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Integral transforms are a versatile mathematical technique that can be applied in a wide range of science and engineering fields. We consider the general integral transform with the Caputo derivative and Constant Proportional Caputo derivative in this work. We present some applications to show the effect of the general integral transform with different fractional derivatives. [ABSTRACT FROM AUTHOR]

• Exact solution of the problem of Eshelby's inhomogeneity in strain gradient elasticity theory. • Green's function technique and integral equation developed for arbitrary boundary conditions. • Analytical expressions of mean strain inside inclusion. • Analytical expressions of effective elastic properties with strain gradient. • Effect of the scale parameter, the contrast of elastic behavior and the volume fraction of inhomogeneity. A new micromechanical approach to deal with the problem of Eshelby's inhomogeneity is developed for the prediction of the effective properties of composite materials according to the strain gradient elasticity theory. The method is based on the Green's function technique leading to an integral equation of the heterogeneous elastic problem. Within the simplified strain gradient elasticity theory, the integral equation for an infinite heterogeneous medium subjected to non-homogeneous boundary conditions is acquired. Thanks to this integral equation, the exact solution of Eshelby's inhomogeneity problem is detailed for spherical inhomogeneity and isotropic elastic behavior. From the expression of strain localization relations, the effective elastic properties of a two-phase composite material are then predicted through Mori Tanaka's homogenization scheme. To test the relevance of the suggested approach, its predictions are compared with results issued from some reference models and experimental data. [ABSTRACT FROM AUTHOR]

Sonine kernel is characterized by the Sonine condition (denoted by SC) and is an important class of kernels in nonlocal differential equations and integral equations. This work proposes a SC with a more general form (denoted by gSC), which is more convenient than SC to accommodate complex kernels and equations. A typical kernel is given, and the first-kind Volterra integral equation under gSC is accordingly transformed and then analyzed. Based on these results, it is finally proved that the gSC is indeed equivalent to the original SC, which indicates that the Sonine kernel may be essentially characterized by the behavior of its convolution with the associated kernel at the starting point. [ABSTRACT FROM AUTHOR]

The aim of this work is to give some fixed point results based on the technique of measure of noncompactness which extend the classical Darbo's theorem. With the help of our Darbo-type theorem, we obtain the existence of solution of implicit fractional integral equations in C (I , ℓ p α) (collection of all continuous functions from I = [ 0 , a ] (a > 0) to ℓ p α ), where ℓ p α is a tempered sequence space. Finally, we present a numerical example to see the validity and practicability of our existence result. [ABSTRACT FROM AUTHOR]

Process of enhancing testing-capacity regarding COVID-19 is a topic of interest. This task of enhancing is constrained by socio-economic background of a country either in favorable or unfavorable ways. In this paper, we investigate timing of enhancing testing-capacity as an optimal problem, where the enhancement is quantified via number of tests as an instant measure and recovered portion as a long-term measure. The proposed work is structured analogous to an optimal machine replacement model based on a non-linear integral equation. Overall model is partially identifiable and compatible parameter estimations are carried out for a specific case study covering an early stage scenario. In addition, scenario development criteria on demand and effort for enhancing testing-capacity are introduced for predictions. In one numerical experiment, it is observed that frequency of enhancing testing-capacity starts decreasing after two increments indicating a favorable direction amidst effort constraints. • Enhancing testing-capacity of COVID-19 is a process that needs monitoring. • Timing for enhanced testing can be modeled as an optimal replacement model. • Parameter estimation under different scenarios paves the way for process planning. [ABSTRACT FROM AUTHOR]

[Display omitted] • A new general integral transform which is covered all class of integral transform in the class of Laplace transform. • We investigated the application of this new transform for solving ODE with constant and variable coefficient. • This new transform can handle easily for fractional order integral equations and fractional order differential equations. • We have discussed the advantage and disadvantage of other integral transformed which is defined during last 2 decades. • We proved the related theorems for this new transform. Integral transforms are important to solve real problems. Appropriate choice of integral transforms helps to convert differential equations as well as integral equations into terms of an algebraic equation that can be solved easily. During last two decades many integral transforms in the class of Laplace transform are introduced such as Sumudu, Elzaki, Natural, Aboodh, Pourreza, Mohand, G_transform, Sawi and Kamal transforms. In this paper, we introduce a general integral transform in the class of Laplace transform. We study the properties of this transform. Then we compare it with few exiting integral transforms in the Laplace family such as Laplace, Sumudu, Elzaki and G\_transforms, Pourreza, Aboodh and etc. A new integral transform is introduced. Then some properties of this integral transform are discussed. This integral transform is used to solve this new transform is used for solving higher order initial value problems, integral equations and fractional order integral equation. It is proved that those new transforms in the class of Laplace transform which are introduced during last few decades are a special case of this general transform. It is shown that there is no advantage between theses transforms unless for special problems. It has shown that this new integral transform covers those exiting transforms such as Laplace, Elzaki and Sumudu transforms for different value of p (s) and q (s). We used this new transform for solving ODE, integral equations and fractional integral equations. Also, we can introduce new integral transforms by using this new general integral transform. [ABSTRACT FROM AUTHOR]

In this paper, integral equation formulations and field-dependent micromechanical modeling of global nonlinear magneto-mechanical behavior of magneto-elastic composites under high strain and magnetic field are elaborated. The modeling is obtained based on an extension of micro–macro transition inclusion problem to nonlinear behavior using field-dependent and highly nonlinear localization tensors. A methodological procedure is elaborated based on newly introduced strain and magnetic field-dependent Green tensors, strain and magnetic field-dependent integral equations linked to tensors of Eshelby and micromechanical approaches. The field-dependent global moduli are predicted based on the Mori–Tanaka approach and self-consistent method. Iterative incremental schemes based on the Newton–Raphson and fixed-point algorithms are examined and established precise semi-analytic equations of the effective magneto-elastic properties of composites that are dependent on the magnetic-strain field for different types of inclusions. A numerical code is elaborated for numerical predictions and the obtained field-dependent effective magneto-elastic coefficients are obtained and presented for various volume fractions of inclusion, types and shapes of the reinforced nonlinear composites. • Nonlinear micromechanical modeling of fully coupled magneto-elastic under large deformation and high magnetic field. • Magnetic-Strain dependent Green tensors and associated localization tensors. • Iterative incremental schemes for semi-analytical magnetic-strain field dependent effective magneto-elastic properties. • Magneto-mechanical field dependent effective properties for several types of matrix and inclusions phases. [ABSTRACT FROM AUTHOR]

This work is concerned with the construction and the hp non-conforming a priori error analysis of a Discontinuous Galerkin DG numerical scheme applied to the hypersingular integral equation related to the Helmholtz problem in 3D. The main results of this article are an error bound in a norm suited to the problem and in the L 2 -norm. Those bounds are quasi-optimal for the h -convergence and the p -convergence. Various formulation choices and penalty functions are theoretically discussed. In particular we show that a penalty function of the shape h 2 p leads to a quasi-optimal convergence of the scheme. Some numerical experiments confirm the expected rates of convergence and the effect of the penalty function. [ABSTRACT FROM AUTHOR]

The paper is concerned with the wavelet-Galerkin method for the numerical solution of Fredholm linear integral equations and second-order integro-differential equations. We propose a construction of a quadratic spline-wavelet basis on the unit interval, such that the wavelets have three vanishing moments and the shortest support among such wavelets. We prove that this basis is a Riesz basis in the space L 2 0 , 1. We adapt the basis to homogeneous Dirichlet boundary conditions, and using a tensor product we construct a wavelet basis on the hyperrectangle. We use the wavelet-Galerkin method with the constructed bases for solving integral and integro-differential equations, and we show that the matrices arising from discretization have uniformly bounded condition numbers and that they can be approximated by sparse matrices. We present numerical examples and compare the results with the Galerkin method using other quadratic spline wavelet bases and other methods. [ABSTRACT FROM AUTHOR]

In this paper, we prove the existence of solutions for the following fractional boundary value problem c D α u (t) = f (t , u (t)) , α ∈ (n , n + 1) , 0 ≤ t < + ∞ , u (0) = 0 , u ′ ′ (0) = 0 , ... , u (n) (0) = 0 , lim t → + ∞ c D α − 1 u (t) = β u (ξ). The considerations of this paper are based on the concept of a new family of measures of noncompactness in the space of functions C n , γ (R +) satisfying the Hölder condition and a fixed point theorem of Darbo type. We also provide an illustrative example in support of our existence theorems. Finally, to credibility, we apply successive approximation and homotopy perturbation method to find solution of the above problem with high accuracy. [ABSTRACT FROM AUTHOR]

The interest of the majorizing (scalar) sequences lies in that, from their convergence, we can deduce the convergence of an iterative method in Banach spaces. We propose a new technique to construct majorizing sequences that generalizes that given by Kantorovich for Newton's method. [ABSTRACT FROM AUTHOR]

This paper mainly studies numerical solution to the Volterra integral equation of the second kind. By using the roots of Chebyshev polynomial as collocation points, a new collocation method is proposed to solve the Volterra integral equation of the second kind. The proposed method firstly interpolates the product of the kernel function and the unknown solution at the roots of Chebyshev polynomial. Then, the Volterra integral equation is transformed into a system of linear algebra equations by properties of Chebyshev polynomials. Finally, the numerical solution of the Volterra integral equation is obtained by the Chebyshev polynomial interpolation. In addition, the error estimates of the proposed method are provided in a semi-posteriori sense; and numerical examples are given to show effectiveness of the proposed method. [ABSTRACT FROM AUTHOR]

Abstract In this study, we derive an asymptotic solution of the integral equation satisfied by the second moment function M 2 t , a. We first find the Laplace transform M 2 L s , a and then obtain M 2 t , a asymptotically by inversion. Further, we have derived the asymptotic expressions of M 2 t , a for some special lifetime distributions such as exponential, gamma, Weibull, lognormal and truncated normal. Finally, the asymptotic solution is compared with the numerical solution to evaluate its performance. [ABSTRACT FROM AUTHOR]

In this paper, we propose a center Lipschitz condition for the second Fréchet derivative together with the use of restricted domains in order to improve the domain of starting points for Newton’s method. In addition, we compare the new result with an older one and see that the former improves the latter. [ABSTRACT FROM AUTHOR]

This paper presents mathematical modeling and predictions of effective nonlinear electro-mechanical behavior of piezoelectric materials under large deformation and high electric field. The heterogeneous inclusion problem is extended to investigate the fields dependent electro-elastic behaviors under high fields. The associated strain field dependent Green tensors are introduced. The field dependent interaction tensors, related to Eshelby's tensors, are explicitly formulated and used to derive micromechanical models based on the Mori–Tanaka approach and the self-consistent model. Due to electric-strain field dependence nonlinear algebraic tensors equations are resulted. Iterative incremental schemes based on the Newton–Raphson algorithm are elaborated and explicit semi-analytical formulations of electric-strain field dependent effective electro-elastic moduli of piezoelectric multi-phase composites are obtained for various inclusion types. Strain and electric field's effects on effective properties can be analyzed for various electro-elastic composites. Numerical results of field dependent effective electro-mechanical properties are given for several inclusion's volume fraction and shapes as well as types of matrix and inclusions phases. [ABSTRACT FROM AUTHOR]

We analyze the semilocal convergence of Newton’s method under a center Lipschitz condition for the second derivative of the operator involved different from that used by other authors until now. In particular, we propose to center the Lipschitz condition for the second derivative in a different point from that where Newton’s method starts. This allows us to obtain different starting points for Newton’s method and modify the domain of starting points. [ABSTRACT FROM AUTHOR]

In this paper, we propose a new systematic procedure of estimating elastic properties of composites constituted of two phases, matrix and inclusions. A class of integral equations based on eigenstrain (or eigenstress) with the matrix as reference material is constructed with an explicit form in Fourier space. Each integral equation belonging to this class can yield estimates of the overall elastic tensor via Neumann series expansion. The best estimates and series are selected based on the convergence rate criteria of the series, i.e the spectral radius must be minimized. The optimized series is convergent for any finite contrast between inclusions and matrix. Applying the optimized series and the associated estimates to different microstructures yields very satisfying results when compared with the related full solution. For the case of a random distribution of spherical inclusions, exact relations between the elastic tensor and n th order structure factors are demonstrated. [ABSTRACT FROM AUTHOR]

An elastoplastic model for the analysis of a driven pile extended at the bottom with a micropile under axial load is presented. The model is an extension of the integral equation method of Poulos and Davis. The finite-difference scheme used to obtain the pile displacements is reformulated to take into account the discontinuity in the stress distribution at the joint between pile and micropile. The results obtained with the proposed method are compared with the outcomes of a more sophisticated finite element simulation, and also with data from full-scale load tests. Reasonably good agreement is obtained in both cases. [ABSTRACT FROM AUTHOR]

We study qualitative properties of solutions of an integral equation associated the Benjamin–Ono–Zakharov–Kuznetsov operator. We establish the regularity of the positive solutions without the assumption of being in fractional Sobolev–Liouville spaces. Moreover we show that the solutions are axially symmetric. Furthermore we establish Lipschitz continuity and the decay rate of the solutions. [ABSTRACT FROM AUTHOR]

This paper presents our study of American floating strike lookback options written on dividend-paying assets. The valuation of these options can be mathematically formulated as a free boundary inhomogeneous Black–Scholes PDE with a Neumann boundary condition, which we, by using a Mellin transform, convert into a relatively simple ordinary differential equation with Dirichlet boundary conditions. We then use these results to derive an integral equation that can be used to calculate the price of American floating strike lookback options. In addition, we also used Mellin transforms to derive the closed-form of the perpetual case. [ABSTRACT FROM AUTHOR]

In the framework of linear elastic continuum mechanics, an analytical formulation is presented for the axisymmetric axial interaction of a rigid disk in frictionless contact with the face of a penny-shaped crack in a transversely isotropic solid. The problem is reduced to an integral equation and is shown to be degenerated to the formulation of isotropic materials in the literature. As the closed-form solution is not possible, by means of a numerical procedure, the obtained integral equation is solved and the accuracy of numerical procedure and mathematical formulation is verified through comparison with the available results for the relevant analysis in isotropic solids. Through several numerical displays, the effect of material anisotropy on the stiffness of the interacting system and the stress intensity factors at the tip of penny-shaped crack is surveyed. [ABSTRACT FROM AUTHOR]

In the paper we consider the possibility of constructing models of objects that have the maximum average values of the characteristics of scattering at a certain sector of angles. For optimization of these characteristics we use genetic algorithm. We developed algorithm on the base of the dependencies of the characteristic dimensions of a hollow structure with a maximum average values of the characteristics of scattering were calculated. [ABSTRACT FROM AUTHOR]

This paper analyzes partial slip contact problems in the theory of linear viscoelasticity under a wide variety of loading conditions, including cyclic (fretting) loads, using a semi-analytical method. Such problems arise in applications like metal-polymer contacts in orthopedic implants. By using viscoelastic analogues of Green’s functions, the governing equations for viscoelastic partial-slip contact are formulated as a pair of coupled Singular Integral Equations (SIEs) for a conforming (pin-plate) geometry. The formulation is entirely in the time-domain, avoiding Laplace transforms. Both Coulomb and hysteretic effects are considered, and arbitrary load histories, including bidirectional pin loads and remote plate stresses, are allowed. Moreover, the contact patch is allowed to advance and recede with no restrictions. Viscoelasticity necessitates the application of the stick-zone boundary condition in convolved form, and also introduces additional convolved gap terms in the governing equations, which are not present in the elastic case. Transient as well as steady-state contact tractions are studied under monotonic ramp-hold, unload-reload, cyclic bidirectional (fretting) and remote plate loading for a three-element solid. The contact size, stick-zone size, indenter approach, Coulomb energy dissipation and surface hoop stresses are tracked during fretting. Viscoelastic fretting contacts differ from their elastic counterparts in notable ways. While they shakedown just like their elastic counterparts, the number of cycles to attain shakedown states is strongly dependent on the ratio of the load cycle time to the relaxation time. Steady-state cyclic bulk hysteretic energy dissipation typically dominates the cyclic Coulomb dissipation, with a more pronounced difference at slower load cycling. However, despite this, it is essential to include Coulomb friction to obtain accurate contact stresses. Moreover, while viscoelastic steady-state tractions agree very well with the elastic tractions using the steady-state shear modulus in load-hold analyses, viscoelastic fretting tractions in shakedown differ considerably from their elastic counterparts. Additionally, an approximate elastic analysis misidentifies the edge of contact by as many as 7 degrees in fretting, showing the importance of viscoelastic contact analysis. The SIE method is not restricted to simple viscoelastic networks and is tested on a 12-element solid with very long time scales. In such cases, the material is effectively always in a transient state, with no steady edge-of-contact. This has implications for fretting crack nucleation. [ABSTRACT FROM AUTHOR]

There is an inaccurate formula in Huang et al. (1996). In fact, a substantial term is missing in their equation (14) for computing the value of an important option hedging parameter, i.e., the vega. We fix it in this note by providing its correct form and characterizing an associated (new) integral equation. Some related explanations and arguments are also corrected. [ABSTRACT FROM AUTHOR]

• Wertheim's thermodynamic perturbation theory were applied to the rose model. • Liquid part of phase diagram were calculated. • Two critical points are observed. Monte Carlo simulations and Wertheim's thermodynamic perturbation theory (TPT) are used to predict the phase diagram and percolation curve for the simple two-dimensional rose model of water. In the rose model of water, the water molecules are modelled as two-dimensional Lennard-Jones disks, with additional rose potentials for orientation dependent pairwise interactions that mimic formation of hydrogen bonds. Modifying both the shape and range of a 3-petal rose function, it was constructed an efficient and dynamical mimic of the 2D Mercedes Benz (MB) water model and experimental water. The liquid part of the phase space is explored using grand canonical Monte Carlo simulations and two versions of Wertheim's TPT for associative fluids. We find that the theory reproduces well the physical properties of hot water but is less successful at capturing the more structured hydrogen bonding that occurs in cold water. In addition to reporting the phase diagram and percolation curve of the model, it is shown that the improved TPT predicts the phase diagram rather well, while the standard one predicts a phase transition at lower temperatures. For the percolation line, both versions have problems predicting the correct position of the line at high temperatures. [ABSTRACT FROM AUTHOR]

In this paper, we describe a Nyström integration method for the integral operator T which is the normal derivative of the double-layer potential arising in problems of two-dimensional acoustic scattering by infinite rough interfaces. The hypersingular kernel and unbounded integral interval of T are the key difficulties. By using a mollifier, we separately deal with these two difficulties and propose its Nyström integration method. Furthermore, we establish convergence of the method. Finally, we apply the method to the scattering problem by infinite rough interfaces and carry out some numerical experiments to show the validity. [ABSTRACT FROM AUTHOR]

We show that, by correctly selecting the probability distribution function p ( s ) for a particle’s distance-to-collision, the nonclassical diffusion equation can be represented exactly by the nonclassical linear Boltzmann equation for an infinite homogeneous medium. This choice of p ( s ) preserves the true mean-squared free path of the system, which sheds new light on the results obtained in previous work. [ABSTRACT FROM AUTHOR]

An inverse coefficient problem of thermal conductivity for a functionally graded hollow cylinder is considered. After applying the Laplace transform, the direct thermal conductivity problem is solved by using two methods: (1) based on a reduction to the Fredholm integral equation of the 2nd kind; (2) by means of the Galerkin method. A comparison of the direct problem solving techniques is provided. The nonlinear inverse problem is solved on the basis of an iterative process; at every step of the latter the linear Fredholm integral equation of the 1st kind is solved. Results of the computational experiments on a reconstruction of variation laws of thermal conductivity and specific volumetric heat capacity are presented. [ABSTRACT FROM AUTHOR]

Absorption spectroscopy is an advanced tool for flow diagnostics in measuring multiple parameters of species. Such kinds of problems can be modeled by some integral equations with known kernel, aiming to the determination of the integrands from their integration values along all possible paths of injected lasers. This paper considers the parameters detection problems in combustion process, with the purpose of recovering the gas temperature and the concentration of burned gas simultaneously using injected lasers along two directions with multiple frequencies. After establishing the nonlinear integral equations describing the energy absorption process, this ill-posed model is transformed into a nonlinear optimization problem with some penalty terms. Then we present an alternative iteration scheme (AIS) to solve this problem. The convergence of the iterative sequence for AIS algorithm together with the estimate on the value of cost functional is established, ensuring that AIS can indeed generate a satisfactory approximate solution to the original optimization problem. Numerical implementations using simulant data are presented to show the validity of the proposed scheme. [ABSTRACT FROM AUTHOR]

In this paper, an integral equation approach is adopted to price American-style down-and-out calls. Instead of using the probability theory as used in the literature, we use the continuous Fourier sine transform to solve the partial differential equation system governing the option prices. As a way of validating our approach, we show that the “early exercise premium representation” for American-style down-and-out calls without rebate can be re-derived by using our approach. We then examine the case that time-dependent rebates are included in the contract of American-style down-and-out calls. As a result, a more general integral representation for the price of an American-style down-and-out call, with the presence of an extra term associated with the rebate, is obtained. Our numerical method based on the newly-derived integral representation appears to be efficient in computing the price and the hedging parameters for American-style down-and-out calls with rebates. In addition, significant effects of rebates on the option prices and the optimal exercise boundaries are illustrated through selected numerical results. [ABSTRACT FROM AUTHOR]

Transient dynamic stresses around three stacked parallel cracks in an infinite elastic plate are estimated for an incident impact stress wave impinging normal to the cracks. Using Fourier and Laplace transform techniques, the boundary conditions are reduced to six simultaneous integral equations in the Laplace domain. The differences in the displacements inside the cracks are expanded in a series of functions that have zero value outside the cracks. The Schmidt method is used to solve the unknown coefficients in the series such that the conditions inside the cracks are satisfied. The stress intensity factors are defined in the Laplace domain, and these are inverted using the numerical method. The stress intensity factors are calculated numerically for some crack configurations. [ABSTRACT FROM AUTHOR]

The differential transform method is extended to solve the Cauchy type singular integral equations (CSIEs) over a finite interval. New theorems for transformation of Cauchy singular integrals are given with proofs. Approximate solutions of CSIEs with two types of kernels, Degenerate and convolution, are obtained. The system of linear equations for characteristic equation is solved analytically. Numerical results are shown to illustrate the efficiency and accuracy of the present method. [ABSTRACT FROM AUTHOR]

The goal of this work is to reduce the cost of computing the coefficients in the Karhunen–Loève (KL) expansion. The KL expansion serves as a useful and efficient tool for discretizing second-order stochastic processes with known covariance function. Its applications in engineering mechanics include discretizing random field models for elastic moduli, fluid properties, and structural response. The main computational cost of finding the coefficients of this expansion arises from numerically solving an integral eigenvalue problem with the covariance function as the integration kernel. Mathematically this is a homogeneous Fredholm equation of second type. One widely used method for solving this integral eigenvalue problem is to use finite element (FE) bases for discretizing the eigenfunctions, followed by a Galerkin projection. This method is computationally expensive. In the current work it is first shown that the shape of the physical domain in a random field does not affect the realizations of the field estimated using KL expansion, although the individual KL terms are affected. Based on this domain independence property, a numerical integration based scheme accompanied by a modification of the domain, is proposed. In addition to presenting mathematical arguments to establish the domain independence, numerical studies are also conducted to demonstrate and test the proposed method. Numerically it is demonstrated that compared to the Galerkin method the computational speed gain in the proposed method is of three to four orders of magnitude for a two dimensional example, and of one to two orders of magnitude for a three dimensional example, while retaining the same level of accuracy. It is also shown that for separable covariance kernels a further cost reduction of three to four orders of magnitude can be achieved. Both normal and lognormal fields are considered in the numerical studies. [ABSTRACT FROM AUTHOR]

A small-angle X-ray scattering has been used to probe protein–protein interaction in solution. Conventional methods need to input modeled potentials with variable/invariable parameters to reproduce the experimental structure factor. In the present study, a model-free method for extracting the excess part of effective interaction potential between protein molecules in solutions over an introduced hard-sphere potential by using experimental data of small-angle X-ray scattering is presented on the basis of liquid-state integral equation theory. The reliability of the model-free method is tested by the application to experimentally derived structure factors for dense lysozyme solutions with different solution conditions [Javid et al., Phys. Rev. Lett. 99 , 028101 (2007), Schroer et al., Phys. Rev. Lett. 106 , 178102 (2011)]. The structure factors calculated from the model-free method agree well with the experimental ones. The model-free method provides the following picture of the lysozyme solution: these are the stabilization of contact-pair configurations, large activation barrier against their formations, and screened Coulomb repulsion between the charged proteins. In addition, the model-free method will be useful to verify whether or not a model for colloidal system is acceptable to describing protein–protein interaction. [ABSTRACT FROM AUTHOR]

In this paper, based on the principle of virtual work, we formulate the equivalent eigenstrain approach for inhomogeneous inclusions. It allows calculating the elastic deformation of an arbitrarily connected and shaped inhomogeneous inclusion, by replacing it with an equivalent homogeneous inclusion problem, whose eigenstrain distribution is determined by an integral equation. The equivalent homogeneous inclusion problem has an explicit solution in terms of a definite integral. The approach allows solving the problems about inclusions of arbitrary shape, multiple inclusion problems, and lends itself to residual stress analysis in non-uniform, heterogeneous media. The fundamental formulation introduced here will find application in the mechanics of composites, inclusions, phase transformation analysis, plasticity, fracture mechanics, etc. [ABSTRACT FROM AUTHOR]

In this study, the frictional contact with partial slide between two dissimilar elastic cylinders is considered. According to the Spence׳s self-similarity condition, a system of singular integral equations is constructed with respect to the normal pressure and the shear traction in the contacting area. Based on the Goodman׳s hypothesis, the preceding system is uncoupled. From this, the tangential load in the central sticking zone is possible to be obtained analytically by means of the theory on the singular integral equation. Besides, a nonlinear equation in regard to the ratio of the adhesive and slip zone sizes is derived on the basis of the continuity of the tangential load. The sticking zone size can thus be determined by solving the nonlinear equation mentioned above iteratively. The problem in question is additionally solved by utilizing numerical method to make verification and validation of the theory and the related prevision found in the present paper. Numerical examples are provided to instantiate the theroy and method proposed. [ABSTRACT FROM AUTHOR]

In this paper, authors demonstrate the efficiency of optimal homotopy asymptotic method (OHAM). This is done by solving nonlinear Volterra integral equation of first kind. OHAM is applied to Volterra integral equations which involves exponential, trigonometric function as their kernels. It is observed that solution obtained by OHAM is more accurate than existing techniques, which proves its validity and stability for solving Volterra integral equation of first kind. [ABSTRACT FROM AUTHOR]

Abstract: In the anti-plane problem about a high-frequency diffraction by an interface crack located between two different elastic materials we propose a new asymptotic approach, which reduces the problem to the Wiener–Hopf integral equations. The key point of the method is a factorization of the symbolic function which is performed in an efficient way. As a result, the leading asymptotic term is written out in an explicit analytical form. [Copyright &y& Elsevier]

Abstract: By making use of asymptotic properties of nonoscillatory solutions, the oscillation behavior of solutions for the integro-dynamic equation and the integral equation on time scales is investigated. Easily verifiable sufficient conditions are established for the oscillation of all solutions. The results are new for both continuous and discrete cases. The paper is concluded by an open problem. [Copyright &y& Elsevier]