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The Generalized Finite Element Method (GFEM) is developed from the Partition of the Unity Method (PUM), which expands the standard finite element space by using non-polynomial function spaces called the enrichment spaces. GFEM has been successfully applied to various problems, but it still has some drawbacks. It lacks robustness in adjusting meshes when solving interface problems, and the condition number of the stiffness matrix will increase dramatically when the interface is close to the mesh boundary. This phenomenon can lead to ill-conditioned linear equations. A stable GFEM called SGFEM is proposed for the Stokes interface problem in this paper, which modifies the enrichment space. The SGFEM space of the velocity is divided into a basic part S F E M and an enrichment part S E N R ∗. The discretization of space (S F E M × Q h) uses Q 1 − Q 0 element or the Taylor-Hood element for the study. S E N R ∗ uses different interpolation functions. Numerical studies show that SGFEM has the optimal convergence order of the error and robustness. The growth rate of the scaled condition number of the stiffness matrix is the same as that of a standard FEM. • The stable GFEM is proposed for the Stokes interface problems. • Error convergence order same as FEM solving Stokes equation without interface. • Scaled condition number of stiffness matrix is the same growth order as standard FEM. [ABSTRACT FROM AUTHOR]

In the engineering design of modern wing panels and their optimal control system, the first step is to find and predict their dynamic characteristics. These characteristics are dependent on material properties, panel geometry, and boundary conditions. In addition, possible fracture existence, imperfections and environmental conditions may unpredictably change the frequency and modes shape of the panel and then the parameters of a control system. In the present paper, the effect of surface fracture depth and orientation on frequency and vibration mode switching of perovskite solar cells-based aircraft wing panels is investigated for the first time. The PSC (perovskite solar cell) panel is made of a polymer substrate reinforced with even-distributed graphene platelets (GPLs) and thin solar layers to enhance the overall electromechanical performance of the whole structure. Equations of motion of the panel are obtained by the Hamilton approach based on the properties of Reddy's third-order shear deformation theory and linear components of the Green-Lagrange strain tensor. The novel aspect is a line-spring model of surface crack by Cartmell et al. adapted to the model of the PSC panel under consideration. The discretization of the established equations of motion for the cracked panel is carried out by the element-free IMLS-Ritz method with mesh-free features. Validation of the obtained solutions is carried out by comparisons with results from previous studies for cracked laminated plates. First, material and geometric properties effect on vibration characteristics of the PSC structure was investigated. Next, partial surface crack depth and orientation angle are examined to show changes in frequency and vibration modes shape of the panel. It is shown, that there is a mode swapping between the second and third modes when transitioning from an intact panel to a cracked one. As the crack length increases, higher-order mode variations become more noticeable, potentially leading to their swapping or the excitation of new ones. Additionally, the findings demonstrate that the surface crack orientation angle significantly impacts the higher-order modes of the panel. When the orientation angle is 90°, the first six modes align with those of the intact plate. Mode shapes and frequency shifts are significant for crack orientation angles between 0° and 90°. [ABSTRACT FROM AUTHOR]

This paper presents the study of the plane strain problem of an infinite isotropic elastic medium subjected to far-field load and containing multiple Gurtin–Murdoch material surfaces located along straight segments. Each material segment represents a membrane of vanishing thickness characterized by its own elastic stiffness and residual surface tension. The governing equations, the jump conditions, and the surface tip conditions are reviewed. The displacements in the matrix are sought as the sum of complex variable single-layer elastic potentials whose densities are equal to the jumps in complex tractions across the segments. The densities are found by solving the system of coupled hypersingular boundary integral equations. The approximations by a series of Chebyshev's polynomials of the second kind are used with the square root weight functions chosen to satisfy the tip conditions automatically. Numerical examples are presented to illustrate the influence of dimensionless parameters and to study the effects of interactions. [ABSTRACT FROM AUTHOR]

This paper extends the recently proposed variant of meshless local Petrov Galerkin (MLPG) method, i.e., MLPG7, for solving time dependent PDEs. As test function, the method uses a novel modification of fundamental solution of Laplace operator that not only the test function itself but also its derivative vanish on boundary of local subdomains. Therefore, more stable local integral equations are obtained by replacing a boundary integral of derivatives of field variables with a domain integral of field variables itself. MLPG7 formulation converts the nonlinear governing equation into a system of first order ordinary differential equations (ODEs). For this system of ODEs, a theoretical stability analysis is carried out for the case of regularly spaced nodal points. Stability and convergence conditions for fully discretized equations by Euler (forward and backward) and Crank–Nicolson methods are investigated. The nonlinear term is treated iteratively. A numerical study investigates the convergence and stability of the method. A comparison is also done with some well-known methods and the results reveal the excellence of MLPG7. The convergence of the iterative approach is numerically verified by another illustrative test example. As nonlinear test problems, Allen–Cahn equations are studied. [ABSTRACT FROM AUTHOR]

In this paper, a linear smoothing scheme over eight-node Reissner-Mindlin plate element under the framework of the CS-FEM is employed to buckling analysis of laminated composite plates based on the first-order shear deformation theory. The modified stain matrix is computed by the divergence theorem between the nodal shape functions and their derivatives using Taylor's expansion. Isoparametric mapping and the computation of interior derivatives of shape function is not required in the proposed method, furthermore, all the computation are based on the global Cartesian coordinates. Some numerical examples are given at the end to demonstrate that the present method has good performance to alleviate the shear-locking phenomenon and improve the quality of the solutions with distorted meshes. [ABSTRACT FROM AUTHOR]

This paper aims to analyze a novel localized radial basis function method known as the 'direct RBF-based partition of unity method' for solving nonlinear fractional parabolic equations. In the proposed method, the weight functions are not operated on by the differential operators, resulting in a decrease in computational cost and algorithmic complexity. Another advantage of the direct RBF-based partition of unity method compared to the standard RBF-based partition of unity method is that in the direct method, it is possible to use discontinuous weights, which is not applicable in the standard method. Furthermore, we conduct a convergence analysis of the method and derive an error bound for the local approximation. This error bound is determined by considering conditions on the eigenvalues of the Laplacian operator matrix. To ensure the accuracy and reliability of our proposed approach, we conducted a numerical simulation and provided two numerical examples for comparison. We evaluated the consistency between the theoretical and numerical results. Our comparison between the standard RBF-based partition of unity and our proposed framework shows that the direct method is more efficient computationally, but both versions have relatively the same accuracy. [ABSTRACT FROM AUTHOR]

Simulating groundwater flows in heterogeneous aquifers is one of the most widely studied problems. The heterogeneity is modeled through random hydraulic conductivity fields log-normally distributed. In this paper, we aim to generate the realization of the log-normal hydraulic conductivity by summing up a finite number of random periodic modes with the Kraichnan algorithm. To address Neumann conditions, we propose a change of variables that results in Dirichlet boundary conditions along all boundaries of the computational domain. Then, we suggest using the radial basis function partition of unity method (RBFPUM) and RBFPUM with QR factorization (RBFPUM-QR) to analyze groundwater flow in the heterogeneous porous medium. We evaluate the performance of the proposed methods by increasing the number of random modes and the variance of the log-hydraulic conductivity fields using Gaussian and exponential correlation. In addition, we verify the effectiveness of the presented approaches in one and two dimensions using the method of manufactured solutions. Furthermore, we provide statistical inferences through Monte Carlo simulations. Numerical results demonstrate that RBFPUM-QR performs well in handling higher heterogeneity in both single realizations and Monte Carlo simulations. • A meshless numerical approach based on RBFs is proposed to solve Darcy's equation with a random hydraulic conductivity. • We use the Kraichnan algorithm to generate the log-normal hydraulic conductivity which is used to describe the heterogeneity. • The proposed approach avoids technical issues encountered in the case of an exponential correlation of the hydraulic conductivity. • The proposed numerical approach performs well to cope with higher heterogeneity. [ABSTRACT FROM AUTHOR]

• Two-phase flow regime identification using ML with enhanced interpretability. • Evaluation of various dimensionality reduction techniques to optimize feature representation. • Kernel Fisher discriminant analysis is the best performing dimensionality reduction technique. • AutoML can impartially select the most effective ML classifier. • Our workflow can develop a virtual flow regime map to automate flow regime recognition. Accurate identification of flow regimes is paramount in several industries, especially in chemical and hydrocarbon sectors. This paper describes a comprehensive data-driven workflow for flow regime identification. The workflow encompasses: i) the collection of dynamic pressure signals using an experimentally verified numerical two-phase flow model for three different flow regimes: stratified, slug and annular flow, ii) feature extraction from pressure signals using Discrete Wavelet Transformation (DWT), iii) Evaluation and testing of 12 different Dimensionality Reduction (DR) techniques, iv) the application of an AutoML framework for automated Machine Learning classifier selection among K-Nearest Neighbors, Artificial Neural Networks, Support Vector Machines, Gradient Boosting, Random Forest, and Logistic Regression, with hyper-parameter tuning. Kernel Fisher Discriminant Analysis (KFDA) is the best DR technique, exhibiting superior goodness of clustering, while KNN proved to be the top classifier with an accuracy of 92.5 % and excellent repeatability. The combination of DWT, KFDA and KNN was used to produce a virtual flow regime map. The proposed workflow represents a significant step forward in automating flow regime identification and enhancing the interpretability of ML classifiers, allowing its application to opaque pipes fitted with pressure sensors for achieving flow assurance and automatic monitoring of two-phase flow in various process industries. [ABSTRACT FROM AUTHOR]

Acoustic simulation in frequency-domain is related to solving Helmholtz equations, which is still highly challenging at high frequency with complex geometries. In this paper, a learning based numerical method (LbNM) is proposed for general boundary value problems of Helmholtz equation. By using Tikhonov regularization, the solution operator is stably learned from various data solutions especially fundamental solutions. The reconstructed solution operator is reusable and enables fast solution updating with new boundary inputs. The present method is featured by strong interpretability and generalizability. Numerical results demonstrate advantages in precision and efficiency. It can be applied to various engineering problems such as cabin acoustic simulations flexibly. [ABSTRACT FROM AUTHOR]

Recently, several numerical methods based on the radial basis functions have been applied to solving differential equations. Many researchers have employed the radial basis functions collocation technique and its improvements to get more accurate and efficient numerical solutions. The Schrödinger equations have several applications in the optic and laser. Accordingly, several numerical procedures have been proposed. In this paper, we present a new numerical algorithm based on the time-split approach, and rational radial basis functions collocation method. First, a second-order time-split approach is used to discretize the time variable. In this stage, the linear and nonlinear terms are separated. The linear term is solved by using a collocation technique based on the rational approach, the radial basis functions, and the partition of unity. The nonlinear term is does not have a differential operator thus we will only insert approximate solutions into it. Finally, several numerical examples have been reported to show the stability, convergence, and accuracy of the proposed numerical algorithm. [ABSTRACT FROM AUTHOR]

This paper presents an adaptive coupling approach between meshless Smoothed Point Interpolation Methods (SPIMs) and the Finite Element Method (FEM) for the physically nonlinear analysis of quasi-brittle media. The nonlinear behaviour is represented by scalar damage and smeared-crack models. In the proposed adaptive coupling approach, the domain is initially discretised with a relatively coarse FEM mesh. During the solution process, FEM patches are replaced by meshless discretisations according to certain selection criteria. Refinement is also considered. • A new meshless-FEM adapting coupling is proposed. • The Kronecker delta property of smoothed point interpolation methods allows a straightforward coupling. • After each mesh modification the iterative process is repeated to guarantee the equilibrium of the model. • The irreversibility of damage is guaranteed throughout the adaptive process. [ABSTRACT FROM AUTHOR]

In this paper, a novel nodal integration technique for meshfree methods is introduced. This technique is based on the idea of the Cartesian transformation method, which prevents the presence of background cells during the numerical integration process. The Gauss–Lobatto quadrature is used instead of the conventional Gaussian quadrature to create the integration points so that the integration points coincide with the field nodes. This development gave rise to an innovative nodal integration method that eliminates the necessity for generating integration cells, thus advancing further toward a genuinely meshfree approach. The meshfree method chosen in this study is the radial point interpolation method due to the satisfaction of the Kronecker delta property. The curved shell is formulated by the first-order shear deformation theory (FSDT). The static and free vibration analyses of different geometry curved shells are conducted. Through several numerical examples, the accuracy and efficiency of the proposed technique are demonstrated and discussed. It is found that the current technique has better performance than the existing numerical integration techniques used in meshfree method. Furthermore, this new nodal integration technique also shows the ability to mitigate the shear-locking phenomenon when analyzing the problem of thin shells using the FSDT formulation. • A novel nodal integration based on the Cartesian transformation method is introduced. • The proposed nodal integration can mitigate the shear-locking phenomenon in thin shells. • A new discretization process is introduced. • High accuracy and computational efficiency are observed. [ABSTRACT FROM AUTHOR]

Piezoelectric materials find widespread applications in high-precision actuators and sensors. However, the traditional finite element method falls short in meeting the simulation needs of piezoelectric structures due to complexities in mesh generation and precision requirements for accurate simulations. This paper focuses on adapting and generalizing the meshless modeling technique based on the radial point interpolation method to address three-dimensional simulations of piezoelectric structures and accurately model positive and inverse piezoelectric behaviors. The piezoelectric body is discretized into point cloud data, and displacement and potential are approximated by interpolating values through the radial point interpolation method. Subsequently, a 3D meshless model for general piezoelectric structures is formulated based on the principle of minimum potential energy. The influences of key factors on interpolation accuracy and computational cost were discussed. Numerical examples, featuring a 3D piezoelectric bimorph beam as actuators and sensors, validate the efficacy of this extended approach. Results affirm that the three-dimensional meshless method, rooted in the radial point interpolation technique, achieves superior accuracy in predicting piezoelectric behavior compared to the traditional finite element method. This adaptation not only enhances the applicability of meshless modeling but also represents a significant advancement in accurately simulating complex three-dimensional piezoelectric structures. • Multi-field coupling meshless modeling for 3D piezoelectric structures based on RPIM. • Valuable insights are provided for selecting meshless model parameters. • Uniformly distributed nodes helps improve the numerical calculation accuracy. [ABSTRACT FROM AUTHOR]

• A new singular boundary element is constructed to model the singular stress fields. • The proposed method can obtain good results without much computational cost. • The first four amplitude coefficients can be determined without post processing. The stress singularity occurs near a V-notch. The conventional boundary element method can only approach to the exact results with gradually refined mesh. In this paper, by introducing the Williams asymptotic expansion, a novel semi-analytical element is proposed. The new element models the geometry and the known physical fields with linear interpolation, while the unknown physical fields will be simulated by the asymptotic expansion. The new element introduces six unknowns: two displacement components at the notch tip and four amplitude coefficients in the asymptotic expansion. By collocating two semi-analytical elements on both sides of the notch, the unknown displacement fields can be depicted with the asymptotic expansion, while the remaining part can be modeled by means of the conventional boundary element. The direct integral method and singular integral separation technique are adopted for the different kinds of singular integrals in the boundary integral equations. After the system of equations are solved, not only the unknown boundary physical quantities, but also the first four amplitude coefficients in the asymptotic expansion can be determined at the same time. The stress intensity factors can be determined from the amplitude coefficients in the asymptotic expansions without any post processing. [ABSTRACT FROM AUTHOR]

In this paper, an updated Lagrangian method based on peridynamic differential operator (PDDO) is proposed to study the natural convection and heat transfer of magnetic nanofluids under the fluid and magnetic coupled filed. The governing Navier-Stokes equations considering the effects of Lorentz force in Magnetohydrodynamics (MHD) and the magnetization intensity in Ferrohydrodynamics (FHD) are reformulated in their integral form using a second-order PDDO. Non-local integration of the discretized material points avoids the discontinuous of the field variables, thus, lead to a more flexible way to simulate natural convection of pure fluid and nanofluid under multi-physical fields for stable and convergent solutions. The efficiency, accuracy and capability of the present method has been demonstrated by several numerical examples. Subsequently, the effects of the Rayleigh number, Hartmann number, and Magnetic number on the convective behavior of magnetic nanofluids are investigated and discussed. [ABSTRACT FROM AUTHOR]

Fractional differential equations (FDEs) offer numerous capabilities for modeling unusual phenomena. So, the study of these models is essential. This paper proposes an efficient meshless technique for obtaining the numerical solution of a space fractional diffusion model with Caputo derivative type. Typically, in a meshless processes based on moving Kriging (MK) interpolation, the MK technique is used to calculate the shape functions and their derivatives with positive integer order against space variables to discretize the governing equation in space variables. However, in this study, we employ the Taylor series of MK interpolation's correlation function to enhance the shape function derivatives for arbitrary order 0. 5 < α < 2. Moreover, the finite difference technique is employed to discretize the model in the time dimension and its characterized by unconditional stability and a rate of convergence of O (τ). This approach converts the primary problem into a system of linear algebraic equations. Finally, we present several examples in one and two dimensions and compare the results with other well-known schemes to show the capability and accuracy of this approach. [ABSTRACT FROM AUTHOR]

This paper presents a polynomial layer-wise model in the framework of Carrera's Unified Formulation for the bending analysis of a magneto-electric shells with variable radii of curvature. A parametric surface is used to model the middle surface of the shell. Lame Parameters and Radius of Curvature are calculated by using Differential Geometry. The mechanical displacement, along with the electric and magnetic scalar potential functions, are expressed and modeled using Chebyshev polynomials of the Second Kind. The shells are exposed to different mechanical, electrical and magnetic loads. The Principle of Virtual Displacement is employed for obtaining the governing equations which are discretized by Chebyshev-Gauss-Lobatto grid distribution and solved in semi-analytical manner by the so-called Differential Quadrature Method (DQM). The basis function selected is the Lagrange polynomial. The DQM is employed for its straightforwardness in tackling complex yet regular shell structures under various multiphysical loads. A simple stress recovery technique based on 3D equilibrium equations is introduced to obtain the out-of-plane shear and normal stresses, transverse electric, and magnetic induction. Close-to-3D solutions have been achieved for classical shell structures. Furthermore, benchmark solutions for complex smart shells featuring variable radii of curvature, such as parabolic and cycloidal shells, are introduced. [ABSTRACT FROM AUTHOR]

In this paper, we present a novel approach for discretizing the derivative Nonlinear Schrödinger (DNLS) equation in an integrable manner. Our proposed method involves discretizing the time variable, resulting in a discrete system that converges to the DNLS equation in a natural limit. Furthermore, the discrete system retains the same set of infinitely conserved quantities as the original DNLS equation. To demonstrate the effectiveness of our proposed method, we designed a numerical simulation scheme using the Fourier Pseudo-spectral Method to discretize the spatial variable. The numerical results confirm that our new discrete integrable scheme can accurately preserve the conserved quantities of the DNLS equation. [ABSTRACT FROM AUTHOR]

In this paper, we present a novel mathematical model of fractional order for epidemic dynamics in plants, this fractional order model (FOM) in the sense of Caputo derivatives governing fractional differential equations (FDEs), introducing modified parameters to coincide both sides dimensions for FOM that means enhancing its accuracy in representing the real-life scenarios to control the spread of the epidemic. Our contributions include a comprehensive qualitative analysis of the proposed FOM, such as proving the existence and uniqueness of the projected solution by transforming our problem into a fixed point problem and applying a Banach contraction principle and Schauder's fixed-point theorem. The positivity and boundedness of this solution are also demonstrated. We precisely evaluate all equilibrium points (EPs), examine their local and global stability by using Routh–Hurwitz conditions and fractional LaSalle's invariance principle (LIP), respectively and shed light on the dynamic behavior of the FOM. Furthermore, we discuss the sensitivity analysis for parameters of the control reproduction number (CRN), with insightful plots to illustrate the model's response to parameter changes (replanting and roguing parameters). From this foundation, we formulate a fractional optimal control problem (FOCP) by incorporating preventive u 1 and curative u 2 controls to effectively eliminate the propagation of the epidemic in plants. The fractional necessary optimality conditions (FNOCs) are derived attribution to Pontryagin's maximum principle (PMP). We employ the forward–backward sweep method (FBSM) based on fractional Euler method (FEM) in order to facilitate control implementation and show the different suggested strategies. By enhancing this method, optimal control efforts are projected onto the FOM and yield three various strategies to impact the epidemic's trajectory. For each strategy, we explain how the presence (i.e. with control cases) and absence (without control cases) of the proposed controls affected the susceptible, protected and infected plants. Some attractive figures with various values of the weight factor and fractional order α for the CRN are also presented, allowing a visual assessment of their impact on the dynamics of the epidemic in plants. In addition, we calculate the objective function for controlled and uncontrolled cases to provide a quantitative measure of its effectiveness in containing the outbreak in the plants. The results gained from our analyses and simulations provide valuable guidance for the management and elimination of epidemics in plants by offering various scenarios to implement the proposed controls. [ABSTRACT FROM AUTHOR]

In this paper, considering the spatial propagation behavior of Internet rumor, we propose a new reaction–diffusion rumor propagation model with time delay in social networks. Based on this, we discuss the stability conditions at the equilibrium point and analyze the Turing instability for the approximation propagation system. Simultaneously, the amplitude equations for the approximation system are derived by multiscale analysis. We use numerical simulation to compare the effect of parameter size, period, and time delay on the patterns. Moreover, we compare the different reactivity of the system in the W S networks and the B A networks. Finally, we plot the corresponding amplitude equations patterns by controlling the parameters and verify the correctness of the theoretical results. [ABSTRACT FROM AUTHOR]

This paper aims to study an HIV model with age structure and latently in a spatially homogeneous environment. By applying the fixed point theorem, we obtain the existence of the global solution and the global attractor for the model. We also identify the explicit formula of the basic reproduction number by the mean of the Laplace transformation, and confirm that this number predicts whether the infection occurs or not. Through analyzing the root distribution of the characteristic equation, the local stability of the equilibrium is obtained. By appealing to the appropriate Lyapunov functionals, we further study the global stability of the equilibrium. Numerical simulations also validate our theoretical results. [ABSTRACT FROM AUTHOR]

In this paper, we propose a multi-domain spectral collocation method for partial differential equations on two-dimensional unbounded domains. Some approximation results on the composite generalized Laguerre-Legendre interpolation and quasi-orthogonal projecting are established, respectively. These results play a significant role in related spectral collocation method. As an application, a multi-domain spectral collocation scheme is provided for the Fokker–Planck equation with absorption or non-homogeneous boundary conditions. The convergence of the proposed algorithm is performed. An efficient implementation is presented. Numerical experiments demonstrate the effectiveness and high accuracy of the algorithm. [ABSTRACT FROM AUTHOR]

Air pollution has a serious impact on the spread of respiratory infectious diseases. However, the concentration of air pollution is not the same in different places. In order to characterize the spatial heterogeneity and the diffusion of pollutants, this paper proposes a spatial heterogeneity model, where the transmission rate is closely related to air pollution. The dynamics are established in terms of the basic reproduction number R 0 : if R 0 ≤ 1 , the disease-free equilibrium is globally asymptotically stable; if R 0 > 1 , there exists at least one endemic equilibrium and the disease is uniformly persistent. Our findings suggest that spatial heterogeneity, increased pollutants emission, and limited movement of infected individuals will all contribute to the spread of the respiratory infectious diseases. Subsequently, the corresponding strategies are introduced into the model to control the prevalence of the diseases. Under the minimization of the objective function, the necessary condition for the optimal control is acquired with the help of Pontryagin's maximum principle. Numerical simulations are applied to validate the theoretical results of threshold dynamics and to demonstrate the influences of inflow and clearance rates of pollutants on threshold dynamics and optimal control. [ABSTRACT FROM AUTHOR]

In this paper, the cubic–quintic nonlinear Helmholtz equation which enables a pulse propagates in a planar waveguide with Kerr-like and quintic nonlinearities is studied. By noticing that the system is a non-integrable one, we could to get the diverse of solitary wave solutions by using standard tan (ϕ / 2) -expansion technique and the exp (− Ω (η)) -Expansion scheme. In particular, four forms of function solution including soliton, bright soliton, singular soliton, periodic wave solutions are investigated. To achieve this, an illustrative example of the Helmholtz equation is provided to demonstrate the feasibility and reliability of the procedure used in this study. The effect of the free parameters on the behavior of acquired figures to some solutions of two nonlinear exact cases was also analyzed due to the nature of nonlinearities. The dynamic properties of the obtained results are shown and analyzed by some density, two and three-dimensional images. We believe that our results will be a way for future research generating optical memories based on the non-paraxial solitons. [ABSTRACT FROM AUTHOR]

In this paper, the cooperative control of fractional-order multi-agent systems with time delay and unknown control direction is studied by combining frequency distributed model and event-triggered mechanism. The Nussbaum function is employed to address the unmeasured control direction. Then, through the event-triggered mechanism, a corresponding controller is designed, which can guarantee that all signals are bounded while saving resources. The proposed theme is convenient for analyzing the stability of fractional-order systems based on a transformation of frequency distributed model. In particular, in order to enhance the universality of the proposed scheme in the fractional-order system stability analysis, a generalized fractional-order generalized lemma with respect to the Nussbaum function is given. Finally, the availability of the proposed method is confirmed by a simulation example. [ABSTRACT FROM AUTHOR]

In this paper, we propose a variance reduction approach for Markov chains based on additive control variates and the minimization of an appropriate estimate for the asymptotic variance. We focus on the particular case when control variates are represented as deep neural networks. We derive the optimal convergence rate of the asymptotic variance under various ergodicity assumptions on the underlying Markov chain. The proposed approach relies upon recent results on the stochastic errors of variance reduction algorithms and function approximation theory. [ABSTRACT FROM AUTHOR]

In this paper, a weak Galerkin finite element method is implemented to solve the one-dimensional singularly perturbed reaction–diffusion equation. This weak Galerkin finite element scheme uses piecewise polynomial with degree k ≥ 1 in the interior part of each element and piecewise constant function at the nodes of each element. The existence and uniqueness of the weak Galerkin finite element solution are proved. Based on the interpolation operator and the corresponding approximation properties, an ɛ − uniform error bound of O (N − (k + 1) + ɛ (N − 1 ln N) k) in the energy-like norm is investigated rigorously. Furthermore, an ɛ − uniform error bound of O (N − (k + 1 / 2) + (N − 1 ln N) k) in the balanced norm is established by the weighted local L 2 projection and its corresponding approximation properties. Finally, numerical experiments validate the theoretical results. Moreover, numerical results show that this weak Galerkin finite element solution achieves the convergence rate of O ( (N − 1 ln N) k + 1) in the L 2 norm and the discrete L ∞ norm uniformly with respect to the singular perturbation parameter ɛ. [ABSTRACT FROM AUTHOR]

In this paper, a modified hybrid B-spline approximation based on spatial regulator tensor network (MHBA-SRTN) is proposed for solving the application and the feasibility problems of fractional calculus-based Burger equation. The main innovation points include: (1) a dual singular kernel based fractional derivative operator is employed on the Burger's equation with external force term to analyze the solutions and properties under perturbation condition for further realism simulation; (2) a modified hybrid B-spline basis function and its corresponding topological tensor network structure are further proposed to solve the provided Burger's equation with more favorable approximation accuracy and solution uniqueness; (3) the tensor network transformed the hybrid B-spline construction process into the tensor solving with the introduced spatially regulated tensor network decomposition for a more robust solution procedure; (4) an acceleration approach for decomposition process is introduced to significantly reduce its computational and storage complexity. Extensive experiments are also carried out to verify the performance of MHBA-SRTN. [ABSTRACT FROM AUTHOR]

This paper proposes a novel L1-based predictor–corrector method for the fractional differential equations involving generalized-Caputo type derivative. A decomposition scheme is used to obtain the three-point predictor–corrector formula. The error and stability of the proposed method are given in detail. A computer virus and a five-dimensional Hopfield neural network models are solved using the proposed approach. [ABSTRACT FROM AUTHOR]

This paper is a computational bifurcation analysis of a non-linear partial differential equation (PDE) characterizing equilibrium configurations in Micro electromechanical Systems (MEMS). MEMS are engineering systems that utilize electrostatic forces to actuate elastic surfaces. The potential equilibrium states of MEMS are described by solutions of a singularly perturbed elliptic nonlinear PDE. We develop a numerical method which couples a finite element approximation with mesh refinement to a pseudo arc-length continuation algorithm to numerically obtain bifurcation diagrams in the physically relevant two dimensional scenario. Several geometries, including a unit disk, square, and annulus, are studied to understand the behavior of the system over a range of domains and parameter regimes. We find that solution multiplicity, and importantly the potential for bistability in the system, depends sensitively on the parameters. In the annulus domain, symmetry breaking bifurcations are located and asymmetric solution branches are tracked. This work significantly extends the envelope for numerical characterization of equilibrium states in microscopic electrostatic contact problems relating to MEMS. [ABSTRACT FROM AUTHOR]

It is well known that the analytical matrices arising from the discretization of distributed parameter systems using the finite element technique are usually symmetric and banded. How to preserve the coefficient matrices of the updated model being of the same band structure is an important yet difficult challenge for model updating in structural dynamics. In this paper, an iterative method for updating mass, damping and stiffness matrices simultaneously based on partial modal measured data is provided. By the method, the optimal updated matrices can be obtained within finite iteration steps by choosing a special kind of initial matrix triplet. The proposed approach not only preserves the physical connectivity of the original model, but also the updated model reproduces the measured modal data, which can be utilized for various finite element model updating problems. Numerical examples confirm the effectiveness of the introduced method. • An iterative updating method for damped structure systems is established. • The proposed approach preserves the physical connectivity of the original model. • The updated model can reproduce the measured modal data. • The deviation between the updated model and the original model is minimal. [ABSTRACT FROM AUTHOR]

In this paper, a predator–prey model with double Allee effect and self-diffusion terms is considered. The stability of the coexisting equilibrium is studied by analyzing the eigenvalue spectrum, and the spatial Turing patterns are investigated by using the method of multiple scale analysis. The amplitude equation is obtained, which shows that the system supports patterns like spots, stripes, hexagonal patterns and mixed-mode patterns. The influence of double Allee effect in system is discussed by the comparison of bifurcation diagrams. Finally, the numerical simulations have verified the correctness of above theoretical analysis and tell how the double Allee effect plays an important role in biological universe. [ABSTRACT FROM AUTHOR]

Iterative approaches have been established to be fundamental for the creation of fractals. This paper introduces an approach to visualize Julia and Mandelbrot sets for a complex function of the form Q (z) = z p + log c t for all z ∈ ℂ , where p ∈ N ∖ { 1 } , t ∈ [ 1 , ∞) , c ∈ ℂ ∖ { 0 } , using a four-step iteration scheme extended with s -convexity. The study introduces an escape criteria for generating Julia and Mandelbrot sets using a four-step iterative method. It investigates how changes in the iteration parameters influence the shape and color of the resulting Julia and Mandelbrot sets. This approach can generate a wide range of captivating fractals and analyze them through numerical experiments. [ABSTRACT FROM AUTHOR]

In this paper, we deal with the analytical and numerical solutions of the Gardner–Kawahara (G-K) equation, known as the extended Korteweg–de Vries equation, which describes solitary-wave propagation in media and occurs in the notion in plasmas and in notion of shallow water waves with surface tension and notion of magneto-acoustic waves. This study is split into two primary parts: in the first part, (G ′ / G) and exp (− ϕ (ξ)) expansion methods are employed to obtain some novel exact solutions of the G-K equation. Furthermore, some hypotheses are used to get different exact traveling wave solutions (including Hyperbolic,Trigonometric and Rational solutions). (G ′ / G) -expansion method is a comprehensible and noncomplex technique used to procure traveling wave solutions of partial differential equations (PDEs). By using the (G ′ / G) -expansion technique, solution of PDEs can be indicated as a series and the coefficients of the series can be procured by solving a set of algebraic equations. This method is highly effective, requiring only the solution of a few algebraic equations to create the traveling wave solution (Ali and Tag-eldin, 2023 [3]). The exp (− ϕ (ξ)) - expansion method is used as the first time to examine the wave solution of a nonlinear dynamical system in a new double-Chain model of DNA and a diffusive predator–prey system. The method also has been used for many other nonlinear evolution equations (Abdelrahman et al., 2015 [1]). In the second part, finite element method is dedicated to deriving numerical solutions of the equation. In the same part, based on septic B-spline approximation, a collocation method has been offered and applied for numerical solutions of G-K equation conceiving different parameter values of test problem. To discretize time and space derivatives, forward finite difference and Crank–Nicolson approximations have been used, respectively. Unconditional stability is proved by using Von-Neumann method. B-spline functions have some reliable properties like smoothness, local support and ability of handling local phenomena, which make them proper to solve linear and nonlinear partial differential equations easily and precisely. Collocation technique has two great superiorities: constituting method does not involve integrations and the resulting matrix system is banded with small band width. For this reason, B-splines when associate with the collocation technique ensure a simple solution process of linear and nonlinear PDEs. An example is successfully solved by calculating the error norms L 2 and L ∞ for illustrating the proficiency and reliability of the method and highlighted the significance of this work. Also to reflect the efficiency of this method for solving the nonlinear equation, the results are depicted both graphically and in tabular form. Calculated results are found to be in good agreement with the exact solutions and other techniques given in the literature. [ABSTRACT FROM AUTHOR]

In this paper, we introduce a generalized rational map to develop a theory of escape criterion via the SP-iteration process equipped with s -convexity. Furthermore, we develop algorithms for the exploration of unique kinds of Mandelbrot as well as Julia sets. We demonstrate graphically the change in colour, size, and shape of images with the change in values of the considered iteration's parameters. The new fractals thus obtained are visually very pleasing and attractive. Most of these newly generated fractals resemble natural objects around us. Moreover, we numerically study the dependence between the iteration's parameters and the set size. The experiments show that this dependency is non-linear. We believe that the obtained conclusions will motivate researchers who are interested in fractal geometry. [ABSTRACT FROM AUTHOR]

In the present paper, a previously introduced numerical method, GDDR (Generalized Differential Dynamic Relaxation), is developed to analyze ratcheting behavior of moderately thick rectangular plates. The validity of the method is verified by comparison with literature data and finite element method results. Classical Plate Theory (CPT) and First-order Shear Deformation Theory (FSDT) are utilized to define the displacement field of plates. Incremental strain equations are derived according to von Karman relations. Prandtl-Reuss flow rule and von Mises yield criterion and are used to express the behavior of the plates in the plastic region. The Armstrong-Frederick nonlinear kinematic hardening model is incorporated for elastic-plastic analyses. Finally the results of ratcheting analysis obtained by GDDR method based on two mentioned theories are compared. Results showed that higher amounts of strains are predicted when FSDT theory is employed. Moreover, the area of hysteresis loops obtained by FSDT analysis are larger than those of CPT analysis, meaning that FSDT approach predicts more energy dissipation during each cycle of loading compared with CPT approach. Finally, comparing the results for different ratios of thickness to width shows that GDDR method can predict more accurate results, especially for higher ratios, when it utilizes FSDT theory. [ABSTRACT FROM AUTHOR]

• The polyhedral formula of the Cosserat theory has been derived. • The precision of the proposed method can approach that of quadratic elements. • The meshing performance is enhanced benefiting efficient octree techniques. The Cosserat continuum offers high accuracy in micro-structure analysis and stress concentration simulation due to its consideration of mechanical factors such as coupled stress and internal length scale. However, existing methods are mainly developed based on the isoparametric conventional continuum framework, with relatively simple element shapes and weak adaptability to complex geometries. Therefore, this paper proposes an improved Cosserat continuum theory based on the three-dimensional (3D) polyhedral scale boundary finite element method (PSBFEM). The main work is as follows: (1) The Cosserat continuum theory formula is deduced within the SBFEM framework by integrating a polygon mean-value shape function; (2) The analytical accuracy of the proposed method is examined through classical micro-structure test examples; (3) An efficient octree algorithm is introduced to evaluate the comprehensive performance of the proposed method for the analysis of complex structures, and the selection and impact of the internal length scale are discussed. The results indicate that the proposed method incorporates the good performance of SBFEM and the Cosserat continuum, and its accuracy can approach the traditional quadratic Cosserat continuum elements; The stress concentration problem can be effectively addressed, leading to a more accurate representation of structural stress characteristics. Additionally, complex polyhedral elements can be directly calculated, and the element library of traditional Cosserat theory has been greatly expanded. [ABSTRACT FROM AUTHOR]

In the present work, analytical numerical solutions of the static behavior of fully clamped functionally graded (FG) doubly-curved panels are presented. The mechanical model is based on the Carrera Unified Formulation (CUF) under the Equivalent-Single-Layer (ESL) approach. The governing equations, in their strong form, are derived from the Principle of Virtual Displacements (PVD). The main novelty of this work is the integration of CUF and the Boundary-Discontinuous solution methodology. This merging yields a remarkably accurate and robust procedure that can handle different shell theories with arbitrary order of expansion. Besides, a new branch of future interesting applications of this purely analytical procedure is discussed. Several ceramic-metal graded shells with two different distribution-law variations subjected to uniform distributed load (UDL) are studied. Various thickness ratios, radius-to-thickness ratios, and order of expansion of the displacement field are considered. The accuracy of the proposed solution in predicting static displacements and stresses of thick and thin FG panels is evaluated and discussed by comparing it with those of the literature for various case problems. As the proposed approach solution presented in this paper is unique, the numerical results might be useful as benchmarks for validating new FG shell theories. [ABSTRACT FROM AUTHOR]

This paper analyzes nonlinear free vibration of plates that move axially and in partial contact with liquid. The von Kármán nonlinear plate theory is employed in the theoretical model. The fluid is characterized as an ideal fluid and represented using the velocity potential and Bernoulli's equation. The fluid pressure exerted on the plate is equivalent to a virtual additional mass, which is considered as part of the total mass of the structure. Employing the Hamilton's principle to derive the immersed moving plates' governing equations. Afterward, the nonlinear frequencies of plates moving axially and contacting with liquid are evaluated by the Chebyshev collocation method combined with the direct iterative technique. The results indicate that the Chebyshev collocation method exhibits excellent convergence and exceptional accuracy. It has been observed that the immersed moving plates' nonlinear frequencies gradually increase with the decrease of the axial velocity. The increase of immersion level or fluid density reduces nonlinear frequencies, but has little influence on nonlinear to linear frequency ratios. [ABSTRACT FROM AUTHOR]

This paper extends the meshless local exponential basis functions to the analysis of doubly curved laminated and sandwich shallow shells. The method discretizes the shell domain by some nodes at which the degrees of freedom are defined. A specific number of nearby nodes, only based on their distance, are selected as a cloud. Within every cloud, the total solution is set in homogeneous and particular parts. The former is adjusted by exponential basis functions to exactly satisfy the homogeneous governing equation, as in Trefftz approaches. Overlapping of adjacent clouds integrates the solution function over the entire domain. While Trefftz methods usually implement trial and error approaches for eigenvalue problems, we introduce a novel technique for the free vibration solution in one step by treating the frequency including part as a loading term. Three deformation theories, namely CST, FSDT and TSDT will be adopted for the formulation. Various boundary conditions such as simple, clamped and free are applied to the edges. It will be shown that the method benefits from high accuracy and fast convergence rate, not adversely affected by irregularity of the nodal grid. Additional results are also provided for interested researchers. [ABSTRACT FROM AUTHOR]

In this paper, a time-domain local radial basis function collocation method (LRBFCM) combined with the Houbolt method is developed for the anti-plane elastic wave propagation analysis in 2D phononic crystals. The periodic boundary conditions are applied to a unit-cell to simplify the analysis. In the first step, the time-dependent displacement field is computed by using the proposed time-domain LRBFCM. Then, the Fast Fourier Transform (FFT) is applied to transform the displacement field from the time-domain to the frequency-domain, from which the frequency band structures can be further obtained. To enhance the stability of the proposed LRBFCM, the direct method is used to calculate the spatial derivatives arising in both periodicity and interface continuity conditions. The proposed time-domain LRBFCM provides an alternative numerical tool to many frequency-domain methods for the band structure analysis of phononic crystals. [ABSTRACT FROM AUTHOR]

Domain decomposition method (DDM) is an efficient tool for solving multi-domain problems. Many kinds of DDM have been proposed in literature, such as the d -N alternating method and optimized Schwarz method, etc. The main difference between these methods lies in the transmission conditions on the interfaces between subdomains. Most DDMs have at least one relaxation parameter which must be properly chosen to ensure convergence. To avoid prior determination of a constant relaxation parameter, dynamic relaxation parameters are devised based on error analysis. Another, difficulty of DDM is that special preprocessor has to be used to handle cases where interior cross-points exist among the domains. In dual interpolation boundary face method (DiBFM), boundary integral equation (BIE) is collocated on nodes inside elements, therefore naturally avoids the interior cross-points problem. In this paper, we have implemented some main kinds of DDM in the frame of DiBFM and conducted comparative studies accounting for different interface conditions and relaxation parameters, using examples that involve different topological connections between subdomains. [ABSTRACT FROM AUTHOR]

Computation of piled-raft interaction effects based on boundary element method (BEM) achieves high accuracy. However, it's not preferred, due to the impractical requirement for huge computational time. In this paper, a simplified coding technique which is generally applicable to any BEM application is demonstrated for pile-soil-pile interaction problems. The proposed simplified technique is based on parallel computing to optimize computational performance. Three parallel computing approaches have been developed in order to take full advantage of the computer's power. The approaches are based on CPU Multithreading, GPU Multithreading, and a hybrid system that combines both. The approaches have been tested on five machines with different hardware (CPU, GPU, and Memory) configurations to analyze four practical examples with different load cases, geometric details, interaction methods, and cases of interactions. The parallelization approaches performance illustrated high efficiency with large-scale computations and the consumed time has been reduced by 95 % for both the GPU and hybrid system approaches and 90 % reduction for the multithreading on the CPU. The analysis results confirmed the validity of the proposed methods without affecting the accuracy of the numerical results. The strength of the proposed technique is in its simplicity and scalability to be used for any BEM code. [ABSTRACT FROM AUTHOR]

This paper presents a two-dimensional fast multipole boundary element method designed for the efficient analysis of large-scale anisotropic elastic problems subjected to centrifugal loads. The expansions and fast multipole operations employed in this study bear resemblances to those utilized in existing formulations of the fast multipole boundary element method, previously proposed for addressing potential and isotropic elastostatic problems. The anisotropic fundamental solutions adopted here are rooted in the Leknitskii formalism and have been suitably expanded to facilitate the implementation of fast multipole operations. The treatment of problems involving centrifugal loads are taken into account by the utilization of the modified boundary condition method. This approach entails augmenting the boundary condition with a specific solution tailored to the problem. Following the solution of the linear system, the particular solution is subsequently subtracted from both displacements and tractions. Importantly, this procedural step obviates the need for generating additional vectors or matrices within the integral equation. To evaluate the accuracy and efficiency of the proposed method, numerical solutions are juxtaposed against analytical solutions and benchmarked against the conventional boundary element method. The formulated approach exhibits quasi-linear computational complexity, rendering it particularly well-suited for tackling large-scale problems. The key inference drawn from this work is that the proposed formulation outperforms the conventional BEM in terms of effectiveness. It offers reduced computational time and memory utilization, thereby enabling the analysis of larger problems within shorter time. • The fast multipole boundary element method is presented for anisotropic elastic plane problems. • The code and the expansions were obtained based on the FMBEM for potential problems. • The formulation has a quasi-linear complexity, ideal for large scale problems. [ABSTRACT FROM AUTHOR]

The fast multipole method (FMM) based on the plane wave expansion is known to suffer from numerical instability in the low-frequency regime. This paper presents a low-frequency fast multipole boundary element method (LF-FMBEM) for acoustic problems in a subsonic uniform flow. First, a hybrid convected boundary integral formula based on the Burton-Miller method is derived to overcome the non-uniqueness difficulty at fictitious eigenfrequencies. The explicit evaluation of hypersingular integrals in the convected boundary integral formulae is also introduced. Then, the formulae of FMM based on the series expansion for convected BEM are derived to improve the calculation efficiency. The recursive calculation method is derived in the expansion of the derivative of the integrands. Besides, the rotation-coaxial translation-rotation back (RCR) technique is employed to accelerate the multipole translation. The numerical implementation process of the developed algorithm is presented in detail. Several numerical experiments are performed to validate the computational efficiency and accuracy of the developed LF-FMBEM. Results show that the proposed algorithm can achieve large-scale computation of one million degrees of freedom (DOF) on a personal computer, and the computational accuracy is still high when the Mach number reaches 0.95. The non-uniqueness problem for convected acoustics problems is also effectively overcome. [ABSTRACT FROM AUTHOR]

Time-fractional diffusion equation is an important transport dynamical model for simulating fractal time random walk. This article is devoted to investigating the a priori and a posteriori error estimates for this model equation. A Petrov–Galerkin spectral method is revisited in this paper to address our problem, which the generalized Jacobi functions and Fourier-like basis functions are utilized as basis for constructing efficient and accurate space–time spectral approximations. Rigorous proofs are given for the stability of our spectral scheme. And then the convergence of the proposed method is proved by establishing an a priori error estimate. Specifically, an efficient and reliable a posteriori error estimator is introduced, and we derive that the residual-based error indicator provides an upper bound and a lower bound for the numerical error. Finally, several numerical experiments are provided to examine our theoretical claims. • An available Petrov–Galerkin spectral method is proposed to approximate the TFDEs. • The spectral accuracy is confirmed by error analysis, along with numerical tests. • An a posteriori error estimate is derived, and numerical tests support our results. [ABSTRACT FROM AUTHOR]

This paper examines the asymptotic stability of nonlinear fractional-order switched systems (FOSSs) under a mode-dependent event-triggered delayed impulsive mechanism (MDETDIM). The impulses and switched signals are asynchronous. A novel MDETDIM is proposed to determine the impulsive sequence, which can prevent the Zeno phenomenon. Lyapunov-based asymptotic stability conditions for general FOSSs are derived using the proposed MDETDIM. The theoretical results are then applied to a fractional-order Hopfield neural network (FOHNN) with event-based delayed impulses and switching effects. Two examples are provided to demonstrate the effectiveness of our proposed results. [ABSTRACT FROM AUTHOR]

The paper is concerned with three types of cubic splines over a triangulation that are characterized by three degrees of freedom associated with each vertex of the triangulation. The splines differ in computational complexity, polynomial reproduction properties, and smoothness. With the aim to make them a versatile tool for numerical analysis, a unified representation in terms of locally supported basis functions is established. The construction of these functions is based on geometric concepts and is expressed in the Bernstein–Bézier form. They are readily applicable in a range of standard approximation methods, which is demonstrated by a number of numerical experiments. [ABSTRACT FROM AUTHOR]

This paper is concerned with the finite-time synchronization (FTS) issue for stochastic complex networks (SCNs) with/without time-delay under impulsive effects subject to denial of service (DoS) attacks. Firstly, a novel distributed dynamic event-triggered controller (DETC) is designed to realize the FTS for SCNs without delays, and the FTS conditions are addressed in the form of the algebraic inequalities. The Zeno behavior can be excluded for the proposed dynamic event-triggered mechanism (DETM). Secondly, a new feedback controller is designed to achieve the FTS for SCNs with time-varying delay. In addition, by applying Lyapunov stability theory and stochastic analysis technology, the FTS conditions are derived. Finally, two numerical examples are provided to illustrate the feasibility of the designed control strategies and the correctness of the stated theoretical results. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the exponential synchronization problem for multi-link and multi-delayed inertial neural networks with Markov jump (MMDINNMJ) using an aperiodically intermittent adaptive control strategy. Different from most research on inertial neural networks, we take multi-link, multi-delay and Markov jump into account. The obstacle caused by the coexistence of Markov jump and multi-delay is avoided by using the delayed integral method while considering the exponential synchronization of MMDINNMJ. Additionally, under graph theory, Lyapunov stability theory and the developed control scheme, some novel sufficient conditions for synchronization at exponential rate in p th (p > 0) moment of underlying networks are determined, which are strongly related to multi-link topological structure, time delay, and Markov jump. Finally, two examples are given to demonstrate the viability of the theoretical conclusions. [ABSTRACT FROM AUTHOR]