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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: 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: 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 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: 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: 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 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 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: 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: 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: We consider a nonlinear optimal control problem with an integral equation as the control object, subject to control constraints. This integral equation corresponds to the fractional moment of a stochastic process involving short-range and long-range dependences. For both cases, we derive the first-order necessary optimality conditions in the form of the Euler–Lagrange equation, and then apply them to obtain a numerical solution of the problem of optimal portfolio selection. [Copyright &y& Elsevier]

Abstract: Adsorption of charged colloidal particles to oppositely charged surfaces is usually an irreversible process. The interaction between a pair of particles can be modeled with an exponentially decaying potential originating from double layer interactions. This work explored the effect of the Debye length on monolayer structures using the integral-equation theory which was successfully developed based on a binary-mixture approximation to include the effect of particle size polydispersity. The theoretical results from the integral equations with a Percus–Yevick closure showed that upon increasing the Debye length, the radial distribution functions, , as well as the structure factor, , decreased, in good agreement with simulation results. When the effect of size distributions was investigated, the prominent peak of the radial distribution function increased non-linearly with the product , which followed the same trend as was reported for the case of the jamming coverage of the monolayer film. [Copyright &y& Elsevier]

Abstract: We present a fast algorithm for the calculation of elastostatic fields in two-dimensional assemblies of elastic grains, separated by sharp grain boundaries. The algorithm uses an integral equation approach, combined with the fast multipole method and recursive compression to resolve stress concentrations also very close to grain boundary junctions. Singular basis functions on analytic form are not required. Accurate results can be obtained at a cost roughly proportional to the number of grains in the assembly. Large-scale problems, with thousands of grains, are solved using modest computational resources. [Copyright &y& Elsevier]

Abstract: In this paper, we consider the renewal risk process under a threshold dividend payment strategy. For this model, the expected discounted dividend payments and the Gerber–Shiu expected discounted penalty function are investigated. Integral equations, integro-differential equations and some closed form expressions for them are derived. When the claims are exponentially distributed, it is verified that the expected penalty of the deficit at ruin is proportional to the ruin probability. [Copyright &y& Elsevier]

Abstract: A general approach is presented for proving existence and uniqueness of solutions to the singular boundary value problem The proof is constructive in nature, and could be used for numerical generation of the solution. The only restriction placed on is that it not be a singular function of the independent variable ; singularities in are easily avoided. Solutions are found in finite regions where , using an integral equation whose Green’s function contains an adjustable parameter that secures convergence of the Picard iterative sequence. Methods based on the theory are developed and applied to a set of problems that have appeared previously in published works. [Copyright &y& Elsevier]

Abstract: In this paper we consider numerical solution methods for two dimensional Fredholm integral equation of the second kind where is smooth and is in . We discuss polynomial interpolation methods for four-variable functions and then use the interpolating polynomial to approximate the kernel function . Based on the approximation we deduce fast matrix-vector multiplication algorithms and efficient preconditioners for the above two dimensional integral equations. The residual correction scheme is used to solve the discretization linear system. [Copyright &y& Elsevier]

Abstract: Analytical solutions are obtained for a class of inherently incremental, similar elastic half-plane frictional contact problems which involve simultaneous application of a changing, remote bulk-stress during oblique loading. The method is based on Abel Integral Equations. The spatial gradient of the slip function is obtained uniquely by the conditions determining the existence of a bounded solution to the shear traction Cauchy SIE and global shear equilibrium, provided there is stick at all points in the contact throughout the loading. Solutions are obtained for both the cylindrical indenter and the flat punch with rounded edges. The method also lends itself easily to generalization to non-linear load paths and a number of such problems are considered. Problems involving the development of slip zones within the contact may be solved in some special cases. [Copyright &y& Elsevier]

Abstract: An exact theory of interfacial debonding is developed for a layered composite system consisting of distinct linear elastic slabs separated by nonlinear, nonuniform decohesive interfaces. Loading of the top and bottom external surfaces is defined pointwise while loading of the side surfaces is prescribed in the form of resultants. The work is motivated by the desire to develop a general tool to analyze the detailed features of debonding along uniform and nonuniform straight interfaces in slab systems subject to general loading. The methodology allows for the investigation of both solitary defect as well as multiple defect interaction problems. Interfacial integral equations, governing the normal and tangential displacement jump components at an interface of a slab system are developed from the Fourier series solution for the single slab subject to arbitrary loading on its surfaces. Interfaces are characterized by distinct interface force–displacement jump relations with crack-like defects modeled by an interface strength which varies with interface coordinate. Infinitesimal strain equilibrium solutions, which account for rigid body translation and rotation, are sought by eigenfunction expansion of the solution of the governing interfacial integral equations. Applications of the theory to the bilayer problem with a solitary defect or a defect pair, in both peeling and mixed load configurations are presented. [Copyright &y& Elsevier]

Abstract: In a piled raft, the length and arrangement of piles has a significant effect on the stresses and deflections of the raft. The use of piles with different dimensions and properties below a raft is an innovative concept and can optimize the design of a piled raft. In this study, an integral equation method with a fictitious pile model was adopted to analyze the piled raft foundation with dissimilar piles. The Fredholm integral equations of the second kind were obtained for this problem. The loads shared by piles and subsoil, the load transfer, and the settlement of the piled raft were obtained using numerical calculation. The results from the present method were compared with those in the literature. An optimization technique was introduced to design piled rafts with dissimilar piles. The stiffening effect of piles on the surrounding soil is also discussed as compared the conventional interaction factor approach. [Copyright &y& Elsevier]

Abstract: The problem of a thin film coated on an elastic layer and subject to a thermal variation is analytically investigated in the present work. The analysis is developed in order to assess the mechanical behaviour of a crystalline undulator designed for obtaining high emission radiations through channelling phenomenon. It consists in a plane silicon wafer alternately patterned with thin films in silicon nitride on both surfaces. The system adopts a periodic curvature as a result of the misfit strain due to the different thermal expansivities of the layer and the film. The problem is governed by an integral equation which can be reduced to a linear algebraic system by approximating the unknown interfacial shear stress via series expansion of Chebyshev polynomials. [Copyright &y& Elsevier]

Abstract: In this paper we prove dominated and monotone convergence theorems for HL integrable Banach-valued functions. These results and a fixed point theorem in ordered spaces are then applied to prove existence and comparison results for integral equations of Fredholm type in ordered Banach spaces involving Kurzweil integrals or improper integrals. Results are used also to solve concrete second-order functional boundary value problems involving discontinuities and singularities. [Copyright &y& Elsevier]

Abstract: In this work, a functional approximation method for calculating the linear functional of the solution of second-kind Fredholm integral equations is developed. When the method is applied to the collocation method or to the multi-projection method, it generates approximations which exhibit superconvergence. [Copyright &y& Elsevier]

Abstract: The main objective of this paper is to establish some new delay integral inequalities, which provide explicit bounds on unknown functions. The inequalities given here can be used as tools in the qualitative theory of certain delay differential equations and delay integral equations. [Copyright &y& Elsevier]

Abstract: Within the framework of two-dimensional linear elasticity, the unilateral frictionless contact between two conformal cylindrical surfaces is governed by an integral equation in which the first and second Dundurs’ constants are involved. In the case of elastic similarity characterized by the zero second Dundurs’ constant, the integral equation is considerably simplified so as to lend itself to a closed-form solution. However, in the case of elastic dissimilarity defined by the non-zero second Dundurs’ constant, the question of obtaining a closed-form solution to the integral equation is a much tougher one. Starting from the integral equation established by To et al. [To, Q.D., He Q.-C., Cossavella, M., Morcant, K., Panait, A., 2007. Closed-form solution for the contact problem of reinforced pin-loaded joints used in glass structures. Int. J. Solids Struct. 44, 3887–3903] for the conformal contact problem originating from a reinforced pin-loaded joint used in tempered glass structures, the present work proposes a new approximate analytical method to solve it in the case of elastic dissimilarity by minimizing an error function. The derived closed-form solution, valid not only for the conformal contact between a pin and an infinite holed plate but also for the one between a pin and a finite holed plate, is shown to be in very good agreement with available numerical results. [Copyright &y& Elsevier]

Abstract: A direct function theoretic method is applied to solve a weakly singular integral equation whose kernel involves logarithmic singularity. This method avoids the occurrence of strong singularity. The solution of this integral equation is then applied to re-investigate the well known problem of water wave scattering by a partially immersed vertical barrier. [Copyright &y& Elsevier]

Abstract: From the axisymmetrical point of view, the dynamic response of single piles vertically embedded in the saturated half space and subjected to harmonic torsional loadings is investigated by means of integral transform and Muki’s methods for the first time. By treating the pile as a one-dimensional bar and the half space as a three-dimensional poroelastic continuum, the dynamic interaction between the pile and poroelastic medium is formulated in a Fredholm integral equation of the second kind. Numerical results are presented to illustrate the influence of pile flexibility, geometrical properties of pile–saturated soil system and frequency of excitation on the pile’s torque, torsional angle and coupled impedances. The conclusions obtained can serve as guidelines for the design and dynamic pile testing of practical engineering. [Copyright &y& Elsevier]

Abstract: An integral equation formulation for steady flow of a viscous fluid is presented based on the boundary element method. The continuity, Navier–Stokes and energy equations are used for calculation of the flow and temperature fields. The governing differential equations, in terms of primitive variables, are derived using velocity–pressure–temperature parameters. The calculation of fundamental solutions and solutions tensor is shown. Applications to simple flow cases, such as driven cavity, forward facing step, deep cavity and channel are presented. Convergence difficulties are indicated, which have limited the applications to flows of low Reynolds numbers. [Copyright &y& Elsevier]

Abstract: In this Letter, an application of homotopy-perturbation method is applied to solve the second kind of nonlinear integral equations such that Volterra and Fredholm integral equations. Comparison are made between the modification of Adomian''s decomposition method and homotopy-perturbation method. The results reveal that the homotopy-perturbation method is very effective and simple and gives the exact solution. [Copyright &y& Elsevier]

In this paper, several integral equations are solved by He’s variational iteration method. Comparison with exact solution shows that the method is very effective and convenient for solving integral equations. [Copyright &y& Elsevier]

Abstract: The purpose of the present note is to establish some new delay integral inequalities, which provide explicit bounds on unknown functions and generalize some results of Li et al. [Some new delay integral inequalities and their applications, J. Comput. Appl. Math. 180 (2005) 191–200]. The inequalities given here can be used to investigate the qualitative properties of certain delay differential equations and delay integral equations. [Copyright &y& Elsevier]

Abstract: Finding a formulation for electromagnetic scattering of surfaces which is both well-posed and produces a well-conditioned linear system is still a challenging problem. We here propose one such formulation valid in the high-frequency regime. The mathematical analysis is provided and numerical results on rather complex geometries show the performance of the method. [Copyright &y& Elsevier]

Abstract: When wavelets are used as basis functions in Galerkin approach to solve the integral equations, Integrals of the form occur. By a change of variable, these integrals can be translated into integrals involving only θ. In this paper, we find quadrature rule on the for the integrals of the formWavelets in this article are those discovered by Daubechies [I. Daubechies, Orthonormal bases of compactly supported wavelets, Comm. Pure Appl. Math. 41 (1988) 909–996], where ϕ is the scaling function and ψ is the wavelet function. [Copyright &y& Elsevier]

Abstract: The crack problem of a magnetoelectroelastic layer bonded to dissimilar half spaces under anti-plane shear and in-plane electric and magnetic loads is considered. Fourier transforms are used to reduce the mixed boundary value problems of the crack, which is assumed to be permeable, to simultaneous dual integral equations, and then expressed in terms of Fredholm integral equations of the second kind. Numerical results show that the stress intensity factors are influenced by the magnetoelectric interactions and the geometry size ratio. [Copyright &y& Elsevier]

Abstract: The singular stress, electric fields and magnetic fields in a rectangular magnetoelectroelastic body containing a moving crack under longitudinal shear are obtained. Fourier transforms and Fourier sine series are used to reduce the mixed boundary value problems of the crack, which is assumed to be permeable or impermeable, to dual integral equations, and then expressed in terms of Fredholm integral equations of the second kind. Results show that the stress intensity factors are influenced by the material constants, the geometry size ratio and the velocity of the crack, and the propagation of the crack possibly brings about branching phenomena. [Copyright &y& Elsevier]

Abstract: In this paper, we bring three theorems that enable us to approximate the solution of Ferdholm integral equations of the second kind. Then we use Coifman wavelets or Coiflets as scaling functions for projection that satisfied the conditions of theorems for approximation. Also we use this projection to convert the integral equation to a Galerkin system, which is the most important of the expansion methods for solving linear integral equations. Finally, by using numerical examples we show that our estimation have a good degree of accuracy. [Copyright &y& Elsevier]

Abstract: Building fires go through a series of stages. They start with a fire plume period during which buoyant fire smoke rises to the ceiling. A second stage is the following enclosure smoke-filling period. In this paper, the first stage is the subject, especially for the fire plume behavior in thermally stratified environments in large volume spaces. In NFPA 92B, Morton''s integral equation was introduced for calculating the maximum plume rise, and beam smoke detectors were recommended for smoke detection design. In this work, experiments and CFD simulations were conducted in a small-scale enclosure and a large space to investigate early fire movements in temperature-stratified ambients. The results show that in a thermally stratified environment, the axial temperature and velocity of a fire plume decrease more quickly along the vertical axis than in uniform environment, and in some cases the fire plume ceases to rise. The previous integral equation was shown to underestimate the actual maximum height of a fire smoke plume, and also was unable to explain the differences of the maximum heights of low-density and high-density smoke plumes with the same stratification and outlet conditions. The integral equation was improved by introducing two correction factors, and extended for non-linear temperature stratified environments. A light section smoke detection method with three space-protected area was suggested and discussed. [Copyright &y& Elsevier]

Abstract: We examine the use of wavelet packets for the fast solution of integral equations with a highly oscillatory kernel. The redundancy of the wavelet packet transform allows the selection of a basis tailored to the problem at hand. It is shown that a well chosen wavelet packet basis is better suited to compress the discretized system than wavelets. The complexity of the matrix–vector product in an iterative solution method is then substantially reduced. A two-dimensional wavelet packet transform is derived and compared with a number of one-dimensional transforms that were presented earlier in literature. By means of some numerical experiments we illustrate the improved efficiency of the two-dimensional approach. [Copyright &y& Elsevier]

Abstract: In this paper, We use the wavelet bases of Hermite cubic splines to solve the second kind integral equationsA pair of wavelets are constructed on the basis of Hermite cubic splines. This wavelets are in C1 and supported on [0,2]. Moreover, one wavelet is symmetric, and the other is anti-symmetric. This spline wavelets are then adapted to the interval [0,1]. The computational results demonstrate the advantage of the wavelet basis. [Copyright &y& Elsevier]

Abstract: In this paper the problem of elastodynamic response of an infinite orthotropic medium containing a central crack under normal impact loading is considered. Laplace and Fourier integral transforms are employed to reduce the dimensional wave propagation problem to the solution of a pair of dual integral equations in the Laplace transform plane. These integral equations are then reduced to integro-differential equations which have been solved in the low frequency domain by method of iteration. To determine time dependence, these equations are inverted to yield the dynamic stress intensity factor (SIF) for normal point force loading. Numerical results of normalized SIF for large normalized time variable and for different concentrated point force loading at arbitrary location of the crack surface have been calculated for graphite–epoxy composite material. In the present paper the method of [A. Papoulis, The method of inversion of Laplace transform, Quart. Appl. Math. 14 (1957) 405–414] is used for the inversion of the Laplace transform. Finally the results obtained are displayed graphically. [Copyright &y& Elsevier]

Abstract: The mechanical quadrature method described for two-dimensional nonlinear singular integral equation with Hilbert kernel in Banach space. The results are compared with the exact solution of the integral equation. [Copyright &y& Elsevier]

Abstract: The problem of differentiating spectral data to yield the third and fourth derivatives is converted into one of solving an integral equation of the first kind. This equation is solved by Tikhonov regularization. The method of General Cross Validation is used to guide the choice of the regularization parameter that keeps noise amplification under control. The performance of this route to third and fourth derivative spectra is demonstrated by applying it to a number of published spectra. A computational problem associated with General Cross Validation has been identified. [Copyright &y& Elsevier]

Abstract: A new integral equation formulation of two-dimensional infinite isotropic medium (matrix) with various inclusions and cracks is presented in this paper. The proposed integral formulation only contains the unknown displacements on the inclusion–matrix interfaces and the discontinuous displacements over the cracks. In order to solve the inclusion–crack problems, the displacement integral equation is used when the source points are acting on the inclusion–matrix interfaces, whilst the stress integral equation is adopted when the source points are being on the crack surfaces. Thus, the resulting system of equations can be formulated so that the displacements on the inclusion–matrix interfaces and the discontinuous displacements over the cracks can be obtained. Based on one point formulation, the stress intensity factors at the crack tips can be achieved. Numerical results from the present method are in excellent agreement with those from the conventional boundary element method. [Copyright &y& Elsevier]

Abstract: The singular stress, electric and magnetic fields in a piezoelectromagnetic strip containing a Griffith crack under longitudinal shear are obtained by the theory of linear piezoelectromagneticity. The crack is situated symmetrically and oriented in a direction parallel to the edges of the strip. Fourier transforms are used to reduce the mixed boundary value problems of the crack, which is assumed to be either impermeable or permeable, to dual integral equations. The solution of the dual integral equations is then expressed in terms of Fredholm integral equations of the second kind. Expressions for strains, stresses, electric fields, electric displacements, magnetic fields and magnetic inductions in the vicinity of the crack tip are derived. For the impermeable crack, the electric field intensity factor (EFIF) and the magnetic field intensity factor (MFIF) depend on the edge loading conditions, whereas, the EFIF and MFIF for the permeable crack are always zero. The results obtained show that the width of the strip have significant influence on the field intensity factors and the energy release rates. [Copyright &y& Elsevier]

Abstract: In this paper we consider Fredholm–Volterra integral equation in one dimension with time dependent. The kernel of the Fredholm term is assumed to be singular while the kernel of Volterra term is continuous. The existence of the solution is proved and a numerical method is presented to get an approximate to such solution. [Copyright &y& Elsevier]