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In this article, we study some anisotropic singular perturbations for a class of linear elliptic problems. A uniform estimates for conforming $ Q_1 $ Q 1 finite element method are derived, and some other results of convergence and regularity for the continuous problem are proved. The arguments are based on the tensorization trick and standard elliptic regularity theory tools. [ABSTRACT FROM AUTHOR]

In this paper, we focus on designing a well‐conditioned Galerkin spectral methods for solving a two‐sided fractional diffusion equations with drift in which the fractional operators are defined neither in Riemann–Liouville nor Caputo sense, and its physical meaning is clear. Based on the image spaces of Riemann–Liouville fractional integral operators on Lp([a, b]) space discussed in our previous work, after a step by step deduction, three kinds of Galerkin spectral formulations are proposed, the final obtained corresponding scheme of which shows to be well‐conditioned—the condition number of the stiff matrix can be reduced from O(N2α) to O(Nα), where N is the degree of the polynomials used in the approximation. Another point is that the obtained schemes can also be applied successfully to approximate fractional Laplacian with generalized homogeneous boundary conditions, whose fractional order α ∈ (0, 2), not only having to be limited to α ∈ (1, 2). Several numerical experiments demonstrate the effectiveness of the derived schemes. Besides, based on the numerical results, we can observe the behavior of mean first exit time, an interesting quantity that can provide us with a further understanding about the mechanism of abnormal diffusion. [ABSTRACT FROM AUTHOR]

In this paper, a high-order direct arbitrary Lagrangian-Eulerian (ALE) discontinuous Galerkin (DG) scheme is developed for compressible fluid flows in two-dimensional (2D) Cartesian and cylindrical coordinates. The scheme in 2D cylindrical coordinates is based on the control volume approach and it can preserve the conservation property for all the conserved variables including mass, momentum and total energy. In this hydrodynamic scheme, a kind of high-order Taylor expansion basis function on the general element is used to construct the interpolation polynomials of the physical variables for the DG discretization. The terms including the material derivatives of the test functions are omitted, which simplifies the scheme significantly. Furthermore, the mesh velocity in the direct ALE framework is obtained by implementing an adaptive mesh movement method with a kind of dimensional-splitting type monitor function. This type of mesh movement method can automatically concentrate the mesh nodes near the regions with large gradients of the variables, which can greatly improve the resolutions of numerical solutions near the specified regions. For removing the numerical oscillations in the simulations, a Hermite Weighted Essential Non-oscillatory (HWENO) reconstruction is employed as a slope limiter. Finally, some test cases are displayed to verify the accuracy and the good performance of our scheme. [ABSTRACT FROM AUTHOR]

In this paper, we study the convergence of adaptive mixed interior penalty discontinuous Galerkin method for H (c u r l) -elliptic problems. We first get the mixed model of H (c u r l) -elliptic problem by introducing a new intermediate variable. Then we discuss the continuous variational problem and discrete variational problem, which based on interior penalty discontinuous Galerkin approximation. Next, we construct the corresponding posteriori error indicator, and prove the contraction of the summation of the energy error and the scaled error indicator. At last, we confirm and illustrate the theoretical result through some numerical experiments. [ABSTRACT FROM AUTHOR]

This article investigates discrete versions of the projection-type and modified projection-type methods for solving Hammerstein integral equations with a weakly singular kernel. The approximating operator utilized is either the orthogonal projection or an interpolatory projection onto a space of piecewise polynomials of degree ≤ r - 1 with respect to a graded partition of [0, 1]. The study reveals that the proposed methods achieve optimal rates of convergence, demonstrating that the numerical quadrature used to estimate the integrals maintains the same rates of convergence as the continuous methods. The theoretical findings are supported by numerical experiments. [ABSTRACT FROM AUTHOR]

In this paper, we construct a reduced order model (ROM) to solve the advection–diffusion-reaction (ADR) equation with variable coefficients. In order to get the desired fidelity solution, we introduce the interior penalty discontinuous Galerkin (IPDG) and implicit–explicit Runge–Kutta (IMEXRK) schemes to construct the full-order model (FOM). The IPDG scheme can achieve higher-order accuracy in spatial discretization. We split the model into the advection-reaction and diffusion terms so as to ensure the stability of the full-discrete scheme. The IMEXRK scheme discrete the advection-reaction term explicitly and the diffusion term implicitly to obtain a higher-order scheme in time discretization. Then, the POD method and Galerkin projection are introduced to construct the ROM. The resulting ROM can maintain the higher-order accuracy of the original IPDG method and greatly reduce the computation cost. Some numerical examples are used to demonstrate the accuracy and effectiveness of the ROM. [ABSTRACT FROM AUTHOR]

An extensive investigation into the bending behavior of nanoshells made of coated functionally graded (FG) carbon nanotubes reinforced composites has been conducted, employing a novel spatial variation in material properties. The study explored Hardcore and Softcore of coated FG nanoshells. The examination includes five distribution patterns: a fully coated tri-directional, two bidirectional patterns, a transverse unidirectional, and an axial unidirectional distribution. The shell is resting on an orthotropic Winkler, Pasternak and Kerr foundations. Galerkin method is employed to establish an analytical solution for the equilibrium equations. This approach is designed to accommodate a variety of different boundary conditions. A comprehensive examination is undertaken to demonstrate the effects of various geometric and material parameters on the deflection and stresses of nanoshells. [ABSTRACT FROM AUTHOR]

In the present work, a novel refined three-variable quasi-3D shear deformation theory incorporates a correction factor is developed to analyze the buckling behavior of multi-directional functionally graded (FG) curved beams. The proposed displacement field is formulated in accordance with the Euler-Bernoulli beam theory. The research investigated two types of coated Functionally Graded (FG) nanobeams: Hardcore (HC) FG curved nanobeams and Softcore (SC) FG curved nanobeams. Three different material distributions are taken into consideration: a bidirectional material distribution referred to as "2D-FG," a unidirectional transverse material distribution known as "T-FG," and a unidirectional axial material distribution called "A-FG." Eringen's nonlocal elasticity theory is employed to capture small-scale effects. The total potential energy principle is utilized to derive the equilibrium equations of curved nanobeams. A novel solution, utilizing Galerkin's method, has been developed to effectively address a range of boundary conditions. The curved FG beam is supported by an elastic foundation following the Winkler/Pasternak/Kerr model. A comprehensive analysis has been conducted to examine the impacts of various FG schemes, curved beam geometry, nonlocal parameter, elastic foundations, and various boundary conditions on the dimensionless critical buckling loads. This analysis aims to provide a comprehensive understanding of how each of these factors influences the critical buckling loads of the nanobeams. [ABSTRACT FROM AUTHOR]

In this study, we investigate a hybrid-type anisotropic weakly over-penalised symmetric interior penalty method for the Poisson equation on convex domains. Compared with the well-known hybrid discontinuous Galerkin methods, our approach is simple and easy to implement. Our primary contributions are the proposal of a new scheme and the demonstration of a proof for the consistency term, which allows us to estimate the anisotropic consistency error. The key idea of the proof is to apply the relation between the Raviart–Thomas finite element space and a discontinuous space. In numerical experiments, we compare the calculation results for standard and anisotropic mesh partitions. [ABSTRACT FROM AUTHOR]

We propose an interior penalty discontinuous Galerkin (IPDG) method for the Maxwell's transmission eigenvalue problem. First we reformulate the problem as a parameterized eigenvalue problem and define the fixed-point function. The well-posedness of the IPDG discretization for the source problem is proved. Then the uniform convergence of the parameterized discrete solution operator is proved. Finally the error analysis for the IPDG method is realized under reasonable assumptions on the fixed-point function. Numerical examples in 2D and 3D are presented for validation. [ABSTRACT FROM AUTHOR]

A modified wavelet interpolation Galerkin method (WIGM) with improved flexibility of nodal distribution is formulated and utilized to solve the Bratu's problem. Such a method treats the nonlinear term as an independent function and directly approximates it through the proposed wavelet interpolation, resulting in a low-cost discretization of the Bratu's equation with exponential nonlinearity. An efficient implementation of Newton's method to find the lower and upper branches of solution as well as the turning point is then presented. Numerical results demonstrate that the present WIGM can accurately estimate both two solution branches and turning point for the Bratu's problem. Compared with many other existing methods, the proposed WIGM also offers marked superiority in terms of accuracy when using the same node distribution. Moreover, a convergent solution to the resulting nonlinear discrete system obtained by the WIGM can be observed in a few iterations even when the initial guess significantly deviates from the exact solution. Finally, a highly accurate approximation of both two-branched solutions and the turning point for the two-dimensional Bratu's problem is obtained by the proposed WIGM. [ABSTRACT FROM AUTHOR]

In this paper we carry out an error analysis for a reduced order method for the Allen-Cahn equation. First, an ensemble of snapshots is formed from the numerical solutions at some time instances of the full order model, which is a time-space discretisation of the Allen-Cahn equation. The reduced order model is essentially a new spatial discretisation method by using low dimensional approximations to the original approximation space. The low dimensional approximation space is generated from the ensemble of snapshots by applying a proper orthogonal decomposition method. To determine the error between the exact solution and the solution of the reduced order model. We consider a time-space discretisation for which an error estimate of the full model solution is available. Specifically, the full discretisation is based on a stabilized auxiliary variable approach for the time stepping and a spectral Galerkin method for the spatial discretisation. The advantages of this full discretisation are its unconditional stability, the availability of error estimates and its ease of implementation. An estimate of the errors in the H 1 seminorm is rigorously derived for both the full order model and the reduced order model, which is then verified by some numerical examples. [ABSTRACT FROM AUTHOR]

The study aims to explore wavelet applications for analyzing nonlinear boundary value problems. Although several wavelet methods are reviewed in the literature, a comparative study of their strengths and limitations has found only a few attempts. This study bridges the gap between two wavelet-based numerical methods, namely, higher order Daubechies waveletbased Galerkin method and Haar wavelet collocation method, by conducting a comparative study. Nonlinear boundary value problems arising in mathematical physics are solved using both schemes, followed by the computation of optimal error estimates. Furthermore, the advantages offered by the Haar wavelet collocation method over the wavelet-Galerkin method and the rate of convergence are also discussed in detail. [ABSTRACT FROM AUTHOR]

In this paper, we study ultra-weak discontinuous Galerkin methods with generalized numerical fluxes for multi-dimensional high order partial differential equations on both unstructured simplex and Cartesian meshes. The equations we consider as examples are the nonlinear convection-diffusion equation and the biharmonic equation. Optimal error estimates are obtained for both equations under certain conditions, and the key step is to carefully design global projections to eliminate numerical errors on the cell interface terms of ultra-weak schemes on general dimensions. The well-posedness and approximation capability of these global projections are obtained for arbitrary order polynomial space based on a wide class of generalized numerical fluxes on regular meshes. These projections can serve as general analytical tools to be naturally applied to a wide class of high order equations. Numerical experiments are conducted to demonstrate these theoretical results. [ABSTRACT FROM AUTHOR]

In this work, an approximate solution is given to compute the gas emissivities and absorptivities for H2O–CO2–CO mixtures. The proposed model is valid for temperatures from 300 to 2700 K and pressure path‐length (PxL) $({P}_{x}L)$ from 0.06 to 40 atm m. In the calculation of the analytical solution (AS), the nonlinear methods of Galerkin and Ritz were implemented based on the residual solution of the differential roots of the Finite Element Method. For each point value (PxL;T) $({P}_{x}L;T)$ using the AS, the spectral absorptivity aλ ${a}_{\lambda }$ and emissivity ελ ${\varepsilon }_{\lambda }$ were determined, while for the gas mixture the emissivity εm ${\varepsilon }_{m}$ and absorptivity am ${a}_{m}$ were computed using the Hottel Graphical Method (HGM) and the proposed method. In the emissivity calculations, the HGM shows less adjustment, with values of ±20% and ±15% for 79.1% and 56.2% of the available analytical data, while the proposed model correlates adequately with the available data, showing an average deviation of ±15% and ±10% for 91.4% and 76.2% of the available data. In the absorptivity estimations, the HGM shows a weaker fit, with values of ±20% and ±15% for 77.3% and 51.4% of the experimental data, respectively, while the proposed model shows good agreement with the available data, with a mean deviation of ±15% and ±10% for 89.8% and 74.1% of the test made. [ABSTRACT FROM AUTHOR]

This paper presents two methods for solving two‐dimensional linear and nonlinear time‐fractional advection–diffusion equations with Caputo fractional derivatives. To effectively manage endpoint singularities, we propose an advanced space‐time Galerkin technique and a collocation spectral method, both employing generalized Bernstein‐like basis functions (GBFs). The properties and behaviors of these functions are examined, highlighting their practical applications. The space‐time spectral methods incorporate GBFs in the temporal domain and classical Bernstein polynomials in the spatial domain. Fractional equations frequently produce irregular solutions despite smooth input data due to their singular kernel. To address this, GBFs are applied to the time derivative and classical Bernstein polynomials to the spatial derivative. A thorough error analysis confirms the technique's accuracy and convergence, offering a robust theoretical basis. Numerical experiments validate the method, demonstrating its effectiveness in solving both linear and nonlinear time‐fractional advection–diffusion equations. [ABSTRACT FROM AUTHOR]

We study space–time isogeometric discretizations of the linear acoustic wave equation that use splines of arbitrary degree p , both in space and time. We propose a space–time variational formulation that is obtained by adding a non-consistent penalty term of order 2 p + 2 to the bilinear form coming from integration by parts. This formulation, when discretized with tensor-product spline spaces with maximal regularity in time, is unconditionally stable: the mesh size in time is not constrained by the mesh size in space. We give extensive numerical evidence for the good stability, approximation, dissipation and dispersion properties of the stabilized isogeometric formulation, comparing against stabilized finite element schemes, for a range of wave propagation problems with constant and variable wave speed. [ABSTRACT FROM AUTHOR]

In this paper, we investigate a fully discrete approximation of stochastic diffusion-wave equation driven by additive tempered fractional Gaussian noise. This additive noise exhibits semi-long range dependence. The model involves two nonlocal terms in time, i.e., a Caputo fractional derivative and a tempered fractional Brownian motion. The space discretization is achieved via the spectral Galerkin method. The Caputo fractional derivative of order α ∈ (1 , 2) is approximated by using the Grünwald–Letnikov formula. Combining Mittag-Leffler function, Laplace transform and z -transform, we establish the error estimates of the full discretization in the sense of mean-squared L 2 -norm. Numerical experiments are presented to confirm the strong convergence rates. [ABSTRACT FROM AUTHOR]

We propose a method to modify a polygonal mesh in order to fit the zero-isoline of a level set function by extending a standard body-fitted strategy to a tessellation with arbitrarily-shaped elements. The novel level set-fitted approach, in combination with a Discontinuous Galerkin finite element approximation, provides an ideal setting to model physical problems characterized by embedded or evolving complex geometries, since it allows us skipping any mesh post-processing in terms of grid quality. The proposed methodology is firstly assessed on the linear elasticity equation, by verifying the approximation capability of the level set-fitted approach when dealing with configurations with heterogeneous material properties. Successively, we combine the level set-fitted methodology with a minimum compliance topology optimization technique, in order to deliver optimized layouts exhibiting crisp boundaries and reliable mechanical performances. An extensive numerical test campaign confirms the effectiveness of the proposed method. [ABSTRACT FROM AUTHOR]

In this paper, the nonlinear post-buckling analysis of imperfect two-dimensional quasicrystal cylindrical shells is performed. The nonlinear governing equations are established by considering initial geometric imperfection, higher-order shear deformation theory and phonon-phason fields coupling effect. The load-shortening curves and buckling modes are obtained by the Galerkin's method. Numerical results are compared with the existing results and excellent agreements are obtained. Effects of geometrical parameters, initial imperfection and phonon-phason fields coupling effect on the critical load and imperfection sensitivity of two-dimensional quasicrystal cylindrical shells are discussed in detail. [ABSTRACT FROM AUTHOR]

This article studied a general analytical method for dynamical behaviors of various FG-GPLRC non-rectangular plates with a piezoelectric layer. For the first time, classical plate theory and Galerkin's method are used to scrutinize the nonlinear vibration of plates with many different shapes. Profiles of new plates are defined by four edges, specifically two parallel opposite edges, and others are assumed to horizontal change according to arbitrary functions. Linear and nonlinear vibration of plates are confirmed with results obtained by Finite Element Method. Effects of environment, material, and geometrical parameters are studied. Besides, periodic and chaotic motions of plates are investigated. [ABSTRACT FROM AUTHOR]

Numerical simulation of high‐speed flows often needs a fine grid for capturing detailed structures of shock or contact wave, which makes high‐order discontinuous Galerkin methods (DGMs) extremely costly. In this work, a characteristic‐compression based adaptive mesh refinement (AMR, h‐adaptive) method is proposed for efficiently improving resolution of the high‐speed flows. In order to allocate computational resources to needed regions, a characteristic‐compression embedded shock wave indicator is developed on incompatible grids and employed as the criterion for AMR. This indicator applies the admissible jumps of eigenvalues to measure the local compression of homogeneous characteristic curves, and theoretically can capture regions of characteristic‐compression which contain structures of shock, contact waves and vortices. Numerical results show that the proposed h‐adaptive DGM is robust, efficient and high‐resolution, it can capture dissipative shock, contact waves of different strengths and vortices with low noise on a rather coarse grid, and can significantly improve resolution of these structures through mild increase of computational resources as compared with the residual‐based h‐adaptive method. [ABSTRACT FROM AUTHOR]

This paper deals with the a posteriori error analysis for an augmented mixed discontinuous formulation for the stationary Stokes problem. By considering an appropriate auxiliary problem, we derive an a posteriori error estimator. We prove that this estimator is reliable and locally efficient, and consists of just five residual terms. Numerical experiments confirm the theoretical properties of the augmented discontinuous scheme as well as of the estimator. They also show the capability of the corresponding adaptive algorithm to localize the singularities and the large stress regions of the solution. [ABSTRACT FROM AUTHOR]

In this article, we present a fully discrete Crank–Nicolson Galerkin finite element method for solving the two-dimensional nonlinear extended-Fisher–Kolmogorov equation: u t + γ Δ 2 u - Δ u - u + u 3 = 0. The boundedness of the numerical solution in the maximum norm, unique solvability, and related convergence results in L 2 and L ∞ -norms are studied in detail. Also, a new linearized Crank–Nicolson Galerkin modification scheme is designed and error estimate without any time step restrictions is established. Finally, some computational experiments in one and two dimension cases are provided to illustrate the efficacy of our method and to confirm the theoretical results. [ABSTRACT FROM AUTHOR]

The goal of this investigation is to achieve the numerical solution of a two-dimensional parabolic partial differential equation(PDE). The proposed method of this paper is based on the discrete Legendre Galerkin method and spectral collocation method to simplify the spatial derivatives and time derivatives. The discrete Galerkin method is a very fast technique compared to the classical Galerkin method since a finite sum is needed for determining the interpolation coefficients. The operational matrix of the discrete Legendre polynomials is introduced to discretize the time derivatives. Using these couple of techniques and the collocation method, the aforementioned problem is transformed into a solvable algebraic system. The results of applying this procedure to the studied cases show the high accuracy and efficiency of the new method. [ABSTRACT FROM AUTHOR]

The aim of this paper is to implement the high-order local discontinuous Galerkin method (LDGM) for solving partial integro-differential equations (PIDEs) in two dimensions. Time marching method and the transform-based method known as the non-time marching method can be used to discretize temporal terms. The combination of a small time step size in time marching methods and a high-order scheme in space with many degrees of freedom requires a considerable amount of computational time. In order to address this limitation, we propose an algorithm based on a combination of the Laplace transform and LDGM for solving fourth-order time-fractional (TF) PIDEs with weakly singular kernels. Unlike time-marching approaches, the transform-based method has a much lower computational complexity and can take advantage of parallel computing. A numerical experiment validates the accuracy and applicability of the proposed temporal discretization approach and also shows that k-degree LDG solutions have a k + 1 convergence rate. [ABSTRACT FROM AUTHOR]

Meshless methods are a relatively modern approach that has garnered significant interest in analyzing and investigating complex structural problems thanks to distinctive features such as high accuracy, flexibility, rapid calculation speed, and cost–effectiveness. In the present study, the element-free Galerkin method is employed as a meshless approach, wherein a set of arbitrary nodes is distributed across the problem's geometry, including its boundaries, to define the problem domain. The Moving Least Squares approximation is also utilized to formulate the shape functions. Given the intricate geometry of the honeycomb core within sandwich panels, the study employs the generalized method to derive the effective mechanical properties of the honeycomb core. Furthermore, to acquire displacement fields and establish relationships for the problem, the classic plate theory and the first-order shear deformation theory are independently applied. These relationships are then formulated using the Galerkin meshless method. Finally, the obtained parameters are evaluated, and the validity of these relationships is confirmed by comparing the results of this study with those presented in existing articles. The outlined procedure has been systematically simulated through a step-by-step MATLAB program. Subsequently, the impact of different boundary conditions, individual layer thicknesses, dimension ratios, and core wall spacing on the panel's displacement, free-vibration, and buckling behaviors is thoroughly investigated. The obtained results substantiate the efficacy of the utilized methodology, demonstrating a favorable combination of accuracy and convergence rate. [ABSTRACT FROM AUTHOR]

In this paper, freezing through a cold storage unit is simulated, featuring a container equipped with branch-shaped fins attached to the lower cold surface. Conduction emerges as the primary influencer of the freezing process, leading to the simplification of governing equations and the adoption of the Galerkin method for numerical modeling. Notably, an adaptive grid is introduced to enhance accuracy, and the discretization of unsteady terms is achieved through implicit formulation. Rigorous validation against benchmark data attests to the reasonable accuracy of the numerical procedure. Innovatively, to enhance the efficiency of cold storage, nanoparticles are dispersed within the water, in addition to the strategic use of fins. Both methods contribute to enhancing the system’s performance by amplifying the conduction mode of heat transfer. Two key variables are considered: the volume fraction (ϕ) of the nanofluid and its shape factor (m). The study reveals that increasing ϕ and m results in a lower period of freezing, showing reductions of approximately 1.81% and 17.59%, respectively. [ABSTRACT FROM AUTHOR]

A collocation method is developed for discrete fracture models in the context of the partition‐of‐unity method. Spline technologies used in isogeometric analysis (IGA) are exploited to provide a smooth inter‐element transition of gradients, thus allowing to get rid of extra flux terms at element boundaries which are generated by Lagrange polynomials. Bézier extraction is utilised to formulate IGA commensurate with a standard finite element data‐structure. The efficacy of the proposed approach is examined through different numerical examples and is compared with other discrete methods for fracture analysis. The proposed approach is competitive in terms of accuracy with the least computational cost, rendering it a suitable candidate for superseding available collocation approaches for fracture simulation. Moreover, the approach naturally assesses the possibility of physics informed neural networks for fracture simulation, to which collocation is central. [ABSTRACT FROM AUTHOR]

In this work, an efficient and stable exponential time difference method is presented for solving boundary layer problems. By combining exponential time difference schemes with spatial direct discontinuous Galerkin discretization based on exponential boundary layer approximations, the proposed algorithm not only may admit large time step sizes but also could provide good spatial approximations even if on rather coarse spatial grids in the boundary layer. Some energy stabilities of the numerical scheme are rigorously derived. Numerical examples illustrate the accuracy, stability and efficiency of the algorithm. [ABSTRACT FROM AUTHOR]

This paper is concerned with a kinetic-fluid model with random initial inputs in the fine particle regime, which is a system coupling the incompressible Navier–Stokes equations and the Vlasov– Fokker–Planck equations that model dispersed particles of different sizes. A uniform regularity for random initial data near the global equilibrium is established in some suitable Sobolev spaces by using energy estimates, and we also prove the energy decays exponentially in time by hypocoercivity arguments, which means that the long time behavior of the solution is insensitive to the random perturbation in the initial data. For the generalized polynomial chaos stochastic Galerkin method (gPC-sG) for the model, with initial data near the global equilibrium and smooth enough in the physical and random spaces, we prove that the gPC-sG method has spectral accuracy, uniformly in time and the Stokes number, and the error decays exponentially in time. [ABSTRACT FROM AUTHOR]

Standard Galerkin methods often perform poorly for problems with low diffusion. In particular for purely convective transport, least-squares (LS) formulations provide a good alternative. While spatial stability is relatively straightforward in a least-squares finite element framework, estimates in time are restricted, in most cases, to one dimension. This article presents temporal stability proofs for the LS formulation of θ -schemes, including unconditional stability for the backward Euler and Crank-Nicolson methods in two or three dimensions. The theory includes also the linear advection-reaction equation. A series of numerical experiments confirm that the new stability estimates are sharp. [ABSTRACT FROM AUTHOR]

This study presents a meshless modeling approach to design variable-stiffness laminates considering manufacturing constraints. The governing equations are derived using lamination parameters and first-order shear deformation theory. The solution approach uses Chebyshev polynomials and Galerkin's method to obtain the discretized equations of motion. The developed framework was used to maximize the fundamental frequency of composite plates. The variable-stiffness designs provided up to 28.4% higher frequencies compared to optimum constant-stiffness laminates, although the actual level of improvement depends on the number of layers. Finally, manufacturable fiber paths were obtained considering the allowed fiber curvature, which can also reduce the frequency values. [ABSTRACT FROM AUTHOR]

A nonlinear parabolic equation with transformation of the spatial variable and periodic conditions on the circle is considered. Using the method of separation of variables, we obtain properties of eigenfunctions and eigenvalues of the corresponding linearized problem. Using the method of central manifolds, we prove the existence and stability of spatially inhomogeneous stationary solutions. Based on the Galerkin method, we analyze approximate solutions of the original problem. [ABSTRACT FROM AUTHOR]

This research uses the Sinc Collocation method to numerically examine the Degasperis–Procesi and Benjamin–Bona–Mahony equations, achieving a high level of precision and accuracy on computational grounds with a variety of mesh points. The proposed technique involves global collocation using Sinc bases function (SBF) as an activation function. Initially, the time derivative discretization has been performed through finite difference scheme and the remaining spatial derivatives are approximated by θ -weighted scheme to minimize the complexity in numerical computations. The main contribution of research is in modifying the proposed scheme to overcome the complexity involved in the whole process of finding solutions with a cardinal expansion of Sinc functions in nonlinear and singular systems. The solutions of Degasperis–Procesi and Benjamin–Bona–Mahony stiff nonlinear systems are illustrated numerically and graphically and the validation and verification of the obtained results are examined through stability analysis by calculating the spectral radius by bounding the value of θ. A comparative study is presented with some of the well-known numerical techniques, which assured the accuracy and reliability of our technique. [ABSTRACT FROM AUTHOR]

This article presents a new approach of shifted Jacobi spectral Galerkin methods to solve weakly singular Fredholm integral equations with non-smooth solutions. We have incorporated the singular part of the kernel into a single Jacobi weight function, by dividing the integration into two parts and using a simple variable transformation. Taking advantage of orthogonal projection operator and weighted inner product with respect to that same Jacobi weight function, we are able to obtain improved convergence rate for iterated shifted Jacobi spectral Galerkin method (SJSGM) and iterated shifted Jacobi spectral multi-Galerkin method (SJSMGM) in both weighted and infinity norms. Further, we obtain improved superconvergence rate for iterated SJSGM and iterated SJSMGM, by improving the regularity of exact solution, using smoothing transformation. Increasing the value of the smoothing parameter we can improve the regularity of the exact solution upto the desired degree. Numerical results with a comparative study of pre and post smoothing transformation are given to illustrate the theoretical results and efficiency of our proposed methods. [ABSTRACT FROM AUTHOR]

In the context of forward uncertainty quantification, we investigate the convergence of the Galerkin projections for random fractional differential equations. The governing system is formed by a finite set of independent input random parameters (a germ) and by a fractional derivative in the Caputo sense. Input uncertainty arises from biased measurements, and a fractional derivative, defined by a convolution, takes past history into account. While numerical experiments on the gPC-based Galerkin method are already available in the literature for random ordinary, partial and fractional differential equations, a theoretical analysis of mean-square convergence is still lacking for the fractional case. The aim of this contribution is to fill this gap, by establishing new inequalities and results and by raising new open problems. [ABSTRACT FROM AUTHOR]

In this study, the nonlinear vibratory responses and primary resonance (PR) of the spiral stiffened functionally graded (SSFG)-toroidal shell segments (TSSs) are investigated. The shell segments considered are with variable thickness and subjected to external excitations. Both the concave and convex TSSs are investigated in this study. The TSSs considered are reinforced with internal spiral stiffeners and are assumed to be continuously graded along their thickness direction. To model the stiffeners, the smeared stiffeners technique is utilized. Also, the improved Donnell shell theory, von-Kármán nonlinear geometric assumptions, and Galerkin's method are applied to derive the discretized nonlinear governing equation of SSFG-TSSs with variable thickness. To examine the PR responses, the multiple scales method is utilized. To investigate the nonlinear vibratory behaviors, the 4th order P-T method is used. Although the results of the present research are verified with previous research available in the literature, the response of nonlinear vibration is compared with the 4th order Runge–Kutta (R-K) method as well. The influences of geometrical and material parameters on the natural frequencies, nonlinear vibration, and PR responses of SSFG-TSSs are also studied. [ABSTRACT FROM AUTHOR]

The present paper aims to analyze the influences of porosities, elevated temperature and elasticity of in-plane boundary conditions on the free vibration and nonlinear transient response of functionally graded material (FGM) doubly curved panels subjected to uniform lateral pressure. Porosities are assumed to exist in the FGM according to even and uneven distributions. Effective properties of porous FGM are determined using a modified rule of mixture. Basic equations in terms of deflection and stress function are derived basing on the classical shell theory taking into account von Kármán–Donnell nonlinearity and initial geometrical imperfection. Analytical solutions for simply supported panels are assumed and Galerkin method along with Runge–Kutta scheme are applied to obtain the deflection–time response for nonlinear dynamic analysis. Numerical results demonstrate that tangential constraints of edges have extremely significant influences on the vibration and transient response of porous FGM panels exposed to an elevated temperature. The analysis also reveals that porosities have negative and positive effects on the dynamic response of FGM panels at reference and elevated temperatures, respectively. Furthermore, when the edges are restrained, initial imperfection importantly influence the behavior tendency of porous FGM panels exposed to preexist thermal environment. [ABSTRACT FROM AUTHOR]

Taper beams are widely used due to its good damping properties in vibration and noise control field. For dynamic response analysis of taper beams in the high frequency, the energy finite element analysis (EFEA) based on wave theory is the most promising method. However, there is no energy finite element model for taper beams. In this paper, energy finite element model for taper beam is built though two steps. Firstly, the approximate displacement solution of the vibration equation of the taper beam is expressed by the geometry-acoustics approximation method. The expression of the relationship between energy density and power flow of the taper beam is obtained by utilizing the displacement solution. Based on the principle of conservation of energy, the governing equation taken energy density as a variable was derived. Secondly, the stiffness matrix of taper beam element and the solution format of this governing equation are obtained using the Galerkin weighting method, which led to the energy finite element model of the taper beam. Taking a taper beam as an example, the energy finite element model presented in this paper is used to calculate the energy density distribution on the taper beam, and the calculation results are consistent with the finite element calculation results, which proving the correctness of the energy finite element model of the taper beam established in this paper. In order to show the advantages of the EFEA model presented in this paper, a comparison with approximate EFEA model based on constant cross-section elements is performed. The results shown that the EFEA model presented in this paper have higher accuracy and less time consumption, which can reflect the dynamic response characteristics of the taper beam better. [ABSTRACT FROM AUTHOR]

In this paper, the simple refined plate theory (S-RPT) is extended for the analysis of the bending and buckling behaviours of functionally graded graphene platelets reinforced composite (FG-GPLRC) plates supported by local elastic foundations under different boundaries. For the first time, an analytical method for determining the location of the local elastic foundation distribution by simple integration is extended to the analysis of the bending and buckling behaviour of plates. The method avoids complex calculations compared to previous methods for determining the position of the elastic foundation. Compared with other simplified plate theories, the displacement pattern of S-RPT is able to reflect the interrelationship between the gradient material properties and the displacement distribution along plate's thickness. The material properties of FG-GPLRC plates are affected by temperature. Using the static equilibrium method, the plate equilibrium equations are derived by directly integrating the three-dimensional elastic equations along the plate cross section. Galerkin's method is used to solve the governing equations. The accuracy of S-RPT under different boundary conditions and the accuracy of the analytical method for determining the location of the elastic foundation are verified by comparing the numerical results with the existing literature. Finally, the effects of different distribution patterns, weight fractions of GPLs, thickness ratios, aspect ratios, boundary conditions, temperature, elastic foundation distribution patterns, and elastic foundation parameters were investigated in detail. [ABSTRACT FROM AUTHOR]

An analytical investigation on the buckling and postbuckling behavior of carbon nanotube-reinforced composite beams integrated with surface-bonded piezoelectric layers under uniform temperature rise is presented in this paper. Carbon nanotubes (CNTs) are reinforced into isotropic matrix through uniform distribution and functionally graded distributions. The properties of material constituents are assumed to be temperature-dependent and effective properties of CNT-reinforced composite are estimated using an extended rule of mixture. Equilibrium equations of the beams are established based on Euler-Bernoulli theory including von Kármán nonlinearity and solved using analytical solutions and Galerkin method. Critical temperatures and postbuckling load-deflection paths are determined using an iteration algorithm. Parametric studies are performed to examine the influences of CNT distribution and volume fraction, applied voltage, in-plane and out-of-plane conditions of the ends, slenderness, and thickness ratio of layers on the critical loads and postbuckling load carrying capacity of beams. Results reveal that CNT volume fraction and degree of in-plane ends constraint have slight and significant influences on the critical temperatures and thermal postbuckling paths, respectively. The study also finds that negative and positive voltages increase and decrease the thermal buckling temperatures of piezoelectric CNT-reinforced composite beams. [ABSTRACT FROM AUTHOR]

The primary goal of this paper is to suggest a fully discrete numerical solution approach for the time‐fractional Burgers' equation. This paper will consider the fractional derivative in the Caputo sense. The time derivative of this equation will be discretized using the L2‐type discretization formula. The spatial variable is approximated by using the nonsymmetric interior penalty discontinuous Galerkin method. The proposed method is globally and unconditionally stable. The accuracy of the solution is evaluated using a convergence analysis. Computational experiments further confirm the accuracy and stability of the suggested strategy. [ABSTRACT FROM AUTHOR]

In this paper, we designed and analyzed a weak Galerkin finite element method on layer adapted meshes for solving the time-dependent convection-dominated problems. Error estimates for semi-discrete and fully-discrete schemes were presented, and the optimal order of uniform convergence has been obtained. A special interpolation was delicately designed based on the structures of the designed method and layer-adapted meshes. We provided various numerical examples to confirm the theoretical findings. [ABSTRACT FROM AUTHOR]

In this paper, we propose two parameter choice rules for the discretizing Tikhonov regularization via multiscale Galerkin projection for solving linear ill-posed integral equations. In contrast to previous theoretical analyses, we introduce a new concept called the projection noise level to obtain error estimates for the approximate solutions. This concept allows us to assess how noise levels change during projection. The balance principle and Hanke–Raus rule are modified by incorporating the error estimates of the projection noise level. We demonstrate the convergence rate of these two modified parameter choice rules through rigorous proof. In addition, we find that the error between the approximate solution and the exact solution improves as the noise frequency increases. Finally, numerical experiments are provided to illustrate the theoretical findings presented in this paper. [ABSTRACT FROM AUTHOR]

In this paper, we obtain the existence of a positive solution for a class of nonvariational fractional elliptic system with concave and convex nonlinearities in two cases. The paper is divided in two parts: In the first one, for general nonlinearity with subcritical or critical growth, we use Galerkin's method and an approximation argument to show the existence of a solution for the system considered. In the second part, for special cases (which include the power case), we remove the restriction on the growth of the nonlinearity and use sub-supersolution, monotone iteration and a comparison argument to obtain a solution for the system considered. [ABSTRACT FROM AUTHOR]

The stochastic food chain model is an important model within the field of ecological research. Since existing models are difficult to describe the influence of cross-diffusion and random factors on the evolution of species populations, this work is concerned with a stochastic cross-diffusion three-species food chain model with prey-taxis, in which the direction of predators' movement is opposite to the gradient of prey, i.e., a higher density of prey. The existence and uniqueness of martingale solutions are established in a Hilbert space by using the stochastic Galerkin approximation method, the tightness criterion, Jakubowski's generalization of the Skorokhod theorem, and the Vitali convergence theorem. Furthermore, asymptotic behaviors around the steady states of the stochastic cross-diffusion three-species food chain model in the time mean sense are investigated. Finally, numerical simulations are carried out to illustrate the results of our analysis. [ABSTRACT FROM AUTHOR]

We analyze a Discontinuous Galerkin method for a problem with linear advection–reaction and p -type diffusion, with Sobolev indices p ∈ (1 , ∞). The discretization of the diffusion term is based on the full gradient including jump liftings and interior-penalty stabilization while, for the advective contribution, we consider a strengthened version of the classical upwind scheme. The developed error estimates track the dependence of the local contributions to the error on the local Péclet numbers. A set of numerical tests support the theoretical derivations. [ABSTRACT FROM AUTHOR]