First Difference Research Articles

Fractional Variational Integrators Based on Convolution Quadrature

Fractional dissipation is a powerful tool to study nonlocal physical phenomena such as damping models. The design of geometric, in particular, variational integrators for the numerical simulation of such systems relies on a variational formulation of the model. In Jiménez and Ober-Blöbaum (J Nonlinear Sci 31:46, 2021), a new approach is proposed to deal with dissipative systems including fractionally damped systems in a variational way for both, the continuous and discrete setting. It is based on the doubling of variables and their fractional derivatives. The aim of this work is to derive higher-order fractional variational integrators by means of convolution quadrature (CQ) based on backward difference formulas. We then provide numerical methods that are of order 2 improving a previous result in Jiménez and Ober-Blöbaum (J Nonlinear Sci 31:46, 2021). The convergence properties of the fractional variational integrators and saturation effects due to the approximation of the fractional derivatives by CQ are studied numerically.

Two-player nonzero-sum and zero-sum games subject to stochastic noncausal systems

This paper studies two-player nonzero-sum and zero-sum games within the context of stochastic noncausal systems (SNSs). These SNSs are transformed into subsystems consisting of forward and backward stochastic difference equations through an equivalent conversion. Subsequently, recurrence equations are introduced to convert stochastic two-player nonzero-sum games into deterministic difference equation solving problems. These recurrence equations are then utilised to derive the relevant equations needed to deduce the saddle-point equilibrium solutions for two-player zero-sum games embedded within linear and nonlinear SNSs. The resolution of these equations yields analytical expressions that encapsulate the saddle-point equilibrium solutions for such types of two-player zero-sum games. To illustrate these findings, an illustrative example is provided.

Exact reachability of discrete‐time stochastic system with multiplicative noise and partial information

AbstractThis paper is concerned with the exact reachability of discrete‐time stochastic system with partial information. The main contribution is to obtain the necessary and sufficient Gramian matrix criterion for the exact reachability of stochastic systems with partial information. The key is to innovatively transform the stochastic system governed by forward stochastic difference equation into an equivalent system governed by backward stochastic difference equation. Simulation examples illustrate the effectiveness of the results.

LQ control of system governed by backward stochastic difference equations and applications

This paper investigates the linear quadratic (LQ) optimal control of system governed by backward stochastic difference equations. In particular, multiple multiplicative noises are involved in the stochastic system, which leads to the situation that the controller uses partial information to complete optimisation. The main contribution is to give the explicitly optimal controller based on introduced Riccati equation. The key technique is to solve the corresponding forward–backward stochastic difference equations by establishing the non-homogeneous relationship, which is completely different from the homogeneous relationship in standard optimal control of system governed by forward stochastic difference equations. Moreover, the derived results are applied to stochastic optimal control problem with terminal constraints as applications.

Scalar Auxiliary Variable (SAV) Stabilization of Implicit‐Explicit (IMEX) Time Integration Schemes for Non‐Linear Structural Dynamics

ABSTRACTImplicit‐explicit (IMEX) time integration schemes are well suited for non‐linear structural dynamics because of their low computational cost and high accuracy. However, the stability of IMEX schemes cannot be guaranteed for general non‐linear problems. In this article, we present a scalar auxiliary variable (SAV) stabilization of high‐order IMEX time integration schemes that leads to unconditional stability. The proposed IMEX‐BDFk‐SAV schemes treat linear terms implicitly using kth‐order backward difference formulas (BDFk) and non‐linear terms explicitly. This eliminates the need for iterations in non‐linear problems and leads to low computational costs. Truncation error analysis of the proposed IMEX‐BDFk‐SAV schemes confirms that up to kth‐order accuracy can be achieved and this is verified through a series of convergence tests. Unlike existing SAV schemes for first‐order ordinary differential equations (ODEs), we introduce a novel SAV for the proposed schemes that allows direct solution of the second‐order ODEs without transforming them into a system of first‐order ODEs. Finally, we demonstrate the performance of the proposed schemes by solving several non‐linear problems in structural dynamics and show that the proposed schemes can achieve high accuracy at a low computational cost while maintaining unconditional stability.

New approaches to numerical differentiation for second and third order

In this article, new numerical methods for calculation of second and third order derivatives are designed by using basic finite difference methods; forward, central and backward finite difference approaches. Those approaches are originally derived from the well-known Taylor series. Main advantage of new numerical formulas (named as Improved Backward Finite Difference Method, Improved Forward Finite Difference Method) is that they produce more accurate numerical results with smaller step size than the well-known backward and forward finite difference methods. For this purpose, some numerical examples are given to compare these new formulas with the traditional finite difference methods; backward and forward. The performance of the new methods in terms of error analysis and elapsed time for both second and third order derivative computations is also presented.

Exploring the Dynamics of Inflation, Interest Rates, and Bond Yields: A Comprehensive Regression Model Comparison

This research presents a comprehensive analysis of the impact of key macroeconomic indicators, including 10- year Government Securities (10Y GSec) yields, 91-day Treasury Bill (TB) rates, interest rates, inflation, exchange rates, foreign reserves, gold prices, equity market indices (NSE Nifty), FDI, and Month-on-Month (MoM) basis returns, on interest rates, inflation, yields and vice-versa in India, using data from 2018 to 2023. The study employs multivariate correlation analysis to identify the relationships among these variables, revealing significant patterns such as the strong correlation between bond yields and interest rates, as well as the inverse relationship between equity market performance and bond yields. This research highlights the influence of monetary policy, external reserves, inflation, FDI, and equity markets in shaping bond yields and interest rates, with gold and equity markets acting as safe-haven assets during times of economic uncertainty. A key focus of this research is a detailed comparison of five regression models— Ordinary Least Squares (OLS), Heteroscedasticity-Corrected (HSC), Tobit, Logistic, and the Transformed First Difference (FD) OLS—used to analyze the relationships among these macroeconomic variables. The study demonstrates that the HSC and Logistic regressions provide the most robust and reliable insights, with the HSC model exhibiting the highest explanatory power in capturing the variance in bond yields, interest rates, and inflation. The analysis underscores the importance of selecting the right regression model to accurately capture the dynamics of financial markets, as model performance varies significantly. This comprehensive study offers a nuanced understanding of the interplay between macroeconomic indicators and financial outcomes, emphasizing the critical role of regression models in enhancing the accuracy of economic forecasting.

Exploring The Dynamics Of Inflation, Interest Rates, And Bond Yields: A Comprehensive Regression Model Comparison

This research presents a comprehensive analysis of the impact of key macroeconomic indicators, including 10- year Government Securities (10Y GSec) yields, 91-day Treasury Bill (TB) rates, interest rates, inflation, exchange rates, foreign reserves, gold prices, equity market indices (NSE Nifty), FDI, and Month-on-Month (MoM) basis returns, on interest rates, inflation, yields and vice-versa in India, using data from 2018 to 2023. The study employs multivariate correlation analysis to identify the relationships among these variables, revealing significant patterns such as the strong correlation between bond yields and interest rates, as well as the inverse relationship between equity market performance and bond yields. This research highlights the influence of monetary policy, external reserves, inflation, FDI, and equity markets in shaping bond yields and interest rates, with gold and equity markets acting as safe-haven assets during times of economic uncertainty. A key focus of this research is a detailed comparison of five regression models— Ordinary Least Squares (OLS), Heteroscedasticity-Corrected (HSC), Tobit, Logistic, and the Transformed First Difference (FD) OLS—used to analyze the relationships among these macroeconomic variables. The study demonstrates that the HSC and Logistic regressions provide the most robust and reliable insights, with the HSC model exhibiting the highest explanatory power in capturing the variance in bond yields, interest rates, and inflation. The analysis underscores the importance of selecting the right regression model to accurately capture the dynamics of financial markets, as model performance varies significantly. This comprehensive study offers a nuanced understanding of the interplay between macroeconomic indicators and financial outcomes, emphasizing the critical role of regression models in enhancing the accuracy of economic forecasting

Analogs of Recurrent Newton Formulas for Systems of Transcendental Equations

Analogs of Recurrent Newton Formulas for Systems of Transcendental Equations

Optimal convergence analysis of second‐order time‐discrete stabilized scheme for the thermally coupled incompressible MHD system

AbstractIn this paper, we construct an optimal convergence analysis of second‐order stabilized scheme for the thermally coupled incompressible magnetohydrodynamic (MHD) system. First, we construct first‐ and second‐order time‐discrete schemes in which the time derivative term is treated by the first‐order backward Euler method and the second‐order backward difference formulation, respectively. And the nonlinear terms are treated by semi‐implicit method. Importantly, we use the Gauge–Uzawa method to decouple velocity and pressure. The proposed schemes have the following two distinct features: they do not need to give an initial value of the pressure, and they do not require artificial boundary conditions on the pressure. Second, the unconditional energy stability of the two schemes is proved. Then, through rigorous error analysis, we provide optimal convergence orders for all unknowns. Finally, some numerical experiments demonstrate the accuracy and effectiveness of the proposed schemes.

Ocular biometric parameters in Chinese preschool children and physiological axial length growth prediction using machine learning algorithms: a retrospective cross-sectional study

ObjectivesTo examine the ocular biometric parameters and predict the annual growth rate of the physiological axial length (AL) in Chinese preschool children aged 4–6 years old.MethodsThis retrospective cross-sectional study included...

Model of sensitivity to the initial conditions of the equations of motion in the electric fieldn

A numerical method for calculating ste¬ady periodic processes of nonlinear differential equations of motion in an electric field is proposed based on their integration under initial conditions that exclude transi¬ent reaction. The calculation of such initial conditions is based on Newton's iterative formula. At each iteration, together with the basic equations of motion, the corresponding equations of the first variation are integrated over the periodicity time interval, which represent a model of sensitivity to the initial conditions, In addi¬tion, the number of unknown variational equations has been reduced, which makes it possible to lower their or¬der, and, at the same time, the order of the monodromy matrix. All this contributes to the simplification of the relevant computational algorithms and computer programs. The method takes into account the finite rate of electric field propagation.

Data-driven tracking control for discrete-time nonlinear systems: a backward difference iterative learning approach

This article investigates the trajectory tracking problem of repeatable discrete-time nonlinear systems with unknown dynamics. The conventional paradigm in this field is to construct an iterative learning controller with the approximately identified model, considering only tracking error information at the current iteration. This leads to the neglect of the error variation trend along historical iterations. To circumvent unmodeled dynamics and capitalise on multi-step iteration error information, a backward difference iterative learning control (BD-ILC) approach is formulated. Initially, a vector data transfer mechanism employing the backward difference iteration error is introduced based on the dynamic data mapping. Within this mechanism, the proportional, differential and quadratic differential control terms are firstly incorporated simultaneously along the iteration domain. Furthermore, a convergence analysis approach is developed using the error deflation spanning tree technique. Simulations validate the enhanced efficacy of the BD-ILC approach, surpassing that of distinct control approaches.

An Enhanced and Highly Efficient Semi‐Implicit Combined Lagrange Multiplier Approach Preserving Original Energy Law for Dissipative Systems

ABSTRACTRecently, a new Lagrange multiplier approach was introduced by Cheng, Liu, and Shen, which has been broadly used to solve various challenging phase field problems. To design original energy‐stable schemes, they have to solve a nonlinear algebraic equation to determine the introduced Lagrange multiplier, which can be computationally expensive, especially for large‐scale and long‐time simulations involving complex nonlinear terms. In this article, we propose an essential improved technique to modify this issue, which can be seen as a semi‐implicit combined Lagrange multiplier approach. In general, the newly constructed schemes keep all the advantages of the Lagrange multiplier method and significantly reduce the computation costs. Besides, the new proposed second‐order backward difference formula (BDF2) scheme dissipates the original energy, as opposed to a modified energy for the classic Lagrange multiplier approach. In addition, we establish a general framework for extending our constructed method to dissipative systems. Finally, several examples have been presented to demonstrate the effectiveness of the proposed approach.

Two linearized difference schemes on graded meshes for the time-space fractional nonlinear diffusion-wave equation with an initial singularity

Abstract In this article, two linearized difference schemes are considered to solve a class of time-space fractional nonlinear diffusion-wave equations with weak singularity near the initial time based on its equivalent integro-differential equation. The Crank-Nicolson method combined with the trapezoidal product integration rule with graded meshes is implemented to approximate the Riemann-Liouville integral. The fractional central difference formula and a novel compact difference method are used to construct fully discrete difference schemes. Using the energy method, the stability and convergence of the proposed schemes are demonstrated in terms of the L 2 norm with the accuracy of O N − min { r σ , 2 } + M − 2 and O N − min { r σ , 2 } + M − 4 , respectively. The theoretical analysis is complemented by numerical results.

Finite-Horizon H∞ State Estimation for Complex Networks With Uncertain Couplings and Packet Losses: Handling Amplify-and-Forward Relays.

This article is concerned with the state estimation problem for a class of complex networks (CNs) with uncertain inner couplings and packet losses over communication networks. The inner couplings are allowed to be uncertain and varying in a specific interval. The amplify-and-forward (AaF) relay protocols are introduced to improve the communication quality and enhance the propagation distance. The Bernoulli random variables are used to characterize the randomly occurring packet losses encountered in communication channels. The focus of this article is on the design of a state estimator for each node of CNs such that a prescribed H∞ performance constraint is satisfied for the dynamical error system over a finite horizon. A sufficient condition is first provided to verify the existence of the desired H∞ state estimator, and the estimator gain is then determined by solving two coupled backward Riccati difference equations (RDEs). Subsequently, a recursive state estimation algorithm is put forward that is suitable for online computation. Finally, a numerical example is given to demonstrate the effectiveness of the proposed estimation method.

Optimizing of selective catalytic reduction urea injection and NOx conversion analysis of diesel engine based on higher-order physical model and sequential quadratic programming algorithm

In order to maximize the NOx conversion rate and minimize NH3 slip in the diesel selective catalytic reduction (SCR) system, first, the high-order SCR model was simplified and solved using the trapezoidal method and second-order backward difference formula, while the chemical reaction parameters were optimized using the Gauss-Newton method. The results of the engine bench test show that the mean absolute errors of NH3 and NOx concentrations under steady-state operating conditions were 2.67 ppm and 1.72 ppm, respectively, and 5.46 ppm and 42.50 ppm under severe operating conditions. Second, urea injection rate is optimized based on a high-order model combined with the sequential quadratic programming (SQP) algorithm. The change in NOx conversion rate at different temperatures and air speeds is analyzed, and the results show that the NOx conversion is best at 650 K and 29,000 h⁻1. Third, the experiment validation show that the optimal NH3/NOx ratio is 0.69 at 554 K and 54,200 h⁻1; 0.8; and 1.37 at 746 K and 116,700 h⁻1. Finally, this paper simulates and analyzes the SCR emission characteristics at different temperatures and air speeds when the upstream NOx concentration is 1,000 ppm.

A novel structure adaptive new information priority grey Bernoulli model and its application in China's renewable energy production

At present, the global energy structure is undergoing major changes. China is in the transition period of energy structure. Accurately anticipating future energy trends is critical for China's energy structure and modernization. Considering the uncertainty and sparsity characteristics of China's energy system sequence, this study examines China's renewable energy scenario using the grey model with limited sample and uncertain system modelling features. Renewable energy is affected by a variety of uncertain factors, exhibiting a range of complicated traits including nonlinearity, periodicity and random volatility. The traditional grey model has been difficult to appropriately predict its future evolution. This paper focuses on the adaptability of the model, optimizes and improves the accumulation operator and model structure, and establishes a fractional-order structural self-adaptation grey Bernoulli model based on new information priority. Firstly, based on the new information priority accumulation operator, it is extended to the fractional order. In terms of model structure, combined with NGBM(1,1) model, SADGM(1,1) model and FPGM(1,1) model, periodic fluctuation term and nonlinear power term are included to improve the model's capacity to capture nonlinear, fluctuating and periodic features, and enhance the adaptability and flexibility of the model. The backward difference technique yields the model's parameter estimation and temporal response sequence. Based on the results of the algorithm comparison experiment, the Improved Grey Wolf Optimization Algorithm is chosen to optimize the structural parameters of the model in an effort to enhance its performance. The performance comparison experiment of the model was designed, and three cases of China's hydropower generation, China's renewable energy power generation installed capacity and China's solar energy quarterly power generation were selected to compare the performance with a variety of grey prediction models. Monte-Carlo simulation and probability density analysis were utilized to confirm the stability and accuracy of the proposed model. The results show that the proposed FSANGBM(1,1) model can handle data series of renewable energy with nonlinear, volatile, and periodic features with high prediction ability. Finally, the model is applied to forecast three cases' future development trends.

Decision-Making of a New Type of Stochastic Space and Its Associated Operator Ideal

We develop and examine the pre-modular space of null variable exponent-weighted backward generalized difference gai sequences of fuzzy functions in this paper. These sequences of fuzzy functions are important contributions to the concept of modular spaces because they have exponent weighting. Using extended s−fuzzy functions as well as this sequence space of fuzzy functions, it has been possible to accomplish an idealization of the mappings. We have presented some topological and geometric properties of this new space, as well as the ideal mappings that correspond to them.

New implicit time integration schemes for structural dynamics combining high frequency damping and high second order accuracy

The time integration schemes, GA-23 and GA-234, recently proposed by the authors for first order problems, are extended to solve second-order problems in structural dynamics. The resulting methods maintain unconditional stability and user-controlled high-frequency damping. They offer improved accuracy and exhibit less numerical damping in the low-frequency regime, outperforming the well-known generalised-α method. When the high-frequency damping is maximised the new schemes can be cast in the format of backward difference formulae, offering more accurate alternatives to the standard second order formula. The effectiveness of the new time integration schemes is validated through a number of numerical examples, including a linear elastic cantilever beam, a nonlinear spring pendulum, and wave propagation on a string.