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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]

In this paper we propose and analyze the alternating direction implicit (ADI) difference schemes in conjunction with the second order backward differentiation formula (BDF2) method with variable time step sizes for solving the two-dimensional parabolic equation. The spatial compact operators are also applied to construct high order ADI scheme. By using the discrete energy method and the positive definiteness of the nonuniform BDF2 approximation, we prove the unconditional H 1 semi-norm stability for the variable-step BDF2-ADI scheme and the variable-step compact BDF2-ADI scheme under the constraint r n ≤ 4.8 , where r n denotes the adjacent step size ratio. Moreover, the optimal second order and the fourth order convergence rates are derived rigorously under this restriction. To the best of our knowledge, this is the first strict theoretical analysis of the variable-step ADI numerical schemes for the multidimensional parabolic equations. Several numerical examples are included to verify the analysis results. The extension to the nonlinear Allen-Cahn equation is also presented, for which the variable-step BDF2-ADI scheme and the corresponding compact scheme combined with the adaptive time-stepping algorithm improve the efficiency of the long time simulation. [ABSTRACT FROM AUTHOR]

• Inequality between countries is decreasing, while inequality within countries is increasing. • As far as energy is concerned, the convergence of consumption began in the 1970s. • Global warming is closely linked to the economic convergence of energy consumption. • Higher energy consumption in developing countries is leading to higher global CO 2 emissions. • The world is becoming more equal in energy use and the technological exploitation of resources. Global inequality is made up of two components: inequality within countries and between countries. Over the last two centuries, the second component has strongly shaped global inequality. However, little is known about its evolution over time, and nothing at all about inequality in energy use and greenhouse gas (GHG) emissions. This article presents a comprehensive reconstruction of international divergence and convergence from the side of energy consumption and CO 2 emissions over two centuries. From 1820 to 1919, inequality between countries increased and so did inequality in GHG emissions. This increase was followed by ups and downs until the 1970s, when a decline of international inequality began. Convergence in both energy consumption and GHG emissions accelerated after 2000. [ABSTRACT FROM AUTHOR]

This paper deals with speeding up the convergence of a class of two-step iterative methods for solving linear systems of equations. To implement the acceleration technique, the residual norm associated with computed approximations for each sub-iterate is minimized over a certain two-dimensional subspace. Convergence properties of the proposed method are studied in detail. The approach is further developed to solve (regularized) normal equations arising from the discretization of ill-posed problems. The results of numerical experiments are reported to illustrate the performance of exact and inexact variants of the method on several test problems from different application areas. [ABSTRACT FROM AUTHOR]

The DPG method with optimal test functions for solving linear quadratic optimal control problems with control constraints is studied. We prove existence of a unique optimal solution of the nonlinear discrete problem and characterize it through first-order optimality conditions. Furthermore, we systematically develop a priori as well as a posteriori error estimates. Our proposed method can be applied to a wide range of constrained optimal control problems subject to, e.g., scalar second-order PDEs and the Stokes equations. Numerical experiments that illustrate our theoretical findings are presented. [ABSTRACT FROM AUTHOR]

We consider a new splitting based on the Sherman-Morrison-Woodbury formula, which is particularly effective with iterative methods for the numerical solution of large linear systems and especially for systems involving matrices that are perturbations of circulant or block circulant matrices. Such matrices typically arise in the discretization of differential equations using finite element or finite difference methods. We prove the convergence of the new iteration without making any assumptions regarding the symmetry or diagonal-dominance of the matrix, which are limiting factors for most classical iterative methods. To illustrate the efficacy of the new iteration we present various applications. These include extensions of the new iteration to block matrices that arise in certain saddle point problems as well as two-dimensional finite difference discretizations. The new method exhibits fast convergence in all of the test cases we used. It has minimal storage requirements, straightforward implementation and compatibility with nearly circulant matrices via the Fast Fourier Transform. Remarkably, the new method was tested against very large matrices demonstrating extremely fast convergence. For these reasons it can be a valuable tool for the solution of various finite elements and finite differences discretizations of differential equations. [ABSTRACT FROM AUTHOR]

The purpose of this study is to introduce a computational method for solving the two-dimensional optimal control problem of the large membrane vibration equation of fractional order. Our proposed approach utilizes Chebyshev cardinal functions in two dimensions to obtain a finite dimensional optimal control problem. This problem is then solved using the Lagrange multipliers method by substituting the expansions of the state and control variables in terms of two-dimensional Chebyshev cardinal functions into the objective function, dynamical system, and initial conditions. The utilization of Chebyshev nodes offers a distinct advantage by effectively mitigating the Runge phenomenon, characterized by the undesirable oscillation and divergence of high-degree polynomial interpolants in close proximity to the interval's endpoints. We extensively discuss the convergence of our method and evaluate its accuracy using three test problems. Our results demonstrate that this technique provides highly accurate numerical solutions for these types of problems. [ABSTRACT FROM AUTHOR]

This paper presents a novel, robust, efficient, and simple optimization algorithm called the Object-Oriented Programming Optimization Algorithm (OOPOA) for tackling constrained and unconstrained optimization problems. The algorithm is inspired by the inheritance concept of Object-Oriented programming languages, where the features of a class are classified into three types according to inheritance probability: public, private, and protected. The object-oriented programming inheritance concept is implemented in the algorithm to update the population for the next generations. The proposed algorithm ensures exploitation by selecting the solution with the highest fitness to be inherited in each iteration. It ensures exploration by applying a mutation process that helps explore wide regions in the search space. The performance of this technique is demonstrated by solving 34 different optimization tasks, including 20 standard benchmark problems, ten IEEE Congress of Evolutionary Computation benchmark test functions, and four constrained real-world engineering design problems. [Display omitted] [ABSTRACT FROM AUTHOR]

In this paper we propose an approximation method based on the classical Schoenberg-Marsden variation diminishing operator with applications to the construction of new quadrature rules. We compare the new operator with the multilevel one studied in [12] in order to characterize both of them with respect to the well known classical one. We discuss convergence properties and present numerical experiments. [ABSTRACT FROM AUTHOR]

In this paper, the error estimates of the Crank-Nicolson-Leapfrog (CNLF) time-stepping scheme for the two-phase Cahn-Hilliard-Navier-Stokes (CHNS) incompressible flow equations based on scalar auxiliary variable (SAV) are strictly proved. Due to the complexity of the multiple variables and the strong coupling of the equations, it is not easy to prove rigorous error estimates. Under the corresponding regularity assumption and the superconvergence of the negative norm estimates of the two quasi-projections, we prove that the error estimates for phase-field ϕ in the H 1 -semi-norm and velocity u in the H 1 -norm are able to achieve second-order convergence rates in time and the O (h r + 1) (r ≥ 1) in space. The nonlocal variables Q and r also achieve the same convergence rate. In addition, the pressure p in the L 2 -norm can only reach first-order convergence rate in time and the O (h r) (r ≥ 1) in space. At the same time, several numerical examples are given to illustrate the accuracy and effectiveness of the numerical scheme. [ABSTRACT FROM AUTHOR]

• A distribution dynamics approach is employed to examine the converge of TFEE across Chinese PAA level cities. • The findings show a clear convergence in TFEE. • Convergence comes more from efficiency drops in high TFEE cities than from efficiency gains in low TFEE cities. • The environmental protection polices in key cities improved energy efficiency. • Large-sized cities show better performance in energy efficiency than small- and medium-sized cities. Improving energy efficiency is considered the most direct route to reducing carbon emissions on a massive scale. To examine the long-run trend of China's energy efficiency, this paper employs a distribution dynamics approach to analyze the convergence of total-factor energy efficiency (TFEE) across 286 Chinese cities between 2002 and 2014. The result suggests the existence of convergence, which comes more from the efficiency declines in the high TFEE cities rather than the efficiency gains in the low TFEE cities. Despite the fact that a few cities have performed well in energy efficiency, the majority of cities have converged to a low-efficiency point. Further analyses show that the inefficiency was attributed to inefficient energy use in small- and medium-sized cities. Our analysis provides strong scientific support for China's ongoing energy conservation and environmental protection policies. [ABSTRACT FROM AUTHOR]

Performance requirements necessitate control designs that assure not only transient response specifications but also steady-state accuracy. Monotonic convergence of the tracking error is crucial for an efficient control design to prevent the performance degradation caused by overshooting. This needs a balanced consideration of both reaching conditions and the monotonic convergence, in the context of sliding mode control. In this paper, the dynamic behaviour of the dead-beat sliding mode control is characterized and the signum function is replaced by employing a non-switching one, in order to reduce chattering. The paper conducts a thorough analysis of monotonic convergence of both the switching and the non-switching error dynamics. By deriving the conditions for monotonic convergence, the control parameters can be strategically chosen to ensure monotonic convergence of the tracking error. Numerical and experimental results are presented to validate effectiveness of the proposed control scheme, which evaluate the tracking performance achieved by both the switching and the non-switching control methods. • The DBSMC approach uses a dead-beat reaching law for the global sliding-mode. • Both the switching and non-switching attractive laws are proposed. • The monotonic convergence is characterized by the MD-region and the SS-band. • The calculation of convergence steps of the DBSMC schemes is provided. [ABSTRACT FROM AUTHOR]

In this article, two difference schemes are proposed to solve numerically a nonlinear strong damped wave equation in two dimensions. The first scheme is nonlinear implicit, the second scheme is linear implicit. It is proved that the two schemes are unique solvable, unconditionally stable. Moreover, the present numerical methods are convergent with second order accuracy both in space and time in the discrete H 1 -norm. Numerical experiments are provided to show the validity and the accuracy of the proposed difference schemes as well as their efficiency. [ABSTRACT FROM AUTHOR]

This study presents some new iterative algorithms based on the gradient method to solve general constrained systems of conjugate transpose matrix equations for both real and complex matrices. In addition, we analyze the convergence properties of these methods and provide numerical techniques to determine the solutions. The effectiveness of the proposed iterative methods is demonstrated through various numerical examples employed in this study. [ABSTRACT FROM AUTHOR]

This paper centers on constructing a lucid and utilitarian approach to tackle linear and non-linear two-dimensional partial differential equations. To test the applicability of the proposed algorithm a variant of the classical two-dimensional unsteady Burgers' equation is set up as a testing ground. The method in a nutshell reduces to solving one-dimensional Burgers' equations resulting from the application of appropriate operator splitting techniques in the temporal direction. In solving these one-dimensional Burgers' equations a refined tailored finite point method in conjunction with an apposite linearization to the purpose is employed. The conditional stability, consistency, and convergence of the method are established theoretically and the method is found to be first-order convergent in time and second-order convergent in space. To illustrate the accuracy of the scheme, divers examples have been solved and the results obtained prove that this method is top-notch in terms of cost-cutting and time efficiency through the sufficiency of coarse meshes. • This work presents a minimal machinery algorithm on an operator splitting technique in conjunction with a tailored FPM for a 2D Burgers' equation. • The tailored FPM is advisable for its efficient adaptation to local solution properties, user-friendly algorithm, and cost-effectiveness. • The manuscript offers a comprehensive mathematical and computational background along with stability, consistency, and convergence analyses. • The algorithm despite being simple and straightforward has performed on a par with other methods based on elite classical formulation techniques. [ABSTRACT FROM AUTHOR]

In this paper, fourth-order compact difference schemes are derived, analyzed and tested at length for both one- and two-dimensional Rosenau equations under the spatial periodic boundary conditions on the basis of the double reduction order method and bilinear compact operator. We prove that these schemes satisfy mass and energy conservation law via the help of the energy method. In addition, the uniquely solvable, unconditional convergence and stability are all obtained with the convergence order four in space and order two in time under the L ∞ -norm. Several numerical examples are presented to support the theoretical results. [ABSTRACT FROM AUTHOR]

• Economic miracles, such as the Asian miracles, are not statistical accident to be ignored, and they are not as rare as commonly assumed i.e., the distribution of cross-country long-term growth follows a Power law. • Disentangling the role of policies from luck implies that economic miracles are alike, and every failure is a failure in its own way. High and sustained growth depends on maintaining the best possible policies while not being too unlucky. • The relatively few successes in achieving high and sustained growth such as the Asian miracles should constitute a focal point for the study of development policies. • In contrast, results derived from a large pool of development failures should be taken with skepticism, especially if it is based on the study of growth in the short to medium run. We show evidence that development "miracles," such as the Asian Miracles, may not be that rare as the distribution of cross-country long-term growth follows a Power Law. We propose a growth theory, disantangling the role of policies from luck, which is consistent with this stylized fact and predicts that high growth depends on maintaining the best possible policies while not being too unlucky. We argue that miracles are all alike and every failure is a failure in its own way, suggesting that the few growth miracles hold more clues about good policies than the vast number of growth failures. We infer important implications for the study of economic development. [ABSTRACT FROM AUTHOR]

In this paper, an implicit finite difference scheme combined with Kansa meshless method is proposed for the two-dimensional time and space fractional Bloch-Torrey equation. The Caputo derivative with respect to time is discretized by finite difference method based on Alikhanov's super convergent approximation while Riesz derivative with respect to spatial variables by Kansa meshless method. The convergence and stability of the time semi-discretization are proven using energy method. The convergence order in time direction is proven to be 2. The implementation of the full discretization is discussed in detail. Especially, the issue of calculation of Riesz derivative of RBF is addressed. Numerical examples are given which verify the feasibility of the proposed method. [ABSTRACT FROM AUTHOR]

In this paper, we present a finite difference scheme for the triple-coupled Schrödinger equations (T-CNLS) in optics. The T-CNLS is approximated by Crank-Nicolson scheme in time and finite difference method in space. Some mathematical characters are investigated, such as structure-preserving properties, unique solvability, convergence in L ∞ norm. Some numerical examples are reported to illustrate the theoretical results. [ABSTRACT FROM AUTHOR]

A high-order accuracy finite difference scheme is investigated to solve the one-dimensional extended Fisher-Kolmogorov (EFK) equation. A three level linearized compact finite difference scheme is proposed. Priori estimates and unique solvability are discussed in detail by the discrete energy method. The unconditional stability and convergence of the difference solution are proved. The new compact difference scheme has second-order accuracy in time and fourth-order accuracy in space in maximum norm. Numerical experiments demonstrate the accuracy, efficiency of our proposed technique. [ABSTRACT FROM AUTHOR]

This paper mainly studies the high-order stable numerical solutions of the time-space fractional diffusion equation. The Fourier spectral method is used for discretization in space and the Spectral Deferred Correction (SDC) method is used for numerical solutions in time. Therefore, a high-precision numerical discretization scheme for solving the fractional diffusion equation is obtained, and the convergence and stability of the scheme are proved. Finally, several numerical examples are given to illustrate the effectiveness and feasibility of the numerical scheme. [ABSTRACT FROM AUTHOR]

The Prebisch-Singer hypothesis implies divergence in the terms of trade between developing and developed countries but does not eliminate the possibility of convergence if productivity in developing countries' exports sufficiently improves. This study constructs a new and unique export sophistication dataset for the productivity of high-technology export products to examine convergence behavior of developed and emerging market countries. Relative and weak σ-convergence tests reveal overall productivity divergence in international trade. However, there are convergence clubs whereby income, R&D expenditures, foreign direct investment, and educational expenditures yield similar effects on the convergence club formation in both developed and emerging market countries. [ABSTRACT FROM AUTHOR]

The solution of convection diffusion problem is a complex task, requiring the solving of linear equations which can take up a significant amount of computational time. This issue can be mitigated through the use of iterative methods. We present a class of fast successive permutation iterative algorithms with relaxation factors (ω -SPI) to solve 1D, 2D and 3D convection diffusion equations. The stability, convergence and optimal value of the relaxation factor are proved. In order to test the efficiency and accuracy of our proposed iterative algorithms, we present several numerical experiments. The results demonstrate that the new algorithms are more accurate and faster than traditional methods such as full-implicit, Gauss-Seidel and SOR algorithms. [ABSTRACT FROM AUTHOR]

A unified framework of nonconforming virtual element methods (VEMs), including C 0 and fully nonconforming virtual elements, are built for the fourth order extended Fisher-Kolmogorov (EFK) equation. For the constructed semi-discrete and fully discrete schemes, the unique solvability is obtained. Then the numerical schemes are proved to be dissipative in the senses of discrete energy, which can be further used to derive the priori bounds of the discrete solutions. In addition, a new analytical technique is used to derive the unconditional convergence of the numerical schemes without any restriction on the grid ratio. Finally, numerical experiments are provided to confirm the theoretical results. [ABSTRACT FROM AUTHOR]

Feature selection (FS) is a crucial task in machine learning applications, which aims to select the most appropriate feature subset while maintaining high classification accuracy with the minimum number of selected features. Despite the widespread usage of metaheuristics as wrapper-based FS techniques, they show reduced effectiveness and increased computational cost when applied to high-dimensional datasets. This paper presents a novel Binary Object-Oriented Programming Optimization Algorithm (BOOPOA) for FS of high dimensional datasets, where the Object-Oriented Programming Optimization Algorithm (OOPOA) is a novel optimization technique inspired by the inheritance concept of Object-Oriented programming (OOP) languages. The effectiveness of this method in solving high dimensional FS problems is validated by using 26 datasets, most of which are of high dimension (large number of features). Seven existing FS algorithms are compared with the proposed OOPOA using various metrics, including best fitness, average fitness (AVG), selection size, and computational time. The results prove the superiority of the proposed algorithm over the other FS algorithms, having an average performance of %92.5, 0.078, 0.084, %38.9, and 8.6 min for classification accuracy, best fitness, average fitness, size reduction ratio, and computational time. The outcomes demonstrate the proposed FS approach's superiority over currently used methods. [Display omitted] [ABSTRACT FROM AUTHOR]

In this paper, a novel auxiliary variable approach is firstly introduced to reformulate the fractional nonlinear Schrödinger equation with wave operator in an equivalent system, which admits a modified energy conservation. For the new system, a linearly implicit energy-preserving scheme with constant-coefficient matrices is constructed by combining Fourier spectral Galerkin method and Crank-Nicolson method. In theoretical aspects, the energy conservation is proved, and the convergence analysis is discussed with the help of mathematical induction. Meanwhile, the proposed auxiliary variable approach is utilized for the coupled fractional nonlinear Klein-Gordon-Schrödinger equation to construct a linearly implicit and decoupled energy-preserving scheme. Extensive numerical experiments verify that the theoretical results are correct and the proposed schemes have high efficiency in long-time computations. [ABSTRACT FROM AUTHOR]

In this paper, we propose a Fourier-Laguerre spectral method for exterior problems of two-dimensional complex obstacles based on the mapping method. We first use a polar coordinate transformation to convert the exterior domains of complex obstacles into the exterior domain of the unit disk. Then we apply the polar coordinate transformation to the exterior problem of an elliptic equation, derive its weak formulation and prove the existence and uniqueness of the weak solution. On this basis, we construct the Fourier-Laguerre spectral Galerkin scheme, describe its numerical implementation and analyze the convergence of the numerical solution under the H 1 -norm. Numerical results indicate that our spectral Galerkin method is easy to implement and possesses high-order accuracy. [ABSTRACT FROM AUTHOR]

In this paper, a generalized parametric iterative method (GPIM) is proposed to solve the nonlinear discrete formulation of the magnetohydrodynamic (MHD) equations. We exploit the idea of decoupling to solve the strongly coupled MHD system, expecting to save storage space and to accelerate the convergence. The noticeable advantage of the proposed algorithm is that it is not necessary to solve the saddle point system for each iteration step and converge geometrically with the contract number. Numerical experiments are given to verify the validity and correctness of the theoretical results. [ABSTRACT FROM AUTHOR]

We discuss goal-oriented adaptivity in the frame of conforming finite element methods and plain convergence of the related a posteriori error estimator for different general marking strategies. We present an abstract analysis for two different settings. First, we consider problems where a local discrete efficiency estimate holds. Second, we show plain convergence in a setting that relies only on structural properties of the error estimators, namely stability on non-refined elements as well as reduction on refined elements. In particular, the second setting does not require reliability and efficiency estimates. Numerical experiments underline our theoretical findings. [ABSTRACT FROM AUTHOR]

This paper presents convergence analysis for a tracking differentiator of an active disturbance rejection control method which is widely applied but lacks theoretical analysis. Since a nonlinear piecewise function is used in the tracking differentiator, the convergence analysis is difficult for tracking errors. Convergence proof processes of the tracking differentiator are divided into three situations based on the nonlinear piecewise function. Tracking errors of the tracking differentiator are proved to be uniformly ultimately bounded considering three situations, and relationships between upper bounds of tracking errors and adjustment parameters are founded by a Lyapunov approach, which provides a basis for parameters adjustment. Finally, simulation and experiment results verify the effectiveness of the proposed convergence analysis. • Convergence analysis is carried out for a tracking differentiator. • Relationships are illustrated between tracking errors and adjustment parameters. [ABSTRACT FROM AUTHOR]

In this paper, we discuss the convergence of an Algebraic MultiGrid (AMG) method for general symmetric positive-definite matrices. The method relies on an aggregation algorithm, named coarsening based on compatible weighted matching , which exploits the interplay between the principle of compatible relaxation and the maximum product matching in undirected weighted graphs. The results are based on a general convergence analysis theory applied to the class of AMG methods employing unsmoothed aggregation and identifying a quality measure for the coarsening; similar quality measures were originally introduced and applied to other methods as tools to obtain good quality aggregates leading to optimal convergence for M-matrices. The analysis, as well as the coarsening procedure, is purely algebraic and, in our case, allows an a posteriori evaluation of the quality of the aggregation procedure which we apply to analyze the impact of approximate algorithms for matching computation and the definition of graph edge weights. We also explore the connection between the choice of the aggregates and the compatible relaxation convergence, confirming the consistency between theories for designing coarsening procedures in purely algebraic multigrid methods and the effectiveness of the coarsening based on compatible weighted matching. We discuss various completely automatic algorithmic approaches to obtain aggregates for which good convergence properties are achieved on various test cases. [ABSTRACT FROM AUTHOR]

In this paper, a fully discrete finite element scheme is established with L 1 approximation for solving the nonlinear time-fractional wave equation with a variable coefficient. The stability of the scheme is analyzed and the optimal error estimates are derived. To improve the computational efficiency, we propose a two-grid algorithm based on the fully discrete finite element scheme. With the proposed technique, a small-scale nonlinear problem is calculated by iteration in a coarse-grid space with mesh size H , and then a linearized problem is solved in a fine-grid space with mesh size h , H ≫ h. This technique not only maintains the numerical precision, but also saves a lot of computing time. The stability and optimal convergence order are considered. Using the Taylor expansion, the second two-grid algorithm is constructed to improve the optimal convergence order. The similar properties are discussed in detail. Numerical experiments are provided to illustrate the theoretical analysis and test the performance of the proposed algorithms. [ABSTRACT FROM AUTHOR]

A novel canonical Euler splitting method is presented for semilinear composite stiff parabolic partial functional differential algebraic equations with initial and Dirichlet boundary conditions. The original partial differential problems are transformed into the semi-discrete problems by spatial discretization, and then the canonical Euler splitting method is employed to solve the resulting semi-discrete problems. Under appropriate assumptions, the stability and convergence theories of this method are established. A series of numerical experiments are given to illustrate the effectiveness of this method and the correctness of theoretical results. Numerical results also demonstrate that the constructed method can significantly improve the calculation speed. [ABSTRACT FROM AUTHOR]

This paper examines the significance of real and monetary integrations for the inflationary dynamics of an emerging country, Turkey. The analysis accounts for 2-digit items of CPI inflation, which can be broadly categorized as tradable versus non-tradable and goods versus services. We find that a fall in the inflation gap between partner countries is mainly related to real integration whereas the co-movement of inflation is prominently driven by monetary policy co-movements. The product-type analysis shows that inflation gap in tradable items between trade partners shrinks and becomes more correlated with the (de)convergence and co-movement of real integration. [ABSTRACT FROM AUTHOR]

In this paper, two efficient high-order physical property-preserving linearly implicit finite difference schemes are firstly developed and analyzed for nonlinear fourth-order wave equation with damping based on the scalar auxiliary variable (SAV) approach and newly developed Lagrange multiplier (LM) approach, respectively. Then the modified energy preservation property, the priori bounds of the numerical solution and convergence analysis of the former scheme are presented in detail. It is pointed out that the former scheme only conserves/dissipates the modified discrete energy, not the original energy while the latter scheme conserves/dissipates the original discrete energy exactly. Also, the latter scheme only demands to solve a decoupled and linear system with constant coefficients at each time step plus a nonlinear algebraic system. Finally, some numerical simulations are presented to demonstrate the accuracy, efficiency and preservation property of the obtained schemes. • Two efficient physical property-preserving difference methods for the nonlinear fourth-order wave equation with damping are proposed. • The modified preservation property and error estimation of the former scheme are presented in detail. • Both schemes demand to solve a sequence of linear decoupled systems with constant coefficients at each time step. • Some numerical results are performed to demonstrate the accuracy, efficiency and preservation property of both schemes. [ABSTRACT FROM AUTHOR]

Discrete time-variant nonlinear optimization (DTVNO) problems are commonly encountered in various scientific researches and engineering application fields. Nowadays, many discrete-time recurrent neurodynamics (DTRN) methods have been proposed for solving the DTVNO problems. However, these traditional DTRN methods currently employ an indirect technical route in which the discrete-time derivation process requires to interconvert with continuous-time derivation process. In order to break through this traditional research method, we develop a novel DTRN method based on the inspiring direct discrete technique for solving the DTVNO problem more concisely and efficiently. To be specific, firstly, considering that the DTVNO problem emerging in the discrete-time tracing control of robot manipulator, we further abstract and summarize the mathematical definition of DTVNO problem, and then we define the corresponding error function. Secondly, based on the second-order Taylor expansion, we can directly obtain the DTRN method for solving the DTVNO problem, which no longer requires the derivation process in the continuous-time environment. Whereafter, such a DTRN method is theoretically analyzed and its convergence is demonstrated. Furthermore, numerical experiments confirm the effectiveness and superiority of the DTRN method. In addition, the application experiments of the robot manipulators are presented to further demonstrate the superior performance of the DTRN method. [ABSTRACT FROM AUTHOR]

In this paper, we develop a novel discontinuous Galerkin (DG) finite element method for solving the Poisson's equation u x x + u y y = f (x , y) on Cartesian grids. The proposed method consists of first applying the standard DG method in the x -spatial variable leading to a system of ordinary differential equations (ODEs) in the y -variable. Then, using the method of line, the DG method is directly applied to discretize the resulting system of ODEs. In fact, we propose a fully DG scheme that uses p -th and q -th degree DG methods in the x and y variables, respectively. We show that, under proper choices of numerical fluxes, the method achieves optimal convergence rate in the L 2 -norm of O (h p + 1) + O (k q + 1) for the DG solution, where h and k denote, respectively, the mesh step sizes for the x and y variables. Our theoretical results are validated through several numerical experiments. [ABSTRACT FROM AUTHOR]

To simulate waves on unbounded domain, absorbing boundary conditions are usually needed to bound the computational domain and avoid boundary reflections as much as possible. In this paper, a new absorbing layer approach is presented to simulate the soliton propagation based on the cubic nonlinear Schrödinger (NLS) equation on unbounded domain, and a two-level finite difference scheme is constructed for solving this NLS problem. Both the analytical solution and the numerical solution are proved to be stable in L 2 -norm and l 2 -norm, respectively, and they decay exponentially in the absorbing (or called lossy) layers. Furthermore, the scheme is shown to be convergent with order O (τ 2 + h 2) , where τ and h are the time step and grid size, respectively. Numerical examples are given to illustrate the method and verify its effectiveness. [ABSTRACT FROM AUTHOR]

How does the human brain develop stable emotion representations? According to recent work by Camacho et al. , neural representations of contextualized emotional cues are distinct and fairly stable by mid-to-late childhood and activation patterns become increasingly similar between individuals during adolescence. Here, I propose a framework for investigating contextualized emotion processing. [ABSTRACT FROM AUTHOR]

In this paper, we establish the convergence and quasi-optimality of an adaptive finite element method for nonmonotone elliptic problems on L 2 errors for a sufficiently fine initial mesh. Although additional refinements are needed to keep the meshes sufficiently mildly graded, it does not affect the convergence and quasi-optimality of the adaptive finite element method presented in this paper. Our theoretical results are verified by some numerical examples. [ABSTRACT FROM AUTHOR]

In the current study, a robust septic Hermite collocation method (SHCM) is proposed to simulate the Kuramoto-Sivashinsky (KS) equation. To approximate the spatial derivatives, the collocation method with the 7 t h -degree Hermite interpolation polynomial is utilized and for the temporal derivative, the Crank-Nicolson (CN) technique is implemented. The nonlinear terms of the KS equation are linearized using the quasi-linearization process. Using the von-Neumann approach, it is demonstrated that the algorithm is unconditionally stable. The convergence analysis of the proposed technique is also given. In the temporal direction, the scheme is observed to be second-order convergent and in space direction, it is found to be sixth-order convergent. The proposed technique's robustness is demonstrated by solving six test problems. The L 2 , L ∞ , and global relative errors are determined and the findings are compared with other methods available in the literature. The behavior of few KS equations, for which their exact solution is not available, is also analyzed. The current findings are better than the results obtained from the other methods and are also matched well with the analytical solution. [ABSTRACT FROM AUTHOR]

This study investigates the green total factor productivity of urban agglomerations, a high-level urban organisational form, by estimating the green total factor productivity (GTFP) of 285 cities in China from 2003 to 2017 using global data envelopment analysis and a super-slack-based measure model with undesirable output. Based on these estimations, we obtain the GTFP of 19 urban agglomerations and analyse the spatiotemporal evolution, dynamic evolution of spatial distribution, and regional differences in the urban agglomeration GTFP using various analytical tools such as the Dagum Gini coefficient, subgroup decomposition, σ and β convergence analysis, and kernel density estimation. Additionally, we identify and analyse the driving factors of GTFP growth in urban agglomerations by employing a panel model from the perspective of improving the efficiency of cities within urban agglomerations and optimising the development of intercity industrial and spatial structures. The results are as follows: First, the GTFP of urban agglomerations generally exhibit a fluctuating upward trend with low values in northwest-central China and high values in the north-southeastern coastal areas. Second, the GTFP differences among the urban agglomerations present an overall convergent trend, with the smallest gap between the national-level urban agglomerations and a stable performance. The gaps between regional and local urban agglomerations undergo considerable narrowing of the evolutionary trend, whereas gaps between local urban agglomerations and other urban agglomerations keep expanding. Third, economic growth, environmental regulation, energy consumption, and foreign direct investment and technological progress are conducive to improving urban GTFP, and a U-shaped relationship exists between cooperative industrial agglomeration and urban agglomeration GTFP. We suggest that strengthening regional collaborative environmental governance can improve the GTFP in urban agglomerations. [ABSTRACT FROM AUTHOR]

Trivially, the time-fractional Black–Scholes (FBS) equation is utilized to describe the behavior of the option pricing in financial markets. This work is intended as an attempt to introduce the ψ -Hilfer fractional Black–Scholes (ψ -HFBS) equation. First, we concentrate on demonstrating the existence of the solution to the ψ -HFBS equations. Second, a numerical scheme is presented for solving the equation given the appropriate initial and boundary conditions. The approximate solutions are considered as linear combinations of the Lagrange functions with unidentified coefficients. By collocating the considered equation together with the boundary and initial conditions at Chebyshev-Gauss-Lobato (CGL) points, it will be converted to a system of linear algebraic equations. Next, we have proved the convergence of the approach. Finally, some test problems are given in order to indicate the suggested method. [ABSTRACT FROM AUTHOR]

In order to solve a time-varying linear matrix equation with periodic noises, an anti-interference dynamic integral neural network (AI-DINN) is proposed. Based on an indefinite unbounded vector/matrix-type error function, the proposed AI-DINN includes an integral structure, a recursive structure, and an adjustment module. It has the excellent ability to overcome the interference of periodic noises. This paper theoretically proves the convergence and robustness of the proposed AI-DINN for solving the time-varying linear matrix equation with the interference of periodic noises. Computer simulation results verify that the proposed AI-DINN method based on different activation functions can achieve convergence within limited time with the interferences of different periodic noises. In addition, the proposed AI-DINN with different activation functions has its own advantages with the interference of different types of periodic noises. Furthermore, comparative simulation experiments verify that the proposed AI-DINN has better convergence and anti-interference performance compared with state-of-the-art methods. [ABSTRACT FROM AUTHOR]