerically or analytically. Results are given in the form of a table of eigenvalues where and . The associated eigenfunctions are given graphically and consistency with certain conservation requirements is demonstrated. A practical example is given, based upon neutron diffusion in a spherical system where these eigenfunctions are required. [Copyright &y& Elsevier]

Abstract: In this paper we consider the asymptotic stability of a generalized linear neutral differential equation with variable delays by using the fixed point theory. An asymptotic stability theorem with a necessary and sufficient condition is proved, which improves and generalizes some results due to Burton (2003) , Zhang (2005) , Raffoul (2004) , and Jin and Luo (2008) . Two examples are also given to illustrate our results. [Copyright &y& Elsevier]

Abstract: Predicting the overall behavior of heterogeneous materials, from their local properties at the scale of heterogeneities, represents a critical step in the design and modeling of new materials. Within this framework, an internal variables approach for scale transition problem in elastic–viscoplastic case is introduced. The proposed micromechanical model is based on establishing a new system of field equations from which two Navier’s equations are obtained. Combining these equations leads to a single integral equation which contains, on the one hand, modified Green operators associated with elastic and viscoplastic reference homogeneous media, and secondly, elastic and viscoplastic fluctuations. This new integral equation is thus adapted to self-consistent scale transition methods. By using the self-consistent approximation we obtain the concentration law and the overall elastic–viscoplastic behavior of the material. The model is first applied to the case of two-phase materials with isotropic, linear and compressible viscoelastic properties. Results for elastic–viscoplastic two-phase materials are also presented and compared with exact results and variational methods. [ABSTRACT FROM AUTHOR]

Abstract: The purpose of this paper is to present some coupled fixed point theorems for a mixed monotone operator in a complete metric space endowed with a partial order by using altering distance functions. We also present an application to integral equations. [ABSTRACT FROM AUTHOR]

Abstract: In a free-electron laser set up as a single-pass amplifier, the evolution of the optical beam in the weak-field regime is ruled by the so-called Integral Equation. We derive this equation by a different analytical technique with respect to the one employed in earlier works. Our approach proves to be mathematically simpler and more physically transparent with respect to the original one. [Copyright &y& Elsevier]

Abstract: 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. [Copyright &y& Elsevier]

Abstract: In this paper, we prove some coupled fixed point theorems for mappings having a mixed monotone property in partially ordered metric spaces. The main results of this paper are generalizations of the main results of Bhaskar and Lakshmikantham [T. Gnana Bhaskar, V. Lakshmikantham, Fixed point theorems in partially ordered metric spaces and applications, Nonlinear Anal. TMA 65 (2006) 1379–1393]. As an application, we discuss the existence and uniqueness for a solution of a nonlinear integral equation. [Copyright &y& Elsevier]

Abstract: This paper presents a mathematical modeling and a numerical methodological approach based on integral equation formulation and radial basis functions (RBF) for the dynamic behaviour damped Timoshenko beams under follower forces. Using the fundamental solution of the main operator and the RBF, the governing non-self-adjoint partial differential equation is transformed into an integral equation. Based on the harmonic assumption and internal concatenation points an eigenvalue problem is obtained and numerically solved. The flutter analysis, complex modes and the frequency load responses are investigated for beams under various subtangential follower loads, aspect ratios, internal and viscous damping. [ABSTRACT FROM AUTHOR]

Abstract: An analytical model is presented for delamination initiation and growth and the resulting response during small mass impact on orthotropic laminated composite plates, which typically is caused by runway debris and other small objects. The solution is obtained by a fast stepwise numerical solution of a single integral equation. Delamination size, load and deflection history are predicted by extension of an earlier elastic impact model by the author. Good agreement is demonstrated in comparisons with finite element simulations and experiments. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, we introduce the -concave–convex operator. Without any compact or continuous assumptions, we prove the existence and uniqueness of fixed points, giving monotone iterative sequences for the unique fixed point for the operator. Finally, we apply the results to an integral equation of polynomial type which possesses items of measurable functions. [Copyright &y& Elsevier]

Abstract: Two integral equations, representing the mechanical response of a 2D infinite plate supported along a line and subject to a transverse concentrated force, are examined. The kernels of the integral operators are of the type and . In spite of the fact that these are only weakly singular, the two equations are studied in a more general framework, which allows us to consider also solutions having non-integrable endpoint singularities. The existence and uniqueness of solutions of the equations are discussed and their endpoint singularities detected. Since the two equations are of interest in their own right, some properties of the associated integral operators are examined in a scale of weighted Sobolev type spaces. Then, new results on the existence and uniqueness of integrable solutions of the equations that in some sense are complementary to those previously obtained are derived. [Copyright &y& Elsevier]

Abstract: We establish the existence of analytic solutions to integral equations in the complex domain when the nonlinearity satisfies either a growth condition or a monotonic type condition. [Copyright &y& Elsevier]

Abstract: In this paper we consider fast numerical solution methods for two dimensional Fredholm integral equation of the second kindwhere a(x, y, u, v) is smooth and g(x, y) is in L 2[α, β]2. Discretizing the integral equation by certain quadrature rule, we get a linear system. To deduce fast approximate solution methods for the resulted linear system, we study the approximation of the four-variable kernel function a(x, y, u, v) by piecewise polynomial: partition the domain [α, β]4 into subdomains of the same size and interpolate the kernel function a(x, y, u, v) in each subdomain. Fast matrix–vector multiplication algorithms and efficient iterative methods are derived. Numerical results are given to illustrate the efficiency of our methods. [Copyright &y& Elsevier]

Abstract: In the study of iterative methods with high order of convergence, Gander provides a general expression for iterative methods with order of convergence at least three in the scalar case. Taking into account an extension of this result, we define a family of iterations in Banach spaces with -order of convergence at least four for quadratic equations. [Copyright &y& Elsevier]

Abstract: We consider the credit risk model of . According to this model, the price of a defaultable bond can be efficiently computed using a variational formulation that consists of an integral relation and a Volterra integral equation. In this integral equation is justified by a probabilistic intuition, but is not proven formally. In this paper we analytically derive the variational formulation used in . This analysis allows to give a correct characterization of the solution of the integral equation. Furthermore the approach proposed in this paper could also be employed for other models of credit risk. [Copyright &y& Elsevier]

Abstract: In this study it is shown that the numerical solutions of linear Fredholm integro-differential equations obtained by using Legendre polynomials can also be found by using the variational iteration method. Furthermore the numerical solutions of the given problems which are solved by the variational iteration method obviously converge rapidly to exact solutions better than the Legendre polynomial technique. Additionally, although the powerful effect of the applied processes in Legendre polynomial approach arises in the situations where the initial approximation value is unknown, it is shown by the examples that the variational iteration method produces more certain solutions where the first initial function approximation value is estimated. In this paper, the Legendre polynomial approximation (LPA) and the variational iteration method (VIM) are implemented to obtain the solutions of the linear Fredholm integro-differential equations and the numerical solutions with respect to these methods are compared. [Copyright &y& Elsevier]