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Fracture toughness is a basic property of ceramic materials, and it is strongly influenced by their microstructures. However, the influences of grain-scale microstructural factors, including grain fracture energy, boundary fracture energy, and grain size of the microstructure, on the fracture toughness remain unclear. To investigate this, a dual-scale finite element method was developed. Models were established to assess the microstructural mechanical properties, with an average grain size of 0.1-100 μm, and different grain boundary fracture energies and grain fracture energies. Based on the microstructural mechanical properties, a macroscale three-point bending model, combined with the extended finite elementmethodwas used to evaluate the fracture toughness. The results indicated that fracture toughness increased with the decrease in the grain size when the grain size was below 5 μm. Conversely, when the grain size was greater than 5 μm, the fracture toughness remained relatively constant. Additionally, the grain boundary fracture energy played a more important role for the fracture toughness than the grain fracture energy. Finally, a formula that quantitatively describes the relationship between the microstructure and fracture toughness was obtained. This work provides a valuable reference for evaluating the performances of structural ceramics. [ABSTRACT FROM AUTHOR]

We study the problem of determination of conditions for the existence of solutions to weakly nonlinear periodic boundary-value problems for systems of ordinary differential equations with switchings and construct these solutions. The critical case where the equation for generating amplitudes of weakly nonlinear periodic boundary-value problem with switchings does not turn into the identity is analyzed. We improve the classification of critical and noncritical cases and construct an iterative algorithm for finding solutions of the weakly nonlinear periodic boundary-value problem with switchings in the critical case on the basis of the generalized Newton–Kantorovich theorem. We investigate the case of a nonlinear equation whose dimension does not coincide with the dimension of the unknown in the case where the rank of Jacobian of the nonlinear equation is complete. As an example of application of the constructed iterative scheme based on the generalized Newton–Kantorovich theorem, we obtain approximations to the solutions of periodic boundary-value problem for the mathematical model of nonisothermal chemical reaction. To check the accuracy of the obtained approximations, we evaluate the discrepancies in the original equation. We establish an estimate for the length of the range of a small parameter, where the iterative procedure aimed at the construction of solutions to the weakly nonlinear periodic boundary-value problem with switchings in the mathematical model of nonisothermal chemical reaction is convergent. To this end, we use the conditions of convergence, in particular, the requirement of contraction for the operator used to obtain the required solution of the original problem under the assumption of applicability of the generalized Newton–Kantorovich theorem. [ABSTRACT FROM AUTHOR]

In this paper we will study a Voronovskaja type theorem and a simultaneous approximation result for a new class of generalized Bernstein operators. The new operators are obtained using a generalization of Kantorovich's method, namely, we will introduce a sequence of operators Kln = Dl ... Bn+l ... Il, where Bn+l are Bernstein operators, Dlf = f(l) + al-1f(n-1) + · · · + a1f' + a0f is a differential operator with constant coefficients aj, j ∈ {0, . . ., l - 1} and Il a corresponding antiderivative operator such that Dl ... Il = Id. [ABSTRACT FROM AUTHOR]

The computation of three-dimensional unsteady non-hydrostatic flows over large domains and/or for long simulation times is frequently conducted in research and practice using approximate methods to avoid the cost of a fully 3D solution. Among these methods are the shallow-water perturbation theories of St. Venant and Boussinesq, and the theory of directed fluid sheets by Green and Naghdi. The latter theory is not limited to shallow-water flows, and it is essentially a weighted-average residual method of Galerkin type, further developed by Shields and Webster for maritime hydraulics. The method is in essence equivalent to the vertically-averaged and moment (VAM) equations developed by Steffler and Jin for open channel flows, although this has not been recognized in the literature. A general framework for constructing VAM models of high-order, by linking maritime and open channel flow developments, is not available in the literature. In this work, a generalized framework for designing weighted-average residual equations for free-surface flow is presented based on the Kantorovich and Krylov method. The development of physically sound expansions for the hydrodynamic variables, and the construction of general systems of VAM equations to determine the unknowns in the expansions by selecting suitable weighting functions, is discussed in detail. The approach produces high-order models, thereby generalizing Steffler and Jin's development. A hierarchy of high-order VAM models is demonstrated to progressively converge to the exact dispersion relation of periodic waves by increasing the vertical resolution. These models are not limited by any shallowness assumption and exhibit more accurate wave dispersion properties compared to the Serre-Green-Naghdi equations. Computational results show that the VAM equations produce an accurate prediction of dam-break waves and dispersive wave effects over submerged bars. [ABSTRACT FROM AUTHOR]

Starting from the decomposition method for operators, we consider Newton‐like iterative processes for approximating solutions of nonlinear operators in Banach spaces. These iterative processes maintain the quadratic convergence of Newton's method. Since the operator decomposition method has its highest degree of application in non‐differentiable situations, we construct Newton‐type methods using symmetric divided differences, which allow us to improve the accessibility of the methods. Experimentally, by studying the basins of attraction of these methods, we observe an improvement in the accessibility of the derivative‐free iterative processes that are normally used in these non‐differentiable situations, such as the classic Steffensen's method. In addition, we study both the local and semilocal convergence of the considered Newton‐type methods. [ABSTRACT FROM AUTHOR]

This work establishes fast rates of convergence for empirical barycenters over a large class of geodesic spaces with curvature bounds in the sense of Alexandrov. More specifically, we show that parametric rates of convergence are achievable under natural conditions that characterize the bi-extendibility of geodesics emanating from a barycenter. These results largely advance the state-of-the-art on the subject both in terms of rates of convergence and the variety of spaces covered. In particular, our results apply to infinite-dimensional spaces such as the 2-Wasserstein space, where bi-extendibility of geodesics translates into regularity of Kantorovich potentials. [ABSTRACT FROM AUTHOR]

We present an iterative nonlinear inverse scattering algorithm for high-resolution acoustic imaging of density and velocity variations. To solve the multi-parameter nonlinear direct scattering problem, the acoustic wave equation for inhomogeneous media in the frequency domain is transformed into a vectorial integral equation of the Lippmann–Schwinger type for the combined pressure and pressure-gradient field. To solve the multi-parameter nonlinear inverse scattering problem, we use the Newton–Kantorovich method in conjunction with matrix-free representations of the Fréchet derivative operators and their adjoints. The approximate Hessian information that is accounted for in our iterative solution of the (nonlinear) multi-parameter inverse scattering problem is essential for the mitigation of multi-parameter cross talk effects. Numerical examples related to seismic and medical ultrasound breast imaging illustrate the performance of the new algorithm for multi-parameter acoustic imaging. [ABSTRACT FROM AUTHOR]

We have proposed a development of the variational iteration method (VIM), or extended Kantorovich method, by studying physically nonlinear (FN) or geometrically nonlinear (GN) Kirchhoff nanoplates as an example. The modified couple stress theory was used for modeling size-dependent factors of the Kirchhoff nanoplates. Nested one into the other iteration procedures of the Birger method of variable elasticity parameters, of the variational iteration method (VIM), and of the Newton–Raphson method for physically nonlinear (FN) Kirchhoff nanoplates were constructed. The solution of problems for geometrically nonlinear (GN) Kirchhoff nanoplates was carried out on the basis of the variational iteration method and the Newton–Raphson method. The validity of the results was ensured by the coincidence of the results obtained via several methods of reducing partial differential equations to ordinary differential equations and via the finite difference method. The computational effectiveness of the proposed iterative procedure was demonstrated in terms of both accuracy and performance. A comparison of the results obtained showed that the variational iteration method (VIM) is the most efficient and fastest of all the methods considered both for problems with physical nonlinearity and for geometrically nonlinear problems. [ABSTRACT FROM AUTHOR]

A problem of stability of cylindrical shells made of a composite material is solved, taking into account the momentum and nonlinearity of their subcritical stress–strain state. In geometric terms, the nonlinear stability problem is solved by finite element methods and the Newton–Kantorovich linearization. Critical loads are determined in the process of solving a nonlinear problem using the Sylvester criterion. The study is carried out using the finite elements of composite cylindrical shells of natural curvature developed on the basis of the Timoshenko hypothesis, with rigid displacements being explicitly distinguished within their displacement approximation, which significantly affects the convergence of the solution. The stability of a circular cylindrical shell made of a polymer composite material under combined loading by torque, bending moment, and internal pressure is investigated. This study also touches upon the influence of the stacking method of monolayers, deformation nonlinearity, and internal pressure on critical loads at which shell buckling is observed. [ABSTRACT FROM AUTHOR]

Methods of reduced order modelling (ROM) of nonlinear PDEs to ODEs based on investigation of flexible arbitrary geometry of the plan nano-plates have been reviewed, modified, and employed. The modified von Kármán equations serve as the paradigmatic object of investigation. They are derived in the framework of the modified coupled stress theory with the inclusion of the von Kármán geometric nonlinearity. The elliptic type PDEs are reduced to a system of ODEs using several methods, and the results are compared in order to quantify them for rectangular nano-plates. The following ROM methods have been used: Fourier method of separation of variables, the Kantorovich-Vlasov method (KVM), the variational iterations method (VIM) known as the extended Kantorovich method (EKM), the Agranovskii–Baglai–Smirnov method (ABSM) in combination with the KVM and VIM. We have found that the solutions obtained by combining VIM with the ABSM can be considered exact solutions of the von Kármán equations. The convergence of the developed iterative procedures is justified by a number of theorems for the static analysis of nanoplates on a rectangular plane. [ABSTRACT FROM AUTHOR]

Conditions for the existence of solutions of nonlinear matrix equations have been found and a method of their construction has been proposed. As an example of the iterative scheme construction, approximations for the solutions of the nonlinear algebraic matrix Riccati equation and their accuracy errors were determined. [ABSTRACT FROM AUTHOR]

A closed‐form bending solution of the multisegmented panels is presented for different support conditions. The panels can have multiple no. of segments along the x‐axis with varying material properties. The governing equations are formulated using the extended Kantorovich method (EKM) in the mixed‐form, and a set of ordinary differential equations have been obtained. The continuity of displacement and stresses are satisfied at the interface of each segment. The panels can have simply‐support, clamp, and free types of support conditions. The panel interface at each segment along the x‐axis is assumed to be perfectly bonded. In this work, the stress and deformation variations for four types of panel configuration have been studied. Two segmented panels having aluminum and Graphite–Epoxy with equal and unequal segments are considered. A four‐segmented Al/SiC panel having gradual material variation is also analyzed. The effect of ply‐angles on the longitudinal variables is also studied for the equal segmented Al/GrEp panel. The deflection and stresses are compared with the finite element solution and found in good agreement with the EKM results. The current development will lead to developing the solution for stepped panels, dissimilar plates, and more complex cases used in aerospace applications. [ABSTRACT FROM AUTHOR]

The present article is an extension of work and provides Voronovskaja-type quantitative estimate for the difference of positive linear operators with different fundamental functions. We also present the applications of such results using Bernstein operators, Bernstein-Kantorovich operators, Bernstein-Durrmeyer operators, Generalized-Bernstein operators based on Pólya distribution and its Durrmeyer and Kantorovich-type modifications. We use Mathematica software to direct massive computations. [ABSTRACT FROM AUTHOR]

This presents a mathematical model proposed by the author for description of deformation process of a shell structure under the action of a load that depends linearly on time. Material orthotropy, geometric nonlinearity, transverse shifts are taken into account. A distinctive feature of the model is the use of a refined discrete method for taking into account stiffeners, proposed by the author earlier. Prior to this, the method was used only in static or isotropic dynamic problems. It is proposed to add correction normalizing factors, which makes it possible to obtain the most accurate values of critical loads. The methodology of the calculation algorithm under dynamic loading is based on the Kantorovich method and the Rosenbrock method, which allows solving rigid ordinary differential equation (ODE) systems. New numerical results for cylindrical panels are obtained. The influence of the number of stiffening elements on the values of the critical load is shown. A comparison with the classical discrete method of taking into account stiffeners is carried out. For the problems considered in this paper, the phase portraits of the system are shown. [ABSTRACT FROM AUTHOR]

The components of the stress-strain state of rubber shock absorbers under viscoelastic deformation are investigated. The contraction of rubber shock absorber under the action of a vertical force and a combination of vertical and shear forces is calculated. The dependence of the contraction of the structure on time was obtained. The specific mechanical properties inherent in rubber, such as the weak compressibility of the material and deformation viscoelasticity deformation, the mathematical modeling of which involves significant mathematical difficulties, were taken into account. In the numerical calculation, the weak compressibility of rubber was modeled using a finite element moment scheme for weakly compressible materials, the essence of which is the triple approximation of displacement fields, strain components and volume change function. For the mathematical modeling of the viscoelastic nature of strains, the hereditary Boltzmann–Volterra theory with exponential relaxation kernel is used. On the basis of the Lagrange variational principle, a system of resolving integral equations of viscoelasticity is obtained taking into account the weak compressibility of the material, and an iterative procedure for its solution is proposed. The modified Newton–Kantorovich method is used to solve a nonlinear boundary value problem. The numerical solution is obtained by the finite element method in the case of viscoelastic deformation of rubber material. The calculation is performed for the case when the rubber layer is vulcanized to metal plates. The influence of the viscoelastic properties of the rubber on the value of rubber shock absorber contraction for two cases of mechanical loading is investigated. Calculations were carried out for two rubber grades: 2959 and 1562. [ABSTRACT FROM AUTHOR]

• A multiscale analysis is presented to study buckling and free vibration of grid-stiffened composite conical panels. • Four types of unit cell configurations including Lozenge, Kagome, Diamond, and triangle grids are considered to construct the conical panel. • Highly accurate closed-form solutions are presented for eigenvalue analyses of the complicated governing partial differential equations. • Other geometries including cylindrical panels and rectangular plates are also studied. • Performance of the method in terms of accuracy, convergence and computational cost is examined through various examples. This paper develops very accurate closed-form solutions for buckling and free vibration of grid-stiffened composite conical panels using a semi-analytical method. On the base of first-order shear deformation theory (FSDT), the governing formulations are derived while an appropriate unit cell is used to derive the material properties of the lattice from their bulk properties. Multiple systems of ordinary differential equations (ODEs) are obtained by applying the Extended Kantorovich Method (EKM) to solve the governing system of partial differential equations (PDEs). The resulting ODEs are then solved using semi-analytical closed-form solutions to study buckling and free vibration of grid-stiffened panels. It is interesting to note that due to the general formulation of the conical panel, it is very easy to obtain other useful geometries including cylindrical panels, and even rectangular plates with general unit cell configurations. Regarding stability, fast convergence, and very low computing cost, the effectiveness of the proposed technique is investigated. The accuracy of the critical buckling loads and natural frequencies for various cases is studied which shows very good agreement in comparison with results obtained from the commercial finite element code ANSYS. [ABSTRACT FROM AUTHOR]

With the increasing penetration of renewable energy, the Power-to-hydrogen technology accelerates the adoption of new energy vehicles, including both hydrogen vehicles (HV) and electric vehicles (EV). Simultaneously, the integration of Vehicle-to-Grid (V2G) technology further promotes the development of electric-hydrogen integrated energy systems coupled with transportation systems. In this study, an optimal scheduling model is developed for an EV-HV coupled electric-hydrogen integrated energy system, taking into account V2G response and carbon trading. Firstly, to address the randomness of renewable energy and loads, the Latin hypercube sampling method in conjunction with the Kantorovich scenario reduction method, is employed for scenario generation and simplification. Secondly, an optimization model is developed for the operation of hydrogen-blended gas turbines and the energy management of EVs and HVs. To encourage the participation of EVs and HVs, a reward mechanism for V2G responses is introduced, complemented by a time-of-use tariff, to guide orderly charging and discharging. Furthermore, the influence of the operating modes of EVs and HVs, as well as the hydrogen blending ratio, on the total system cost and carbon emissions is examined. An industrial park is selected as an illustrative example to showcase the effectiveness of the proposed method, demonstrating satisfied economic and environmental performances. [ABSTRACT FROM AUTHOR]

In this paper, the Variational Iteration Method (VIM) or Extended Kantorovich Method (EKM) is formulated for the first time for flexible porous functionally graded (PFGM) plates with different boundary conditions and geometric nonlinearity according to the theory of Theodore von Karman. Its accuracy and efficiency are demonstrated. The plate is subjected to a uniform transversal load and temperature field. The displacement field of the plate is approximated based on the classical plate theory (CTP) or Kirchhoff's plate theory. The governing equations are derived using Hamilton's principle. Modified Coupled Stress Theory (MCST) is used to account for size-dependent effects. The material properties vary with thickness and are temperature dependent. Four porosity distribution patterns are considered in this study. Several examples are solved to demonstrate the proposed algorithm. The results obtained are compared with solutions obtained by the Bubnov-Galerkin Method (BGM) in higher approximations, the Finite Difference Method (FDM) of second order accuracy, as well as with results obtained by the Finite Element Method (FEM) of other authors. The results include an analysis of the effect of size dependent parameters, porosity type pattern, porosity index, functionally graded index, temperature field and different types of boundary conditions on the stress–strain state and bending deflection of plates. [ABSTRACT FROM AUTHOR]

We establish necessary and sufficient conditions of solvability of the nonlinear boundary-value problem in the critical case. By using the Newton–Kantorovich method, we propose a new iterative scheme for the construction of solutions of the weakly nonlinear boundary-value problem for a system of ordinary differential equations in the critical case. [ABSTRACT FROM AUTHOR]

• Developed a novel mathematical model to study stress-strain behavior PFG size-dependent plates. • Numerical results are obtained using the developed methods: Bubnov-Galerkin method (BGM) in higher approximations, second order finite difference method (FDM), variational iteration method (VIM) both in the first (VIM 1) and in the second (VIM 2) approximation. • Consistent results across all methods affirm their reliability and precision. • Comprehensive analysis includes nanoscale factors, porosity, loading types, and boundary conditions impacting the stress-strain state. In this study, a mathematical model of flexible porous functionally graded (PFG) Kirchhoff size-dependent plates on a rectangular plane is constructed. Hamilton's principle provides new size-dependent governing equations, taking into account geometrical nonlinearity according to the model of Theodor von Kármán, as well as boundary conditions and initial conditions for the displacement of the plates. Modified Coupled Stress Theory (MCST) is used to account for size-dependent effects. Numerical investigations analysing the stress-strain state of flexible Kirchhoff functionally graded size-dependent porous plates on a rectangular plane are based on the application of the Variational Iteration Method (VIM) or the Extended Kantorovich Method (EKM). The efficiency and high accuracy of the VIM method are demonstrated. The validity and reliability of the solutions obtained by VIM are discussed and compared with solutions obtained by other methods, such as the BGM in higher approximations, FDM of second-order accuracy and the FEM. Additionally, the solutions are compared with results obtained by other authors in the study of porous functionally graded macro, micro, and nanoplates. The solutions obtained are considered accurate. The results include an analysis of the influence of size-dependent parameters, type of material porosity, porosity index, functionally graded index, and different types of boundary conditions on the stress-strain state of plates under the action of different types of transverse loading. [ABSTRACT FROM AUTHOR]

A 2-D closed-form analytical solution for the multi-segmented piezoelectric panel has been developed using the extended Kantorovich method (EKM). In this work, a study on the bending behavior of axially segmented piezoelectric panel with different materials have been carried out. This work consider two equally segmented piezoelectric panels made of piezoelectric fiber-reinforced composite (PFRC-2) and lead zirconate titanate (PZT-5A). Furthermore, a particular case of four equal segmented piezoelectric panel is also considered, which imitates an axially graded piezoelectric actuator. The panels are subjected to various boundary conditions and an uniform pressure loading at the top surface. Along the X -axis, interfaces are assumed to be perfectly bonded to satisfy the continuity of displacement and stress at each segment interface. 2-D finite element (FE) solutions using ABAQUS are also obtained to validate displacement and stresses. A good agreement between the analytical and 2-D FE results is observed for all the cases. The current development can be used as a benchmark solution for axially stacked plates or other complex problems. [ABSTRACT FROM AUTHOR]

We give a survey of applications of the ideas and methods of Leonid Kantorovich (1912–1986) to the classical problems of approximation theory. [ABSTRACT FROM AUTHOR]

We establish necessary and sufficient conditions for the solvability of the nonlinear boundary-value problem in the critical case and develop a scheme for the construction of solutions of this problem. By using the Newton–Kantorovich method, we propose a new iterative scheme for the determination of solutions to the weakly nonlinear boundary-value problem for a system of ordinary differential equations in the critical case. As examples of application of the constructed iterative scheme, we obtain approximations to the solutions of a periodic boundary-value problem for the Duffing and Liénard equations. To check the accuracy of the established approximations to the solutions of the periodic boundary-value problem for the Duffing and Liénard equations, we use discrepancies in the original equations. [ABSTRACT FROM AUTHOR]

• Derivation of integro-differential non-linear system for bending vibrations of a rotating slender flexible structure. • New stochastic method developed by combining integral transforms with generalized probabilistic approach of uncertainties. • Easy to combine with parametric and nonparametric probabilistic approaches for both model-parameter and model uncertainties. • Robust dynamic analysis in both time- and frequency-domains under both types of uncertainties. • Stochastic model updating under both types of uncertainties using as observed data modal quantities. The current work combines the generalized probabilistic approach of uncertainties recently developed by Soize (2010)—with the generalized integral transform technique in order to take into account two types of uncertainties: (i) model-parameter uncertainties and (ii) model uncertainties induced by modelling errors. The generalized integral transform technique (or GITT) is a powerful meshless hybrid analytical-numerical approach based on eigenfunction expansions to solve systems of partial differential equations and has been progressively advanced during the last three decades. The current work advances the state-of-the-art of the GITT approach by rigorously taking into account both model-parameter uncertainties and model uncertainties to improve the predictive accuracy of computational models developed in the context of computational structural dynamics. The developed stochastic GITT method is flexible enough to handle parametric and non-parametric probabilistic methods to quantify both types of uncertainties. It is applied to the governing equations derived with the extended Hamilton's variational principle for predicting the in-plane and out-of-plane bending vibrations of a slender flexible structure connected to a rotating rigid shaft, resembling many complex engineered structures. Stochastic dynamic analyses in both time- and frequency-domains under both types of uncertainties are firstly carried out. Finally, stochastic model updating is carried out with the maximum likelihood and Bayesian statistical methods in order to construct both the optimum prior stochastic models of uncertainties and the posterior stochastic model of model-parameter uncertainties, using the first natural frequencies as observed data. [ABSTRACT FROM AUTHOR]

The delamination/debonding failure in laminated structures is often caused by development of concentrated stress fields near the edges. Because of their three-dimensional (3D) nature, it is important to develop accurate elasticity solutions for these fields, which should include the possibility of non-rigid layer interfaces, to have realistic predictions. This article presents an accurate 3D piezoelasticity-based analytical solution for smart composite panels featured with piezoelectric sensor/actuator layers, having interfacial bonding imperfections and showing edge effects. The displacement and electric field discontinuities at the imperfectly bonded layer interfaces are represented by adopting the generalized spring–layer model. The model is implemented into the Reissner-type mixed variational principle for the piezoelectric medium. The partial differential equations thus developed are solved using the mixed-field multiterm Kantorovich method, which exactly satisfy all the mechanical and electric boundary conditions at the panel edges, top and bottom surfaces of the panel and interlaminar continuity/jump conditions. The accuracy and convergence of the present results are established in comparison with the available exact piezoelasticity solutions for simply supported panels. The effect of compliance and imperfection location on the electromechanical response of piezolaminated composite panels with edge effects is investigated. The influence of actuator debonding on the interlaminar stress transfer and actuation authority is also shown. [ABSTRACT FROM AUTHOR]

The optimized design of composite adaptive structures puts forward higher requirements and challenges to the actual configuration of the structural section. In this paper, a trapezoidal laminate model of composite materials is established. Based on the classical laminates theory, the bending problem of trapezoidal laminates is solved by using the Kantorovich method and the principle of minimum potential energy. The analytic form of the solution is found to satisfy the Euler equation and displacement boundary conditions. Taking the wing of a jet transport aircraft as an example, the accuracy of the analytical solution is verified by the finite element method. The Differential Evolution algorithm is used to realize the multi-objective optimal design of the bending–twisting coupled trapezoidal laminates, and the hygrothermal stability of laminates is verified by the finite element method. Finally, based on the Monte Carlo method, the robustness analysis of the bending–twisting coupling effect of laminate is realized; meanwhile the feasibility and reliability of the design scheme are verified. The stress and strain functions at each point of the trapezoidal laminate can be further obtained by the analytical solution, making it more convenient to analyze the stresses and calculate the static forces of the trapezoidal laminate and its composite structures, which is of great significance to effectively improve the comprehensive mechanical properties of the cross-section structure of composite materials. [ABSTRACT FROM AUTHOR]

In this paper, analytical bending solution of a rectangular thin plate with different boundary conditions was obtained by using the two‐dimensional generalized finite integral transform method. During the solution procedure, the vibrating beam functions satisfying different boundary conditions of the plate are taken as integral kernels to form the integral transform pairs. This two‐dimensional integral transform is used to the partial differential governing equations of the plate, to transform them to a system of linear algebraic equations from which the accurate analytical solution is obtained easily. The superiority of the present method is that it does not need to seek the deformation function in advance, and avoids the complex superposition processes. Thus, the proposed method is reasonable and feasible. The good agreement of present results and the analytical solution from the literature shows the validity of the method. [ABSTRACT FROM AUTHOR]

The main purpose of this paper is to use a power series summability method to study some approximation properties of Kantorovich type Szász–Mirakyan operators including Sheffer polynomials. We also establish Voronovskaya type result. [ABSTRACT FROM AUTHOR]

The analytical solution of the bending problem for a clamped rectangular plate with a variable in-plane stiffness is found by using the method of superposition. The flexural rigidity of the plate varies across its width according to an exponential function. First, the analytical solution for a simply supported rectangular plate with a variable in-plane stiffness is obtained, and then the bending problem for the plate clamped at its four edges is solved analytically by the superposition of one simply supported plate under the transverse load and two simply supported plates under pure bending. The influence of the variable in-plane stiffness, aspect ratio, and different boundary conditions on the deflection and bending moment is studied by numerical examples. The analytical solution presented here may be helpful for the design of rectangular plates with a variable in-plane stiffness. [ABSTRACT FROM AUTHOR]

The paper examines a static bending of porous functional plates (FGP) and rectangular plate solutions, based on an underlying high-order shear deformation theory. The proposed high-order shear deformation theory, as opposed to other theories, includes four unknowns. For this reason, a new shear strain function is considered. The technique of Navier is used in closed-form FGP solutions. Results of deflections and stresses are presented for simply supported border conditions. Current figures are contrasted with the non-poreous plate deflecting solutions and the literature's stresses. Effects of different parameters, including thickness, gradient index and porosity of FGM plates, are discussed. [ABSTRACT FROM AUTHOR]

We consider the classical Szász-Mirakyan and Szász-Mirakyan-Durrmeyer operators, as well as a Kantorovich modification and a discrete version of it. The images of exponential functions under these operators are determined. We establish estimates involving differences and quotients of these images. [ABSTRACT FROM AUTHOR]

The classical equations of plate bending in various fastenings along the contour and various loads are considered. Using the decomposition method, three auxiliary problems are formulated to determine the deflection of the plate. The task is to determine the individual terms of the decomposition coefficients a (i)nm from the individual terms of the load series. Then the system of algebraic equations is obtained to determine a (i)nm. If the plate contour is pivotally fixed along the contour, then the system of equations takes a known form and the expression for the deflection function is explicitly found. Approximate analytical solutions are obtained for the functions of deflections, internal force factors and stresses under various boundary conditions of the plate. [ABSTRACT FROM AUTHOR]

This article presents a novel linear programming (LP) based two-stage stochastic approach for microgrids (MGs) under uncertainties. In this regard, the day-ahead programming of dispatchable resources in MG was modeled, considering the uncertainties in demand loads, and upstream network electricity price. Moreover, the inherent stochastic nature of wind and solar resources as well as the environmental aspects were modeled to find a realistic solution. Also, real-time pricing (RTP), demand response (DR) program considering the energy storage system (ESS) as a DR option was implemented on an MG as a smart customer. The extensive form of the two-stage stochastic recourse model was properly implemented for dispatchable and non-dispatchable resources. Furthermore, scenario generation and reduction procedures were realized with autoregressive and moving average (ARMA) model-based time-series (TS), and backward reduction (BR) method by the Kantorovich distance (KD), respectively. The simulations on a grid-connected MG, including micro-turbine (MT), fuel-cell (FC), wind-turbine (WT), photovoltaic module (PV), and ESS were reported in operational cases based on one month of real recorded data for wind speed, solar irradiance, demand load, and upstream network electricity price. The results for the next day in real-time confirmed the accuracy of the developed optimization methodology. [ABSTRACT FROM AUTHOR]

We present a numerical solution in the form of a three-dimensional vortex soliton in unmagnetized plasma in the model of the generalized Zakharov equations with saturating exponential nonlinearity. To find the solution with a high accuracy, we use a two-step numerical method combining the Petviashvili iteration procedure and the Newton–Kantorovich method. The vortex soliton with the topological charge m = 1 turns out to be stable, provided that the nonlinear frequency shift exceeds a certain critical value. The stability predictions are verified by direct simulations of the full dynamical equation. [ABSTRACT FROM AUTHOR]

• Interlaminar stresses of laminated piezoelectric plates are obtained using analytical solution. • Electromechanical coupling effect of the piezoelectric plate is taken into account. • The solution is obtained using 3-D multi-term extended Kantorovich method. • Coupling influence on the edge effects of piezolaminated plates with various layups is investigated. In this paper, the interlaminar stresses of generally laminated piezoelectric (PZT) plates are presented. The electromechanical coupling effect of the piezoelectric plate is considered and the governing equations and boundary conditions are derived using the principle of minimum total potential energy. The solution procedure is a three-dimensional multi-term extended Kantorovich method (3DMTEKM). The objective of this paper is to study coupling influence on the edge effects of piezolaminated plates with finite dimensions and arbitrary lay-ups under uniform axial strain. These results can provide a benchmark for checking the accuracy of the other numerical methods or two-dimensional laminate theories. To verify the accuracy of the 3DMTEKM, special cases such as cross-ply or symmetric laminates are investigated and the results are compared with other analytical solutions available in the literature. Excellent agreement is achieved and then other numerical results are presented for general cases. Numerical examples imply on the singular behavior of interlaminar normal/shear stresses and electric field strength components near the edges of the piezolaminated plates. The coupling influence on the free edge effect with respect to the lay-ups of piezoelectric plate is studied in several examples. [ABSTRACT FROM AUTHOR]

• A novel finite strip/finite element formulation is proposed to deal with buckling and vibration of imperfect nanoplates. • The nonlinear equations are solved by the Kantorovich method. • The imperfection is modelled either as a thickness variation or as a planarity lackness. • The imperfection can be both localized and distributed. • Numerical examples are carried out to show the sensitivity of the results. In the present paper a coupling finite strip–finite element procedure is developed to investigate the buckling and vibration behaviour of imperfect nanoplates via nonlocal Mindlin plate theory. The imperfection can be either a thickness variation or a lack of planarity and it can be either localized or distributed on an entire edge of the nanoplate. The resulting nonlinear equations are solved exactly by applying the Kantorovich method. A finite element approach is proposed for coupling the in-plane and the out-of-plane buckling equations to describe properly the imperfections. Some numerical examples are carried out in order to show the sensitivity of the results to the nonlocal parameter and to the imperfection. [ABSTRACT FROM AUTHOR]

• The article presents the technique for numerically investigation of thin shells. • The most accurate model of dynamic deformation of thin orthotropic shells are used. • The computational algorithm is based on the L.V. Kantorovich method. • Rosenbrock method for solution of a stiff ODE system are used. • The methodology is accompanied by examples of calculation. Thin-walled orthotropic shell structures are used in various fields of engineering, and the process of their deformation is essentially non-linear, which significantly complicates their analysis. A technique for numerically investigation of the process of deformation of such structures under dynamic loading is proposed in this article. The computational algorithm is based on the L.V. Kantorovich method and Rosenbrock method for solution of a stiff ODE system. A new variation of the mathematical model is based on the functional of total deformation energy, takes into account geometrical nonlinearity, transverse shears, orthotropy of the material, and rotational inertia, and may be used for a study of shells of different geometric forms. Dimensionless parameters are used. The proposed model and the use of mentioned methods allows to produce the most accurate calculations of the buckling of shell structures. The applicability of this technique is shown: the results of calculations are given for orthotropic shallow shells of double curvature having a square base with cylindrical and conical panels. The load values are derived for local and total stability loss, for structures made of four different materials. The effect of the rate of loading on the obtained values is shown. [ABSTRACT FROM AUTHOR]

Based on the movement mechanism of strip steel and main drive system during the continuous rolling process, the main drive system of each frame was simplified into a two-degree-of-freedom discrete model and the strip steel was simplified into an axially moving Euler beam. Then, the coupled mechanical model of transverse and longitudinal vibrations of the strip steel with torsional vibration of the main drive system was established. Meanwhile, the nonlinear vibration differential equations of the coupled model were derived with Hamilton's principle, and the solution method of the mathematical model was studied. In fact, the solution method of the mathematical model was a joint solution process of discrete matrix differential equations and continuous nonlinear differential equations. Therefore, the established model could be called the discrete-continuous coupled nonlinear vibration model. Then, the discrete-continuous coupled differential equations were jointly solved with the modified iteration method and the Kantorovich averaging method. Finally, by using the MATLAB simulation, the amplitude-frequency response curves of strip steel vibrations coupled with torsional vibration of main drive system were obtained to explore the influences of axial velocity and tension of strip steel on coupled vibration characteristics. The results improve the theoretical study on the vibration of rolling mill for engineering applications. [ABSTRACT FROM AUTHOR]

In this article we present a new semilocal convergence analysis for the two step Kurchatov method by using recurrence relations under Lipschitz type conditions on first-order divided difference operator. The main advantage of this iterative method is that it does not require to evaluate any Fréchet derivative but it includes extra parameters in the first-order divided difference in order to ensure a good approximation to the first derivative in each iteration. The detailed study of the domain of parameters of the method has been carried out and the applicability of the proposed convergence analysis is illustrated by solving some numerical examples. It has been concluded that the present method converges more rapidly than the one step Kurchatov method and two step Secant method. [ABSTRACT FROM AUTHOR]

In recent years, the renewable energy sources are used in the power system for reducing environmental pollution. One of these renewable energy sources is wind farm that is used widely. However, they increase the uncertainty of the power system. To solve this problem, plenty researches have proposed several methods that have some disadvantages. This paper tries to decline the effects of uncertainty of wind turbines by using of the energy storage systems and optimal demand response programs. A stochastic planning of energy storage systems and wind generation are presented in the short term electricity markets considering the trading day-head and intraday market of demand response. In order to formulate an accurate model of various components of system, all needed constraints are considered. In this problem, wind turbines, energy storage systems, and demand response aggregator work coordinately that is called virtual power plant. For better comparison, two states consist of coordinated and uncoordinated working of all components are considered and assessed. The various scenarios for electricity market price, adjustment market price, imbalance energy price, and the generated power of wind turbine are considered. These scenarios are generated with hybrid neural network and coati optimization algorithm. Also, the iterative fast progressive Kantorovich method is used for scenario reduction. The results show the better response of the proposed method in optimal operation of virtual power plant. • Considering the ESS unit, responsive loads aggregators, and RES unit as a VPP. • Present a stochastic method of energy exchange of ESSs and WT. • Creating scenarios for day-ahead market, adjustment, and imbalance markets, and WT. • Providing value and price of energy to obtain maximum profit in short-term market. • Maximizing the benefit of WT and elimination of uncertainties. [ABSTRACT FROM AUTHOR]

The present paper develops a size-dependent Kirchhoff microplate model with surface effects by using simplified strain gradient elasticity theory and surface elasticity theory. This new model is able to capture size-dependent behaviors and surface effects. The most noticeable difference of the proposed model from the existing plate models about microplates is that not only strain gradient and surface stress are taken into account, but also the surface-induced internal residual stresses are considered. Based on whether the plates having surface-induced internal residual stress or not, their governing equations have distinct differences. An extended Kantorovich method is employed to provide approximate closed-form solution for the rectangular microplate with simply supported boundary conditions. For the microplate with biaxial initial residual surface stress, the numerical results reflect that when the simply supported microplates do not have surface-induced internal residual stresses, internal length scale and biaxial surface residual stress have significant influence on the static bending of the microplates. However, when the simply supported microplates have nonzero surface-induced internal residual stresses, the effects of internal length scale and biaxial surface residual stress become very weak. It indicates that the effect of surface-induced internal residual stresses can counteract most of the effects of internal length scale and surface residual stress. This work provides a more general model for the analysis of microplate problems. [ABSTRACT FROM AUTHOR]

In this paper we study the semi-local convergence of the Inverse Weierstrass iterative method for simultaneous approximation of polynomial zeros. We present a semi-local convergence theorem with computationally verifiable initial conditions. Several numerical examples are provided to show the practical applications of the presented theoretical results. [ABSTRACT FROM AUTHOR]

This article includes semilocal convergence of Homier type method in Banach spaces by considering recurrence relation technique under w-continuity condition which is more generalized than Lipschitz continuity condition. The existence and uniqueness theorem has been formed along with an error bound. A numerical example has been presented to vindicate the theoretical results. [ABSTRACT FROM AUTHOR]

In this paper we obtain estimations of the errors in approximation by positive linear operators which fix certain functions. We use both the first and the second order classical moduli of smoothness and a generalized modulus of continuity of order two. Some applications involving Bernstein type operators, Kantorovich type operators and genuine Bernstein-Durrmeyer type operators are presented. [ABSTRACT FROM AUTHOR]

The Graetz–Nusselt problem for Bingham liquid is solved by the approximate Kantorovich method. The solution is obtained in the form of a Fourier–Bessel series. The internal heat release at the expense of viscous dissipation is taken into account. Expressions for the temperature and local Nusselt number are obtained. The results of numeric analysis of the solution are presented. [ABSTRACT FROM AUTHOR]