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There is a need for further studies on students' learning of Differential Equations (DEs), especially in advanced undergraduate and graduate courses. Research on the mathematical education of engineers shows a conflict between students' demands for practical, contextualized pedagogies and the need for abstract reasoning and appropriate use of mathematical results. Few papers focus on engineering students' interpretation of theorems and their use as tools in argumentation and problem-solving. This paper takes a sociocultural stance on learning and employs dialogical inquiry – a methodology rooted in Bakhtinian theory, newly developed for collaborative inquiry and qualitative data analysis – to investigate the meanings that senior engineering students made while working on a task designed to evaluate their understanding of Existence and Uniqueness Theorems (EUTs) of solutions of DEs. We identified two important epistemological disconnections that explain the difficulties that some of our students faced in making meaning of solutions of DEs and the EUT. • There is a need for more studies on the learning of Differential Equations (DEs). • Few research papers focus on students' meaning-making of theoretical results in DEs. • Dialogical inquiry methodology was used to analyze students' meaning-making processes. • Two epistemological disconnections were found that explain students' difficulties. • The use of warrants in students' dialogue was important in their meaning-making. [ABSTRACT FROM AUTHOR]

Abstract: Two players of Rock-Paper-Scissors are modeled as adaptive agents which use a reinforcement learning algorithm and exhibit chaotic behavior in terms of trajectories of probability in mixed strategies space. This paper demonstrates that an external super-agent can exploit the behavior of the other players to predict favorable moments to play against one of the other players the symbol suggested by a sub-optimal strategy. This third agent does not affect the learning process of the other two players, whose only goal is to beat each other. The choice of the best moment to play is based on a threshold associated with the Local Lyapunov Exponent or the Entropy, each computed by using the time series of symbols played by one of the other players. A method for automatically adapting such a threshold is presented and evaluated. The results show that these techniques can be used effectively by a super-agent in a game involving adaptive agents that exhibit collective chaotic behavior. [Copyright &y& Elsevier]

Abstract Our reply on the above comment [1] shows that the new mapping method which proposed by Zeng and Yong [2] is absolutely right. Consequently, the solutions of the nonlinear evolution equations obtained in Zayed and Al-Nowehy [3,4] are true. Unfortunately, the author of this comment [1] failed in using this method and wrote a wrong comment. [ABSTRACT FROM AUTHOR]

We assessed 6122 environmental modelling papers published since 2005 to determine whether the number of citations each paper had received by September 2014 could be predicted with no knowledge of the paper's quality. A random forest was applied, using a range of easily quantified or classified variables as predictors. The 511 papers published in two key journals in 2008 were further analysed to consider additional variables. Papers with no differential equations received more citations. The topic of the paper, number of authors and publication venue were also significant. Ten other factors, some of which have been found significant in other studies, were also considered, but most added little to the predictive power of the models. Collectively, all factors predicted 16–29% of the variation in citation counts, with the remaining variance (the majority) presumably attributable to important subjective factors such as paper quality, clarity and timeliness. [ABSTRACT FROM AUTHOR]

Abstract: In this paper we will establish some new oscillation criteria for the second-order retarded differential equation of the formThe results obtained essentially improve and extend those of Džurina and Stavroulakis [Oscillation criteria for second-order delay differential equations, Appl. Math. Comput., 140 (2003) 445–453]. An open problem is proposed at the end of this paper. [Copyright &y& Elsevier]

Abstract: This paper considers certain neutral differential equation with the distributed delayWithout imposing a nonnegative restriction on coefficients of the equation, we establish a delay-dependent asymptotic stability condition for the equation by introducing a new Lyapunov function. The condition presented here is different from that of Park [J.H. Park, LMI optimization approach to asymptotic stability of certain neutral delay differential equation with time-varying coefficients, Appl. Math. Comput. 160 (2005) 355–361] in the sense that it is dependent on all delays of the equation. Numerical examples illustrate effectiveness and sharpness of our theoretical result. [Copyright &y& Elsevier]

Abstract: In this technical note, we give a short extension application for nonlinear sum of ratios problem (P) considered in [Y.J. Wang, K.C. Zhang, Global optimization of nonlinear sum of ratios problem, Applied Mathematics and Computation 158 (2004) 319–330]. Actually our result is slightly more general, since we do not specify additional positive coefficient for each ratio. In this note, we use different equivalent problem as done in Wang and Zhang (2004). Our method introduce p variables less than other method (Wang and Zhang, 2004), and our approach need not additional special program to obtain the upper and lower bound of numerator and denominator for each ratio in the objective function. [Copyright &y& Elsevier]

The present comment concerns some doubtful results included in the above papers. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the problem of source recovery in a dynamical system utilizing space-time samples. This is a specific issue within the broader field of dynamical sampling, which involves collecting samples from solutions to a differential equation across both space and time with the aim of recovering critical data, such as initial values, the sources, the driving operator, or other relevant details. Our focus in this study is the recovery of unknown, stationary sources across both space and time, leveraging space-time samples. This research may have significant applications; for instance, it could provide a model for strategically placing devices to measure the number of pollutants emanating from factory smokestacks and dispersing across a specific area. Space-time samples could be collected using measuring devices placed at various spatial locations and activated at different times. We present necessary and sufficient conditions for the positioning of these measuring devices to successfully resolve this dynamical sampling problem. This paper provides both a theoretical foundation for the recovery of sources in dynamical systems and potential practical applications. [ABSTRACT FROM AUTHOR]

This paper employs a numerical approach to investigate the impact of liquidity shocks on European options in modeling markets. To accurately capture the behavior of European options under liquidity shocks, a coupled system of differential equations is employed, consisting of a degenerate parabolic equation and a diffusion-free equation. The primary focus is on developing and analyzing implicit-explicit methods for numerically simulating European option pricing, specifically considering the presence of liquidity shocks while ensuring the positivity of the solution. The paper also includes convergence analysis and establishes the discrete comparison principle for the developed methods. Numerical experiments are conducted using both uniform and nonuniform meshes to validate the theoretical findings, demonstrating the efficiency and accuracy of the proposed methods. [ABSTRACT FROM AUTHOR]

Abstract: This Letter comments on two papers of chaotic synchronization, namely [Phys. Rev. Lett. 76 (1996) 1232] and [Phys. Lett. A 321 (2004) 50]. We find that some statements in the two papers are incorrect by numerical simulations. The consequence of the incorrectness is analyzed as well. [Copyright &y& Elsevier]

Boundary value problems (BVPs) in ordinary differential equations (ODEs) with singularities arise in numerous mathematical models describing real-life phenomena in natural sciences and engineering. This motivates vivid research activities aiming to characterize the analytical properties of singular problems, to investigate convergence of the standard numerical methods when they are applied to simulate differential equation with singularities, and to provide software for their efficient numerical treatment. There are two well-known, high order numerical methods which we focus on in this paper, the finite difference schemes and the collocation methods. Those methods proved to be dependable and highly accurate in the context of regular differential equations, so the question arises how do they preform for singular problems. While, there is a strong evidence for the collocation schemes to be a robust method to solve singular systems in a stable and efficient way, finite difference schemes are still considered less suitable for this problem class. In this paper, we shall compare the performance of the code HOFiD_bvp based on the high order finite difference schemes and bvpsuite2.0 based on polynomial collocation, when the codes are applied to singular problems in ODEs. We are fully aware of the difficulties in a code comparison, so in this paper, we will try to only diagnose the potential improvements, we could address in the next update of the codes. [ABSTRACT FROM AUTHOR]

On the occasion of the centennial of his mentor Alvin Hansen, Paul Samuelson published in 1988 a modified version of his seminal 1939 multiplier-accelerator model to address aspects of Hansen's secular stagnation hypothesis. The "Keynes-Hansen-Samuelson" model (or KHS, as he called it) was built to analyse the effects of population growth on the economy's trajectory. Several changes were then made. Instead of difference equations and a tight accelerator, as in his 1939 model, Samuelson deployed differential equations and a flexible accelerator to produce a nonlinear limit cycle. Despite Samuelson's strong claims for the analytical contributions of his 1988 paper, it has – in contrast with the 1939 model – received only scant attention by macroeconomists and historians of economics alike. Samuelson's 1988 paper was his last published macroeconomic model, based on his long-established tradition of non-optimising macro-dynamics. Our paper provides a close reading of that article and some analytical results that shed new light on the formal aspects of Samuelson's 1988 model. We also discuss how it historically links up with business cycle models advanced by John Hicks, Nicholas Kaldor, Roy Harrod and Richard Goodwin and examine how far Samuelson's use of the term secular stagnation differs from Larry Summers's recent reconstruction of it. [ABSTRACT FROM AUTHOR]

A tracking control scheme is proposed for complex dynamic network (CDN), where the CDN is regarded wholly as a dynamic composite system which consists of two mutually coupled subsystems. One subsystem consists of all nodes and the other consists of all links, and consider the weights of the links to be state variables in the latter subsystem. There are two parts in the structure of the proposed tracking control scheme: the designed controller of nodes and the synthesis of the coupling term in links. These two parts can guarantee two subsystems to asymptotically track the given reference targets, respectively. This approach originates from the communication transmission network (CTN). In CTN, for the network optimization management, the reference network topology (NT), such as the star topology, is required as a target for tracking in communication transmission (links) when each node in network needs to track its own target. The control scheme provided of this paper coincides with above requirement. And finally, a comparative simulation example is given for illustrating the effectiveness of the provided control scheme. • The dynamic equations of nodes and NT are represented by the matrix differential equations, this can contain many models in other papers and simplify the control scheme, which is rarely discussed. • The tracking targets of nodes and NT given in this paper are independent of each other, which is novel. • When all nodes track the reference targets, the eventual network layout is shown by the desired NT, which is novel. [ABSTRACT FROM AUTHOR]

Abstract: In recent years, many results on oscillation criteria of second-order nonlinear neutral delay differential equations with distributed deviating argument have been obtained, but some of these criteria are incorrect due to a mistake in a crucial auxiliary result. We give two counterexamples. [Copyright &y& Elsevier]

This paper proposes a fractional-order time-varying sliding mode control method with predefined-time convergence for a class of arbitrary-order nonlinear control systems with compound disturbances. The method has global robustness and strongly predefined-time stability. All state errors of the system can converge to zero at a desired time, which can be set arbitrarily with a simple parameter. The strongly predefined-time convergence of the system is clearly demonstrated by the analytic expression of state error, which is obtained by solving fractional-order differential equations corresponding to the sliding mode function. The simulation results show that the proposed method still has good control performance in the presence of input saturation and external interference. • The main contributions of this paper are summarized as follows: • A predefined-time fractional-order time-varying sliding mode control scheme is proposed for a class of arbitrary order nonlinear control systems with complex disturbances. All state errors are simultaneously converged to zero by the scheme at a time, which can be presented by a simple parameter. • Different from most of the existing methods, this paper obtains the analytical expression of the state errors by solving the fractional-order differential equation to prove the predefined-time convergence of the system. • The fractional-order term and time-varying term are introduced into the proposed control scheme, which improves the control performance and makes the system have global robustness. The simulation results show that the proposed controller has strong robustness and resistance to input saturation. [ABSTRACT FROM AUTHOR]

This paper focuses on a reduction technique to discover exact solutions for the generalized Camassa-Choi equation with temporal local M-derivative. The paper presents various types of exact solutions along with their corresponding first integrals. Furthermore, the interactions between the orders of α and β in the M-derivative are taken into account and depicted graphically for the derived solutions. Remarkably, the paper demonstrates that in certain situations, exact solutions can be obtained for any value of n, which holds significant mathematical intrigue. The authors note that Nucci's reduction technique has not previously been employed for differential equations with M-derivative, to the best of their knowledge. [ABSTRACT FROM AUTHOR]

In this paper, novel energy-preserving finite difference schemes are presented and analyzed for one-dimensional and two-dimensional nonlinear wave equations with variable coefficients, respectively. Initially, the original differential equations are reformulated as corresponding Hamiltonian systems. Subsequently, the semi-discrete schemes are constructed using the sixth-order finite difference formula. The discrete energy method is employed to rigorously prove the energy conservation laws, convergence, and stability of the solutions for the spatial semi-discrete scheme. Next, the time derivatives in the semi-discrete schemes are discretized with the third-order strong stability preserving Runge-Kutta method, then the local truncation error of the finite difference scheme is provided. The accuracy and effectiveness of the constructed schemes, along with the theoretical analysis, are demonstrated through numerical experiments. [ABSTRACT FROM AUTHOR]

In this paper, we propose a novel methodology for studying the dynamics of epidemic spread, focusing on the utilization of fundamental mathematical concepts related to piecewise differential and integral operators. These mathematical tools play a crucial role in the process of modeling epidemic phenomena, enabling us to investigate the behavior of infectious diseases within defined time intervals. Our primary objective is to enhance our understanding of epidemic dynamics and the underlying influencing factors. We introduce the Susceptible–Infectious–Recovered (SIR) model as the foundational framework, which is formulated as a system of differential equations. Our approach involves discretizing time and employing interpolation techniques for integrals, specifically utilizing the collocation method with Bernoulli wavelets. By incorporating piecewise derivatives, we are able to conduct comprehensive simulations and analyses of epidemic spread under various intervention strategies, including social distancing measures. The outcomes of our numerical simulations serve to validate the efficacy of our proposed methodology, offering valuable insights into the intricate dynamics of real-world epidemic scenarios. This contribution significantly advances the field of epidemic control optimization, providing an integrated framework that seamlessly integrates fractional calculus, piecewise differential derivatives, and the capabilities of wavelets. Our findings provide crucial guidance for policymakers and healthcare leaders, offering a deeper understanding of the effectiveness of different control strategies. By considering our innovative approach, we can better inform and shape epidemic control measures, ultimately enhancing public health and fortifying our defenses against infectious diseases. [ABSTRACT FROM AUTHOR]

This article presents a novel approach using the Fibonacci wavelet collocation method (FWCM) for the numerical solution of Emden-Fowler-type equations. The Emden-Fowler equations are a class of nonlinear differential equations that arise in various fields of science and engineering, particularly in astrophysics and fluid dynamics. Due to their nonlinear and singular nature, the conventional approaches to solving these equations encounter difficulties. This method is particularly effective in handling problems with singularities, as it adapts naturally to the local behaviour of the solution. Here, we introduced the Fibonacci wavelet collocation method as a powerful numerical technique for tackling Emden-Fowler-type equations. The Fibonacci wavelet basis functions possess remarkable properties, including compact support, making them well-suited for approximating solutions to differential equations. The main advantage of this approach lies in its ability to reduce the computational complexity associated with solving Emden-Fowler equations, resulting in accurate and efficient solutions. Comparative analyses with other established numerical methods reveal its superior accuracy and convergence rate performance. Further, several examples demonstrate the method's flexibility when dealing with different singularity levels. This paper contributes to numerical analysis by introducing the Fibonacci wavelet method as a robust tool for solving Emden-Fowler-type equations. [ABSTRACT FROM AUTHOR]

• This paper presents a drained expansion solution of a cylindrical cavity with fabric change effect in the sand. • The problem is formulated as a system of first-order differential equations with the known variables. • A user-defined SANISAND model is implemented as a subroutine, which is applied to FEM simulation. Sandy soil exhibits inherently anisotropy due to its microstructure, also known as 'fabric'. However, the drastic change in fabric observed during the dilatant phase is often overlooked in current cavity expansion research. This paper presents a drained expansion solution of a cylindrical cavity with fabric change effect in the sand using a simple non-associated and anisotropic model, SANISAND. The problem is formulated as a set of first-order differential equations with the unknown variables as functions of an auxiliary coordinate, which can be solved as an initial value problem. A subroutine is implemented into the FEM simulation to verify the proposed method. Additionally, cavity expansion solutions in sand are compared with those based on state-dependent dilatancy. The anisotropic fabric of the sand is studied to investigate the impact of the initial void ratio, initial mean stress, and at-rest coefficient on the void ratio path, stress distributions and paths, and bounding surfaces. The proposed solution provides a framework for the potential use of cavity expansion in sand, considering the fabric change effect, and a benchmark for further developments and numerical calculations by using the SANISAND model. [ABSTRACT FROM AUTHOR]

This paper is about the relationship between R.A. Fisher's fundamental theorem of natural selection (FTNS) and the two major pieces that Fisher wrote on indeterminism, "Indeterminism and Natural Selection" (1934) and Creative Aspects of Natural Law (1950). I argue that the FTNS presents a picture of natural selection that is interestingly different from what we find in these two indeterminism pieces, pace some recent work that advances the opposite conclusion. I also identify as the source of this difference both the mathematical form of the FTNS (i.e., a differential equation) and Fisher's meta-scientific commitment to advancing "general" claims about evolution. [ABSTRACT FROM AUTHOR]

The mealybug (Paracoccus marginatus) is one of the most important pests of papaya (Carica papaya L.). The high potential damage of this pest threatens papaya production. The objective of this paper is to explore the impact of biocontrol and temperature variations on the population dynamics of Paracoccus marginatus (P. marginatus). We first propose and study a mathematical model for the dynamics of P. marginatus within a papaya field. This model consists of a time-delayed non-autonomous system of differential equations that takes into account the egg incubation period, temperature variations, plant compensation and predators that are natural enemies of the papaya pest. We provide the basic properties of the model, compute the basic offspring number N 0 and show that there exists two threshold parameters N _ 0 and ξ 0 such that N _ 0 ≤ N 0 ≤ ξ 0 ≤ 1 and N 0 ≥ N _ 0 > 1 provide conditions for the extinction and persistence of the pest within the papaya field, respectively. After, we investigate the impact of biocontrol on the pest population. We found that the use of a control strategy based on constant releases of predators could reduce the pest population by 35.62 percent and increase the biomass by 77.50 percent, while the use of impulsive releases of predators result in a decrease of 96.31 percent of the pest population and an increases of 55.74 percent of the biomass. We also combine the use of continuous predator releases and bio-pesticides and discovered that a spray applied 90 days after germination could completely eliminate the pest population and boost the biomass by 81.58 percent. • The population dynamics of Paracoccus marginatus , papaya pest is modeled. • The model includes the egg incubation period, temperature variations and biocontrol. • The warm temperatures are favorable to the development and survival of the pest. • The use of impulsive releases of predators can reduce the pest by 96.31%. • The use of continuous releases of predators and bio-pesticides can eradicate the pest. [ABSTRACT FROM AUTHOR]

Energy process system positively contributes to the energy utilization efficiency, the energy complement, and the construction of a low-carbon sustainable energy system. The multiple energy subsystems are deep interdependent, therefore, significant operational risks exist in the energy process system. To avoid system risk and fulfill operation requirement, we propose the Safety-Index based model predictive control that coordinates the controller design and system safety in this work. The innovation of the method is to integrate the state-space model of process system with advanced controller design while accounting for system safety. Since the state-space model of the energy process system is described by coupled partial differential equations and ordinary differential equations, which leads to difficulties in modeling and solving, the main contribution of this work is to simplify the constrained optimization problem formed by model predictive controller. The paper addresses the compressed air energy storage system as case study. From the numerical simulations of the safety controller performance, it shows that the system safety can be guaranteed by control strategy which realizes the system operation target and reject the system external disturbances, which caused by environmental or operation change. • The paper addresses the modeling and dynamic safety control of compressed air energy storage system. • A control loop for safety operation that consists of controllers, system models, system constraints, and safety index is proposed. [ABSTRACT FROM AUTHOR]

In this paper, we propose linearly implicit and arbitrary high-order conservative numerical schemes for ordinary differential equations with a quadratic invariant. Many differential equations have invariants, and numerical schemes for preserving them have been extensively studied. Since linear invariants can be easily kept after discretisation, quadratic invariants are essentially the simplest ones. Quadratic invariants are important objects that appear not only in many physical examples but also in the computationally efficient conservative schemes for general invariants such as scalar auxiliary variable approach, which have been studied in recent years. It is known that quadratic invariants can be maintained relatively easily compared with general invariants, and can be preserved by canonical Runge–Kutta methods. However, there is no unified method for constructing linearly implicit and high order conservative schemes. In this paper, we construct such schemes based on canonical Runge–Kutta methods and prove some properties involving accuracy. [ABSTRACT FROM AUTHOR]

A simplified inverter–motor stator system model is presented in the paper. The model is based on elementary dynamic blocks described by differential equations. The paper presents the results of research aimed at verifying the model on different devices with different types of inverters and motors. Model accuracy coefficients have been developed for all examined cases, confirming the high conformity of the model to the measured speeds and torques of asynchronous motors. The model was also used to simulate the operation of a drive system with a synchronous motor, obtaining high compliance of speed and torque mapping. The presented cases of verification of a simplified model confirm the possibility of using it to model different devices fed by frequency inverters, controlled both manually and automatically. The presented model, using simple description, ensures high accuracy of speed and torque mapping. It should be emphasized that most parameters in the presented model are calculated for motors and inverters based on data sheets. • The simplified model of the system: inverter–motor stator is presented. • Experimental verification confirming good mapping of speed and torques of motor. • Verification performed on different types of machines, motors and inverters. • Model can be used to modeling both synchronous and asynchronous motors. • Inputting only parameters from data sheets of motors and inverters. [ABSTRACT FROM AUTHOR]

In this paper, we address the integration problem of the isomonodromic system of quantum differential equations (qDE s) associated with the quantum cohomology of P 1 -bundles on Fano varieties. It is shown that bases of solutions of the qDE associated with the total space of the P 1 -bundle can be reconstructed from the datum of bases of solutions of the qDE associated with the base space. This represents a quantum analog of the classical Leray–Hirsch theorem in the context of the isomonodromic approach to quantum cohomology. The reconstruction procedure of the solutions can be performed in terms of some integral transforms, introduced in [17] , called Borel (α , β) -multitransforms. We emphasize the emergence, in the explicit integral formulas, of an interesting sequence of special functions (closely related to iterated partial derivatives of the Böhmer–Tricomi incomplete Gamma function) as integral kernels. Remarkably, these integral kernels have a universal feature, being independent of the specifically chosen P 1 -bundle. When applied to projective bundles on products of projective spaces, our results give Mellin–Barnes integral representations of solutions of qDE s. As an example, we show how to integrate the qDE of blow-up of P 2 at one point via Borel multitransforms of solutions of the qDE of P 1. [ABSTRACT FROM AUTHOR]

The Transformer has emerged as the predominant model in Natural Language Processing due to its exceptional performance in various sequence modeling tasks, particularly in handling long-term dependencies. However, the traditional absolute and relative position encoding methods, which do not learn from data, tend to ignore the inherent structure of natural language sequences due to the position embedding layer at the input end of the Transformer model. This paper introduces a novel learnable neural Ordinary Differential Equation Position Encoding (ODEPE) method that can implicitly capture the natural position relationships within a sequence without requiring additional position embeddings. ODEPE can model continuous sequences and leverage differential equations to simulate the evolution of position information along the sequence, enabling position information to flow seamlessly between sequences. Additionally, a highly effective recurrent attention framework is proposed, which hybridizes attention with the ODEPE method to improve model performance. Compared to the Transformer-based sequence modeling network, our framework demonstrates a performance improvement of 24.0 points on the WikiText-103 dataset, while also achieving a performance improvement of 1.06 points on the Enwik8 dataset. This corresponds to an improvement of 4.9% and 0.17%, respectively. • Novel ODE method for implicit local position encoding in Transformers. • Capture natural sequence position without extra embeddings. • Recurrent attention network captures longer dependencies. • Our method is adaptable, applicable in long-term language modeling. [ABSTRACT FROM AUTHOR]

Due to the non-homology problem and the simple structural characteristics, a grey prediction model will have defects in modeling. In this paper, the structure of the GM (1 , 1 , x (1) ) model is deformed, and additional parameters are added. A novel four-parameter grey prediction model NFGM(1,1) is established to avoid the non-homology problem. The accumulation order of the NFGM(1,1) model is optimized to enhance its performance. This paper first introduces a nonlinear term and a linear term into the GM (1 , 1 , x (1) ) model to compensate for its structural defects, which can enhance the accuracy of the model in modeling complex modeling sequences. Secondly, a simplified basic formula of the model is proposed to estimate its parameters and iteratively establish the model, which can avoid the problem of non-homologous errors during modeling. Then a novel four-parameter grey prediction model NFGM(1,1) is constructed. Thirdly, the unbiasedness of NFGM(1,1) is proved and verified by matrix theory. Fourthly, by optimizing the order of the NFGM(1,1) model, the model is more flexible and adjustable, and a novel fractional-order four-parameter grey prediction model FNFGM(1,1) can be obtained. Finally, the FNFGM(1,1) model is applied to the prediction of natural gas production in China. The model results show that the FNFGM(1,1) model exhibits superior performance compared to the NFGM(1,1), TWGM(1,1), TDGM(1,1), DGM(1,1), and GM(1,1) models, with the mean relative simulation/prediction/comprehensive percentage errors of 0.92%/1.42%/1.07%, respectively. According to the predicted results, China's natural gas production will reach 3542.9 × 108 m3 in 2027 and some relevant policy recommendations are put forwarded. • On the basis of GM (1 , 1 , x (1) ) , a nonlinear term and a linear term are introduced to expand the model structure, and a novel four-parameter grey prediction model is proposed. • In order to avoid the non-homology problem of the differential equation and difference equation in traditional modeling, this paper uses the simplified basic formula of the model to estimate parameters, and does not directly derive the time response, but directly uses the iterative recurrence method to model. • The new model is proved to be unbiased by the matrix theory method and verified by arithmetic examples. • According to the objective function and constraints of fractional-order solving, PSO algorithm is used to optimize the order of the new model, which makes the model parameters changeable and improves the model performance. • The comprehensive performance of the new model is verified by a case study, the new model is applied to forecast China's natural gas production in the context of Xi Jinping's new era, and the results show that the new model has better accuracy, and the results obtained can help the government to formulate relevant policies. [ABSTRACT FROM AUTHOR]

Different from the existing multiple Mittag-Leffler stability or multiple asymptotic stability, the multiple exponential stability, which has explicit and faster convergence rate, is investigated in this paper for fractional-order impulsive control Cohen–Grossberg neural networks. First, by using the definition of Dirac delta function, the fractional-order control Cohen–Grossberg neural networks are translated into fractional-order impulsive neural networks, in which pulse effect relies on the fractional order of the addressed system. Then, based on maximum norm, 1 − norm and general q − norm (q = 2 n) , a series of novel criteria are obtained respectively to ensure that such n − neuron neural networks can have ∏ i = 1 n (2 L i + 1) total equilibrium points and ∏ i = 1 n (L i + 1) locally exponentially stable equilibrium points, by utilizing the known fixed point theorem, the method of average impulsive interval, the theory of fractional-order differential equations, and the method of Lyapunov function. This paper's investigation reveals the effects of impulsive function, impulsive interval, and fractional order on the dynamic behaviors. Finally, theoretical results are shown to be effective by four illustrative examples. [ABSTRACT FROM AUTHOR]

Abstract: Based on a recent duality theory for linear differential inclusions (LDIs), the condition for stability of an LDI in terms of one Lyapunov function can be easily derived from that in terms of its conjugate function. This paper uses a particular pair of conjugate functions, the convex hull of quadratics and the maximum of quadratics, for the purpose of estimating the domain of attraction for systems with saturation nonlinearities. To this end, the nonlinear system is locally transformed into a parametertized LDI system with an effective approach which enables optimization on the parameter of the LDI along with the optimization of the Lyapunov functions. The optimization problems are derived for both the convex hull and the max functions, and the domain of attraction is estimated with both the convex hull of ellipsoids and the intersection of ellipsoids. A numerical example demonstrates the effectiveness of this paper''s methods. [Copyright &y& Elsevier]

In the field of scientific computation, orthogonal iteration is an essential method for computing the invariant subspace corresponding to the largest r eigenvalues. In this paper, we construct a flow that connects the sequence of matrices generated by the orthogonal iteration. Such a flow is called an orthogonal flow. In addition, we show that the orthogonal iteration forms a time-one mapping of the orthogonal flow. A generalized orthogonal flow is constructed that has the same column space as the orthogonal flow. By using a suitable change of variables, the generalized orthogonal flow can be transformed into the solution of a Riccati differential equation (RDE). Conversely, an RDE can also be transformed into a flow that can be represented by a generalized orthogonal flow. [ABSTRACT FROM AUTHOR]

• Mathematical modelling – generalization of the analytical model of sandwich beam with an asymmetric structure analytical solution. • The individual nonlinear deformation function of a plane beam cross section – analytically developed. • Dimensionless deformation functions are determined based on the classical shear stress formula called the Zhuravsky stress. This paper is devoted to a sandwich beam with an asymmetric structure under uniformly distributed load. The first end of this beam is simply supported, while second end is clamped. The individual nonlinear deformation function of a plane beam cross section – the shear effect with consideration of the classical shear stress formula is analytically developed. Displacements, strains and stresses taking into account the individual nonlinear deformation function for successive layers of this beam are elaborated in detail. The system of two differential equation of equilibrium of this beam, based on the principle of stationary potential energy, is obtained. This system of equations is precisely solved analytically, according to the theory of differential equations. The individual nonlinear deformation functions, shear stresses and deflections of two cases of sample beams are determined in detail: the first with variable core porosity in the direction of the beam depth, the second with constant core porosity – standard sandwich structure. The calculations results specified in Tables and graphically presented in Figures. [ABSTRACT FROM AUTHOR]

The feasibility of utilizing Deep Neural Networks for generating guidance commands in spacecraft landing scenarios has garnered recent research interest. A primary challenge lies in the inefficiency of training sample generation. This paper proposes a new approach to address it by utilizing nonlinear homotopy methods, which involve embedding the homotopic parameter either into the performance index or the right-hand side of the differential equations. Unlike the existing approaches that overlooked the potential value of the intermediate solutions during homotopy process, the proposed method incorporates these discarded solutions as supplementary training data, which treats the homotopic parameter and state as inputs, and the control as the output. Two examples, namely a minimum-fuel lunar landing and a minimum-time asteroid landing, are provided to illustrate the effectiveness and advantages of the proposed method. • New sample generation method for Deep Neural Networks in spacecraft landing guidance. • Fully utilization of intermediate solutions within the nonlinear homotopy process. • Drastic reduction in the number of optimization problems requiring solving. • Validating the proposed method in lunar and asteroid landing scenarios. [ABSTRACT FROM AUTHOR]

The physics-informed neural network (PINN) has garnered significant attention in the domain of solving differential equations (DEs) owing to its adaptability to diverse DEs. Nonetheless, the PINN method encounters challenges in effectively solving multi-material diffusion problems due to the discontinuity at the interface. In this paper, we propose a PINN-based approach, termed as physics-specialized neural network (PSNN), which tailors a specialized neural network according to the distinctive characteristics of the problem. The PSNN integrates the continuity conditions of solution and flux as hard constraints into the mathematical form of the neural network, thereby perfectly addressing the discontinuity issues at axis-aligned interface. Moreover, this paper outlines the requirements for the specialized functions in PSNN, presents the analytical solving method for specialized functions under axis-aligned interface conditions, and compares PSNN with other enhanced PINN methods in the resolution of multi-material diffusion problems across multiple scenarios. The computational results reveal that PSNN demonstrates high computational accuracy and convergence speed, and the application of hard constraints for boundary conditions (BCs) in PSNN yields superior performance as compared to utilizing soft constraints for BCs. [ABSTRACT FROM AUTHOR]

This paper explores the application of Physical-informed Neural Networks (PINNs) for reconstructing ship propeller wake fields, introducing a novel approach known as self-adaptive loss balanced physics-informed neural networks (LB-PINNs). It can enhance the accuracy of the reconstruction process by incorporating an adaptive weight to balance the loss term within the network. The initial sections of the paper present the foundational principles and framework of PINNs. To validate the efficacy of PINNs in solving partial differential equations, the well-known Burgers equation is applied and then these results with those obtained through LB-PINNs are compared. This comparative analysis highlights the superior performance of LB-PINNs in achieving accurate predictions. Moving forward, the open water characteristics of the KVLCC2 propeller are simulated by using computational fluid dynamics (CFD) software STAR CCM+, and the flow field information of the KVLCC2 propeller in open water is obtained. This simulation provides crucial flow field information, forming the basis for constructing a training sample set to train the neural network. The trained PINN and LB-PINN are used to infer approximate solutions of the governing equations at arbitrary time and space coordinates. The velocity and pressure distributions obtained by PINN and LB-PINN were compared with those simulated by STAR CCM+. The results confirm the applicability of both PINN and LB-PINN in the reconstruction of propeller wake fields, with LB-PINN demonstrating superior performance. • A novel method for solving governing differential equations of fluid mechanics based on LB-PINN is introduced. • LB-PINN's adaptive weight balances the loss term, reducing mean square error by an order of magnitude compared to PINN. • Sparse numerical simulation data suffices for LB-PINN training, enabling ship propeller wake reconstruction. • Comparing flow field reconstructions using PINN, LB-PINN, and CFD shows LB-PINN's validity and accuracy. [ABSTRACT FROM AUTHOR]

• New methods for planar and spatial jerk and jounce were developed and validated for multibody dynamics. • Precision for jerk and jounce is equivalent to velocity and acceleration. • These new equations are significantly more accurate than finite difference approximations. • Assembly tolerance of 1e-12 is recommended. • Kinematics developed for this paper are required to explore hyper-dynamic differential algebraic equations created by first and second derivatives of equations of motion for multibody systems using joint constraint methods. Planar and spatial joint constraint equations for jerk and jounce were derived for multibody dynamics. Exemplar derivations are provided. Results from kinematically driven numerical simulations of these new equations were compared to explicit geometric solutions for planar four-bar and inverted slider-crank mechanisms as well as spatial revolute-spherical-universal-revolute and revolute-spherical-prismatic-universal mechanisms. Root-mean-square-error between velocity, acceleration, jerk and jounce simulations versus explicit solutions were normalized by maximum absolute values for comparison. Relative precisions for new jerk and jounce computations were equivalent to relative precisions for extant velocity and acceleration computations. The new equations were significantly more accurate than finite difference approximations previously required for jerk and jounce. Third and fourth order kinematics developed for this paper are required to explore hyper-dynamic differential algebraic equations created by first and second derivatives of equations of motion for multibody systems using joint constraint methods. [ABSTRACT FROM AUTHOR]

• The static response of a generally restrained stress-driven nonlocal Euler-Bernoulli nanobeam (GRNB), subjected to arbitrary loads is theoretically and numerically investigated. • It is shown that the gradient elasticity theory is a stress-driven nonlocal model with unsymmetrical exponential kernel normalized along the finite beam. • The stress-driven nonlocal and the strain gradient elasticity theories are ruled by a similar 6th order differential equation, with distinct higher-order boundary conditions. • The Green's function method is handled to formulate the problem in a systematic integral approach. • Both theories give very close results and are associated to a stiffening effect of the small-scale effect. This paper focuses on the static analysis of a generally restrained Euler-Bernoulli nanobeam (GRNB), which is subjected to arbitrary distributed or concentrated loads. The small length scale effect is introduced through a strongly nonlocal integral elastic law called the stress-driven nonlocal Euler-Bernoulli beam model, and a weakly nonlocal elastic law called the strain-gradient Euler-Bernoulli beam model. The stress-driven nonlocal theory and the strain gradient elasticity theory are both employed, to formulate the differential equation governing the Euler-Bernoulli nanobeams. Both theories are ruled by a similar 6th order differential equation, with distinct higher-order boundary conditions. The paper demonstrates that the strain gradient theory can be also reformulated as a stress-driven nonlocal theory with unsymmetrical kernel, in contrast to the initial stress-driven nonlocal theory which has symmetrical exponential kernel. Additionally, some theoretical backgrounds about both nonlocal elasticity theorems are provided. Exact solutions for both nanobeam models are presented, using Green's function method. The parameters of the Green's functions are derived for each beam theory, and then utilized to establish the static displacement function of both nanobeam theories. Exact solutions are provided for the bending of both nanobeams with various end boundary conditions, that account for the small length scale effects. The paper also compares the responses of the stress-driven Euler-Bernoulli beam and the strain gradient Euler-Bernoulli beam, showing close deflections and similar effects of increasing the nonlocal parameter on beam stiffness. A major difference highlighted between the two theories is that the strain gradient theory predicts a homogeneous strain state for homogeneous stress state, as opposed to the stress-driven theory. [ABSTRACT FROM AUTHOR]

To investigate the vibration characteristics of the vector propulsion shaft system (VPSS) of underwater vehicles, this paper establishes a numerical model applicable to the transverse vibration analysis of the VPSS. Based on the coupling effect between the VPSS and the motor, the integrated vibration effects of mass eccentricity and bearing nonlinear oil film force on the system are considered, and the Lagrangian method is used to establish the 6-degree-of-freedom nonlinear transverse vibration differential equations of the coupled system. The vibration equations of the system are solved numerically using the Runge-Kutta algorithm, and the vibration characteristics of the VPSS are investigated at different shaft angles and different working speeds. Meanwhile, for the interference of mass eccentricity and the nonlinear oil film force of the bearing on the system, the linear active disturbance control (LADRC) method is used to decouple and control them, and the linear active disturbance multiparameter regulation problem is solved by the bandwidth method. The results show that the control method exhibits strong robustness in resisting interference and regulating the modulation of the transverse vibration amplitude of the VPSS, which is of positive significance for improving the safety and stealth of underwater vehicles. • This paper establishes a numerical model for the transverse vibration analysis of the vector propulsion shaft system. • The combined vibration effects of mass eccentricity and bearing nonlinear oil film force on the system are considered. • For the interference of mass eccentricity and nonlinear oil film force, a LADRC method is used for decoupling control. [ABSTRACT FROM AUTHOR]