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Due to their several attractive properties, BDF-type multistep methods are usually the method-of-choice for solving stiff initial value problems (IVPs) of ordinary differential equations (ODEs) and differential-algebraic equations (DAEs). Recently, a class of BDF-type methods based on linear barycentric rational interpolants (LBRIs), referred to as RBDF methods, was introduced for solving ODEs. In the present paper, we are going to introduce a new family of LBRIs-based BDF-type formulas for the numerical solution of ODEs and DAEs with desirable stability properties and smaller error constants than those of the RBDF methods. Numerical experiments of the proposed methods on some well-known IVPs for ODEs and DAEs with index ≤ 3 illustrate the efficiency and capability of the methods in solving such problems. [ABSTRACT FROM AUTHOR]

The article deals with the problem of a dynamic model of the truck crane considering the elasticity of the rope and the ground. The dynamic model of a truck crane in the vertical plane has been established. When considering the oscillation of a truck crane, it is a dynamic system with a large number of degrees of freedom. This model will allow us to evaluate the dynamic response of the machine, analyze the vibration of the heavy object being lifted and lowered, and subsequently study stability and control problems. In this paper, the dynamic models of the cranes in the working plane have been established, built by Lagrange's multiplier Equations, which are systems of differential-algebraic Equations with the generalized coordinates of the machine's motions and the elastic coordinates. The Lagrange Equation provides a simple method for solving dynamic problems. The advantage of this Equation is that the form and number of Equations do not depend on the number of objects in the investigated system, nor do they depend on the way the objects move. The number of Lagrange Equations depends only on the number of degrees of freedom of the system. [ABSTRACT FROM AUTHOR]

The presence of impermeant molecules within a cell can lead to an increase in cell volume through the influx of water driven by osmosis. This phenomenon is known as the Donnan (or Gibbs-Donnan) effect. Animal cells actively transport ions to counteract the Donnan effect and regulate their volume, actively pumping Na+ out and K+ into their cytosol using the Na+/K+ ATPase (NKA) pump. The pump-leak equations (PLEs) are a system of algebraic-differential equations to model the membrane potential, ion (Na+, K+, and Cl-), and water flux across the cell membrane, which provide insight into how the combination of passive ions fluxes and active transport contribute to stabilizing cell volume. Our broad objective is to provide analytical insight into the PLEs through three lines of investigation: (1) we show that the provision of impermeant extracellular molecules can stabilize the volume of a passive cell; (2) we demonstrate that the mathematical form of the NKA pump is not as important as the stoichiometry for cell stabilization; and (3) we investigate the interaction between the NKA pump and cation-chloride co-transporters (CCCs) on cell stabilization, showing that NCC can destabilize a cell while NKCC and KCC can stabilize it. We incorporate extracellular impermeant molecules, NKA pump, and CCCs into the PLEs and derive the exact formula for the steady states in terms of all the parameters. This analytical expression enables us to easily explore the effect of each of the system parameters on the existence and stability of the steady states. [ABSTRACT FROM AUTHOR]

In pursuing quick and precise progressive flooding simulations for decision-making support, the linearised method has emerged and undergone refinement in recent years, becoming a reliable tool, especially for onboard decision support. This study consolidates and enhances the modelling approach based on a system of differential-algebraic equations capable of accommodating compartments filled with floodwater. The system can be linearised to permit analytical solutions, facilitating the utilization of larger time increments compared to conventional solvers for differential equations. Performance enhancements are achieved through the implementation of an adaptive time-step mechanism during the integration process. Furthermore, here, a correction coefficient for opening areas is introduced to enable the accurate modelling of free outflow scenarios, thereby mitigating issues associated with the assumption of deeply submerged openings used in governing equations. Experimental validation is conducted to compare the method's efficacy against recent model-scale tests, specifically emphasising the improvements stemming from the correction for free outflow. [ABSTRACT FROM AUTHOR]

We are considering differential-algebraic equations with embedded optimization criteria (DAEOs), in which the embedded optimization problem is solved by global optimization. This leads to differential inclusions for cases in which there are multiple global optimizers at the same time. Jump events from one global optimum to another result in nonsmooth DAEs and reduce the order of convergence of the numerical integrator to first-order. Implementation of event detection and location as introduced in this work preserves the higher-order convergence behaviour of the integrator. This allows the computation of discrete tangent and adjoint sensitivities for optimal control problems. [ABSTRACT FROM AUTHOR]

A system of equations with nonlinearity in the electric field potential and temperature is proposed for describing the heating of semiconductor elements on an electrical board with thermal and electrical breakdowns possibly arising over time. A method for numerical diagnostics of solution blow-up is considered. In the numerical analysis of the problem, the original system of partial differential equations is reduced to a differential-algebraic system, which is solved using a single-stage Rosenbrock scheme with complex coefficients. The blow-up of the exact solution is detected using an asymptotically sharp a posteriori error estimate obtained by computing approximate solutions on sequentially refined grids. The blow-up time is numerically estimated in the case of various initial conditions. [ABSTRACT FROM AUTHOR]

The aim of this paper is to study solvability and stability of stochastic differential-algebraic equations (SDAEs) with delay. It is difficult to investigate these properties for SDAEs with delay because of the singularity of the leading coefficient matrix. An index-ν concept is derived for solvability of these equations. Stability of SDAEs with delay is studied by using the method of Lyapunov functions and comparison principle. An example is given to illustrate the results. To the best of our knowledge, these results are novel for SDAEs with delay. [ABSTRACT FROM AUTHOR]

This paper studies the output regulation problem of time-varying descriptor systems and the problem of designing state feedback and dynamic measurement output feedback control laws which asymptotically achieves output regulation and disturbance rejection is considered. New regulator equations are proposed for time-varying descriptor systems in the form of differential-algebraic matrix equations. The unique solution of the proposed regulator equations is given as well. We prove that the output regulation problem of time-varying descriptor systems is solvable if and only if the given regulator equations are solvable. Based on the solution of the regulator equations, the state feedback and dynamic measurement output feedback control laws are designed to solve the output regulation problem. The work extends the existing results of output regulation problem for time-varying linear systems to the time-varying descriptor systems. Numerical examples are given to show the effectiveness of our methodology. [ABSTRACT FROM AUTHOR]

We deduce a procedure to apply balanced truncation to parameter-dependent differential-algebraic systems. For that we solve multiple projected Lyapunov equations for different parameter values to compute the Gramians that are required for the truncation procedure. As this process would lead to high computational costs if we perform it for a large number of parameters, we combine this approach with the reduced basis method that determines a reduced representation of the Lyapunov equation solutions for the parameters of interest. Residual-based error estimators are then used to evaluate the quality of the approximations. After introducing the procedure for a general class of differential-algebraic systems, we turn our focus to systems with a specific structure, for which the method can be applied particularly efficiently. We illustrate the efficiency of our approach on several models from fluid dynamics and mechanics.

Proton Exchange Membrane Fuel Cells are a favorite technology for decarbonizing the transportation sector. However, their large-scale democratization is hampered by their high cost compounded by their unsatisfactory lifespan. To anticipate potential degradation while keeping improving performance, it is essential to maintain an acceptable humidity range inside the cells, especially at the membrane level. However, membrane humidity level is not directly measurable, alternative techniques must be considered to recover this key variable. Here, we develop a real-time software sensor of the membrane water content at the fuel cell's heart. We build a model describing the membrane water balance, electrochemical behavior, and species mass balance. We then reduce the model and perform an Adaptive Extended Kalman Filter. We perform sensitivity analyses in both steady-state and transient conditions. We validate the filter on a "Worldwide Harmonized Light Vehicles Test Cycles" test procedure. Finally, we obtain a fast and accurate model-based software sensor. • Real-time capable software sensor for PEMFC membrane water content. • Observer based on membrane model accounting for most of major transport phenomena. • Adaptive Extended Kalman filter applied to differential-algebraic equations. • Sensitivity analysis of membrane water content shows a 5% coefficient of variation.

This paper proposes an improved expanding interior algorithm (EIA) to estimate the region of attraction (ROA) of power systems with wind power generation based on sum of squares (SOS) programming. An ordinary differential equation (ODE) model is derived for the doubly‐fed induction generator‐based wind turbine (DFIGWT), which is named as an enhanced synchronous‐generator‐mimicking (ESGM) model. The ESGM model bridges the gap between the requirement of an ODE model in ROA estimation and the conventional differential‐algebraic equation (DAE) model of the DFIGWT system. The ESGM model is able to accurately reflect the low frequency dynamics of the DFIGWT. Moreover, an improved EIA is designed to estimate the ROA based on SOS programming, which has higher efficiency than the existing ROA estimation algorithms based on SOS programming. It is able to adaptively search for the Lyapunov function and obtain an optimal estimation of the ROA in an iterative process. The accuracy and efficiency of this algorithm are verified in three test systems composed of DFIGWTs and synchronous generators (SGs). The morphological changes in the ROA of the test systems caused by the penetration of DFIGWT are examined. [ABSTRACT FROM AUTHOR]

In a finite-dimensional space, we consider a linear stochastic differential equation in Itô form with a singular constant matrix on the left-hand side. Taking into account various economic applications of such equations, they are classified as Leontief type equations, since under some additional assumptions, a deterministic analog of the equation in question describes the famous Leontief input–output balance model taking into account reserves. In the literature, these systems are more often called differential–algebraic or descriptor systems. In general, to study this type of equations, one needs higher-order derivatives of the right-hand side. This means that one must consider derivatives of the Wiener process, which exist in the generalized sense. In the previous papers, these equations were studied using the technique of Nelson mean derivatives of random processes, whose description does not require generalized functions. It is well known that mean derivatives depend on the -algebra used to find them. In the present paper, the study of this equation is carried out using mean derivatives with respect to a new -algebra that was not considered in the previous papers. [ABSTRACT FROM AUTHOR]

The relation between passive and positive real systems has been extensively studied in the literature. In this paper, we study their connection to the more recently used notion of port-Hamiltonian descriptor systems. It is well-known that port-Hamiltonian systems are passive and that passive systems are positive real. Hence it is studied under which assumptions the converse implications hold. Furthermore, the relationship between passivity and KYP inequalities is investigated. [ABSTRACT FROM AUTHOR]

Synchronous generator system is a complicated dynamic system for energy transmission, which plays an important role in modern industrial production. In this article, we propose some predictor-corrector methods and structure-preserving methods for a generator system based on the first benchmark model of subsynchronous resonance, among which the structure-preserving methods preserve a Dirac structure associated with the so-called port-Hamiltonian descriptor systems. To illustrate this, the simplified generator system in the form of index-1 differential-algebraic equations has been derived. Our analyses provide the global error estimates for a special class of structure-preserving methods called Gauss methods, which guarantee their superior performance over the PSCAD/EMTDC and the predictor-corrector methods in terms of computational stability. Numerical simulations are implemented to verify the effectiveness and advantages of our methods. [ABSTRACT FROM AUTHOR]

We present a straightforward way to solve a model predictive control problem for a power network system given as a nonlinear differential‐algebraic equation (DAE) in a distributed way using the consensus alternating directions method of multipliers (consensus ADMM) algorithm. While no convergence‐ or stability results are available for fully nonlinear DAE models, this gives unprecedented experimental evidence that power network systems of the presented structure allow to be controlled in this way, unlocking the numerous combined advantages of distributed and predictive control schemes in the context of energy distribution networks, as well as broadening the field of use for the consensus ADMM algorithm to nonlinear DAE models. [ABSTRACT FROM AUTHOR]

Transient stability preventive control (TSPC), a method to efficiently withstand the severe contingencies in a power system, is mathematically a transient stability constrained optimal power flow (TSC-OPF) issue, attempting to maintain the economical and secure dispatch of a power system via generation rescheduling. The traditional TSC-OPF issue incorporated with differential-algebraic equations (DAE) is time consumption and difficult to solve. Therefore, this paper proposes a new TSPC method driven by a naturally inspired optimization algorithm integrated with transient stability assessment. To avoid solving complex DAE, the stacking ensemble multilayer perceptron (SEMLP) is used in this research as a transient stability assessment (TSA) model and integrated into the optimization algorithm to replace transient stability constraints. Therefore, less time is spent on challenging calculations. Simultaneously, sensitivity analysis (SA) based on this TSA model determines the adjustment direction of the controllable generators set. The results of this SA can be utilized as prior knowledge for subsequent optimization algorithms, thus further reducing the time consumption process. In addition, a naturally inspired algorithm, Aptenodytes Forsteri Optimization (AFO), is introduced to find the best operating point with a near-optimal operational cost while ensuring power system stability. The accuracy and effectiveness of the method are verified on the IEEE 39-bus system and the IEEE 300-bus system. After the implementation of the proposed TSPC method, both systems can ensure transient stability under a given contingency. The test experiment using AFO driven by SEMLP and SA on the IEEE 39-bus system is completed in about 35 s, which is one-tenth of the time required by the time domain simulation method. [ABSTRACT FROM AUTHOR]

Electrical circuits are present in a variety of technologies, making their design an important part of computer aided engineering. The growing number of parameters that affect the final design leads to a need for new approaches to quantify their impact. Machine learning may play a key role in this regard; however, current approaches often make suboptimal use of existing knowledge about the system at hand. In terms of circuits, their description via modified nodal analysis is well‐understood. This particular formulation leads to systems of differential‐algebraic equations (DAEs), which bring with them a number of peculiarities, for example, hidden constraints that the solution needs to fulfill. We use the recently introduced dissection index that can decouple a given system of DAEs into ordinary differential equations, only depending on differential variables, and purely algebraic equations, that describe the relations between differential and algebraic variables. The idea is to then only learn the differential variables and reconstruct the algebraic ones using the relations from the decoupling. This approach guarantees that the algebraic constraints are fulfilled up to the accuracy of the nonlinear system solver, and it may also reduce the learning effort as only the differential variables need to be learned. [ABSTRACT FROM AUTHOR]

Challenges in the operation of power systems arise from several factors such as the interconnection of large power systems, integration of new energy sources and the increase in electrical energy demand. These challenges have required the development of fast and reliable tools for evaluating the operation of power systems. The load margin (LM) is an important index in evaluating the stability of power systems, but traditional methods for determining the LM consist of solving a set of differential-algebraic equations whose information may not always be available. Data-Driven techniques such as Artificial Neural Networks were developed to calculate and monitor LM, but may present unsatisfactory performance due to difficulty in generalization. Therefore, this article proposes a design method for Physics-Informed Neural Networks whose parameters will be tuned by bio-inspired algorithms in an optimization model. Physical knowledge regarding the operation of power systems is incorporated into the PINN training process. Case studies were carried out and discussed in the IEEE 68-bus system considering the N-1 criterion for disconnection of transmission lines. The PINN load margin results obtained by the proposed method showed lower error values for the Root Mean Square Error (RMSE), Mean Square Error (MSE) and Mean Absolute Percentage Error (MAPE) indices than the traditional training Levenberg-Marquard method. [ABSTRACT FROM AUTHOR]

The energy transition relies on the integration of non-conventional renewable energy sources. These disruptive technological developments alter the functioning and operation of the electric power system. This paper examines the impacts of wind power on transient stability indicators of the power system, using an implicit formulation and the nine-bus test system. The research findings indicate that transient stability indicators are sensitive to the location and duration of faults. Furthermore, there is an observed trend of increasing maximum rotor speed deviation and oscillation duration, indicating reduced stability margins. [ABSTRACT FROM AUTHOR]

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The object of research is the behavior of the load of the tower crane during the break of the sling. One of the most problematic areas is the safety of work and the prevention of emergency situations. Despite the presence of mandatory safety measures, during cargo transportation, one of the sling branches may be destroyed due to the presence of a dynamic component during the operation of the crane, or errors of the slinger when securing the cargo. Also, the presence of hidden internal or unnoticed defects in the sling construction itself cannot be ruled out. Also, one of the most problematic places is the chaotic fluctuations of the load, which negatively affect the stability of the crane and safety. The paper describes the case of the destruction of one of the branches of a two-rope sling during the transportation of a long product by a tower crane. The proposed method of cargo behavior analysis is based on the use of a dynamic description of cable system failure modes within the framework of setting and solving differential-algebraic equations. This makes it possible to more accurately describe the behavior of the cargo when the sling breaks. The obtained results show that the application of the proposed method makes it possible to bring the mathematical model of the two-link mathematical pendulum significantly closer to the actual mutual oscillations of the load during the sling break. This is due to the fact that the proposed method has a number of features, in particular, high sensitivity to changes in the behavior of the cargo and a quick reaction to a rope break. These results can be used in practice in the design and operation of tower cranes. Thanks to the application of the proposed method, it is possible to obtain accurate values of cargo behavior indicators and timely detection of a rope break. Compared to similar known methods, this method has such advantages as high efficiency, reliability and safety of operation. [ABSTRACT FROM AUTHOR]

Working with a general class of regular linear Hamiltonian systems, we show that renormalized oscillation results can be obtained in a natural way through consideration of the Maslov index associated with appropriately chosen paths of Lagrangian subspaces of C 2 n . We verify that our applicability class includes Dirac and Sturm–Liouville systems, as well as a system arising from differential-algebraic equations for which the spectral parameter appears nonlinearly. [ABSTRACT FROM AUTHOR]

The article deals with a numerical and analytical approach to solving the equations of motion, which enables to treat the considered problems with the change of system structure or number of degrees of freedom without interrupting the numerical integration process. The described methodology allows effectively incorporate switchable constraints in the systems in accordance with their flexible structures. The crucial idea is based on the formulation of the resulting differentialalgebraic equations into a saddle point system, where the switchable constraints are represented by a sign matrix with variable rank. In connection with this property, a pseudoinversion is applied to eliminate algebraic variables and transform the problem to the first order system of ordinary differential equations. Moreover, the time independent case leads to linear autonomous systems with non-diagonalizable matrices, as is proved. The relevant numerical scheme is based on Runge-Kutta methods, that correspond to the power series of the resulting matrix exponential for time independent problems. The methodology presented is illustrated on the idealized two-mass oscillator with a switchable constraint. The numerical experiments performed range from initial stages, through simple transient cases to damped intentional control. The advanced applications can be found in robotics, active and controlled systems, and in the simulations of complex systems in biology and related areas. Moreover, the methodology can also be applied in the simulation of transport systems, especially in relation to vehicle technology, a quarter car suspension system, a vibration control mechanism, a torsion system with a clutch, and machine balancing and storage should to be highlighted. [ABSTRACT FROM AUTHOR]

In this paper, a stage structured predator-prey model with Holling type II functional response is formulated, considering the juvenile prey as the favorite food for the generalist predator. Further, the existence of predators is ensured by the sufficient amount of alternative food available in the habitat. The proportional harvesting of the adult prey is incorporated with the assumption that only the adult prey are of economic worth. The existence and local stability of the distinct equilibrium points of the system are investigated. The bifurcation from origin to predator free equilibrium state is obtained for the bifurcation parameter-effort on harvesting. The occurrence of Hopf bifurcation about the interior equilibrium state is established for arbitrary model parameters and the supercritical nature of this bifurcation is proved by fixing these parametric values. An algebraic equation is included to this modified model to analyze the economic benefits resulted from the harvesting of adult prey. The singularity-induced bifurcation (SIB) about the coexisting equilibrium state of differential-algebraic system is deduced along the parameter v at v = 0 , v being the profit/loss due to harvesting. The state feedback controller is recommended to eliminate the SIB about the coexisting equilibrium state for the differential algebraic system. Adopting an appropriate feedback control would ensure the stability of co-existence interior equilibrium state along with economic profit from harvesting. Numerical examples are used to elaborate the analytical results obtained. [ABSTRACT FROM AUTHOR]

A problem of solvability with parameter for a differential-algebraic equation is considered. For solving the problem is applied Weierstrass canonical form. Problem is reduced to an initial value problem with parameter for differential equations. Conditions for the existence and uniqueness of the problem with parameter for differential-algebraic equations are established. [ABSTRACT FROM AUTHOR]

The stability of nonlinear systems in the control domain has been extensively studied using different versions of the algebraic Riccati equation (ARE). This leads to the focus of this work: the search for the time-varying quaternion ARE (TQARE) Hermitian solution. The zeroing neural network (ZNN) method, which has shown significant success at solving time-varying problems, is used to do this. We present a novel ZNN model called 'ZQ-ARE' that effectively solves the TQARE by finding only Hermitian solutions. The model works quite effectively, as demonstrated by one application to quadrotor control and three simulation tests. Specifically, in three simulation tests, the ZQ-ARE model finds the TQARE Hermitian solution under various initial conditions, and we also demonstrate that the convergence rate of the solution can be adjusted. Furthermore, we show that adapting the ZQ-ARE solution to the state-dependent Riccati equation (SDRE) technique stabilizes a quadrotor's flight control system faster than the traditional differential-algebraic Riccati equation solution. [ABSTRACT FROM AUTHOR]

This paper concerns a class of monotone eigenvalue problems with eigenvector nonlinearities (mNEPv). The mNEPv is encountered in applications such as the computation of joint numerical radius of matrices, best rank-one approximation of third-order partial-symmetric tensors, and distance to singularity for dissipative Hamiltonian differential-algebraic equations. We first present a variational characterization of the mNEPv. Based on the variational characterization, we provide a geometric interpretation of the self-consistent field (SCF) iterations for solving the mNEPv, prove the global convergence of the SCF, and devise an accelerated SCF. Numerical examples demonstrate theoretical properties and computational efficiency of the SCF and its acceleration. [ABSTRACT FROM AUTHOR]

While the kinematic and dynamic modeling of mechanical systems is well developed, the numerical solution of the governing equations of many of these models is still a large field under investigation. Since in any constrained mechanical system, the connections of the connected bodies restrict their relative movement, therefore the governing equations of the model usually form a system of differential-algebraic equations that includes both differential and algebraic equations. It should be noted that, unlike the analytical and numerical solution of differential equations, the analytical and numerical behavior of differential-algebraic equations is more complex and completely different from differential equations. In this paper, a high accuracy method for solving the system of differential-algebraic equations arising in mechanical systems is presented. The presented method is based on the use of a pseudospectral method that will transform the solution of the system of differential-algebraic equations into the solution of a system of algebraic equations. Then, an optimization technique is utilized to facilitate solving the obtained system of algebraic equations. At the end, some numerical experiments are performed on several benchmark problems to show the accuracy and applicability of the method. [ABSTRACT FROM AUTHOR]

Background: This paper aims to present open-source computer simulation programs developed to simulate, track, and estimate the COVID-19 outbreak. Methods: The programs included two separate parts, one set of programs built in Simulink with a block diagram display, and another one coded as a script in MATLAB R2020b. The mathematical model used in this package was the suspectable-infected-removed (SIR), suspectable-exposed-infected-removed (SEIR), and susceptible-exposed-infected-recovereddeceased (SEIRD) models represented by a set of differential-algebraic equations. It can be easily modified to develop new models for the problem. A generalized method was adopted to simulate worldwide outbreaks in an efficient, fast, and simple way. Results: To get a good tracking of the virus spread, a sum of sigmoid functions was proposed to capture any dynamic changes in the data. The parameters used for the input (infection and recovery rate functions) were computed using the parameter estimation tool in MATLAB. Several statistical methods were applied for the rate function, including linear, Mean±SD and root mean square (RMS). In addition, an adaptive neuro-fuzzy inference system (ANFIS) was employed and proposed to train the model and predict its output. Conclusion: This procedure is presented in such a way that it can be generalized and applied in other parts and applications of estimating the scenarios of an event, including the potential of several models, including suspectable-infected-removed (SIR), which is sensitive to pollution, etc. This program can be used as an educational tool or for research studies and this article promises some lasting contributions in the field of COVID-19. [ABSTRACT FROM AUTHOR]

Integration of a vibration absorber and an energy harvester is a promising approach to achieve simultaneously vibration reductions and vibratory energy harvesting. The present work investigates random responses of a nonlinear energy sink (NES) combined with a giant magnetostrictive energy harvester (GMEH). When a structure is modeled as a one-degree-of-freedom oscillator subjected to a random excitation, the structure with the integrated NES-GMEH is governed by two simultaneous nonlinear stochastic differential equations coupled with a nonlinear algebraic equation. The Ferrari formula is applied to decouple the algebraic equations from the stochastic differential-algebraic equations. The resulting stochastic algebraic differential equations are transformed into the steady Fokker-Planck-Kolmogorov equations. A fourth finite difference scheme is used to solve numerically the FPK equations. The FPK equations based solutions are qualitatively verified via Monte Carlo simulations. The parametric effects on vibration suppression and electricity production are examined, where the electricity is indicated by the square expectations of voltage and power. The responses of the structure, the structure with the NES only and the structure with both NES-GMEH are compared under the same Gaussian white noise and system parameters. It is found that the integrated NES-GMEH system achieves the best vibration suppression. It is demonstrated that a properly designed NES-GMEH device can simultaneously suppress vibration and produce electricity under Gaussian white noise excitations. [ABSTRACT FROM AUTHOR]

FE2-approaches are chosen to incorporate micro-mechanical information into macroscopical computations. Although the approach is applied for several years, there are some open theoretical aspects in the algorithmic description. In this contribution the entire concept is brought into the method of vertical lines to solve initial boundary-value problems based on the homogenization concept under consideration. This requires a clear description of known and unknown quantities in the space and time discretization procedure. Dependent on the constitutive model, systems of algebraic or differential-algebraic equations (DAE) arise. Here, we concentrate on models of evolutionary type. Frequently, the resulting DAE-systems are solved using the Backward-Euler method, which are embedded in the more general class of diagonally, implicit Runge-Kutta methods. After the time discretization the resulting systems of non-linear equations have to be solved. This can be done using various schemes to obtain more efficient computations. Here, the three-level Newton algorithm, the Newton-Raphson-Schur method, numerical differentiation and a Chord-version are derived in the context of FE2 and discussed. Finally, step-size-controlled FE2-problems are solved, highlighting the importance of time-adaptive computations, particularly, in combination with the non-linear solution scheme, where the Multilevel-Newton-Chord method shows high efficiency. This contributes to an additional reduction of the overall computational times. [ABSTRACT FROM AUTHOR]

In this study, we introduce higher‐order approximation schemes for a 1D shallow‐water model with a moving boundary and arbitrary cross‐section. The model equations are formulated using Lagrange coordinates to handle the time‐varying spatial domain. By discretizing the action functional on a material‐fixed grid and applying an appropriate quadrature scheme, we derive a finite‐dimensional model. This model, taking mass conservation into account as an auxiliary condition, results in a system of semi‐explicit differential‐algebraic equations (DAE). Unlike previous work, we employ higher‐order quadrature formulae to enhance numerical accuracy, albeit at the cost of more complex nonlinear DAE. In order to compare the performance of the resulting models obtained from using different quadrature schemes, a comprehensive simulation study is conducted. [ABSTRACT FROM AUTHOR]

In this paper, we study the robust stability of implicit dynamic equations with causal operators on time scales. First, we investigate the solvability of these dynamic equations and then consider the preservation of stability under small perturbations. An L p version of Bohl–Perron principle for implicit dynamic equations is also studied. [ABSTRACT FROM AUTHOR]

2.1], the set HT ht is Euclidean dense in HT ht , whence HT ht is Euclidean dense in HT ht . The proof goes through analogously if the Cartesian factors HT ht are omitted. Then, the set HT ht is relative generic in HT ht . [Extracted from the article]

In this paper we derive error estimates for Runge–Kutta schemes of optimal control problems subject to index one differential–algebraic equations (DAEs). Usually, Runge–Kutta methods applied to DAEs approximate the differential and algebraic state in an analogous manner. These schemes can be considered as discretizations of the index reduced system where the algebraic equation is solved for the algebraic variable to get an explicit ordinary differential equation. However, in optimal control this approach yields discrete necessary conditions that are not consistent with the continuous necessary conditions which are essential for deriving error estimates. Therefore, we suggest to treat the algebraic variable like a control, obtaining a new type of Runge–Kutta scheme. For this method we derive consistent necessary conditions and compare the discrete and continuous systems to get error estimates up to order three for the states and control as well as the multipliers. [ABSTRACT FROM AUTHOR]

In this paper, we formulate a fractional differential‐algebraic model with two discrete delays and double linear species harvesting. One of the delays denotes the time taken for digestion of the prey, and the other reflects the negative feedback of the predator's density. Taking into account of the economic profit, linear predator harvesting and prey harvesting have been incorporated into the proposed predator–prey system. From the biological viewpoint, we discuss the sufficient conditions for the existence of nontrivial positive equilibrium point. By jointly using the stability theory of fractional differential‐algebraic equations and the bifurcation theory, we study the delay‐induced instability and Hopf bifurcations. Finally, the correctness of the theoretical analysis is verified by numerical simulations, and the effects of delays, economic profit, and fractional exponents on the stability of the system are explored. [ABSTRACT FROM AUTHOR]

This paper considers linear differential-algebraic equations (DAEs) representing a system of ordinary differential equations with an identically singular matrix at the derivative in the domain of its definition. The matrix coefficients of DAEs are assumed to depend on the uncertain parameters belonging to a given admissible set. For the parametric family under consideration, structural forms with separate differential and algebraic parts are built. As is demonstrated below, the robust stability of the DAE family is equivalent to the robust stability of its differential subsystem. For the structure of perturbations, sufficient conditions are established under which the separation of DAEs into the algebraic and differential components preserves the original type of functional dependence on the uncertain parameters. Sufficient conditions for robust stability are obtained by constructing a quadratic Lyapunov function. [ABSTRACT FROM AUTHOR]

This paper introduces the novel application of the mass and force lumping technique to enhance the finite element discretization of the fully intrinsic beam formulation. In our aeroelastic system model, 2-D unsteady aerodynamics were incorporated alongside simple calculations for thrust and gravity. Through the central difference discretization method, the discretized system was thoroughly examined, shedding light on the advantages of the mass and force lumping approach. With the use of a first-order lumping method, we successfully reconstructed the inertia matrices, external forces, and moments. The resulting equations are more systematically structured, facilitating the extraction of a regular state-space linear system using the direct index reduction method post-linearization. Numerical results further confirm that the proposed techniques can effectively capture the nonlinear dynamics of aeroelastic systems, enabling equation reconstruction and leading to significant benefits in system order reduction and flight dynamical analysis. [ABSTRACT FROM AUTHOR]

In this paper, a reduced-order interval observer (R-IO) design technique is developed for continuous-time descriptor linear-parameter-varying systems with unknown-but-bounded uncertainties. First, by introducing an intermediate variable, the R-IO design amounts to estimating the linear functional of the system states. Then, with a parameter-dependent Luenberger-like structure, the R-IO existence condition is eventually formulated into a set of differential-algebraic equations, not involving any linear transformation process. Further, a parametric solution to such an R-IO is derived through solving the equation set, which clearly shows the design degrees of freedom. Finally, the correctness of the proposed results is verified by a numerical system and an electrical circuit system. [ABSTRACT FROM AUTHOR]

Conditions for the existence, uniqueness and boundedness of global solutions, as well as ultimate boundedness of solutions, and conditions for the blow-up of solutions of nonregular semilinear differential-algebraic equations have been obtained. An example demonstrating the application of the obtained results has been considered. Isothermal models of gas networks have been proposed as applications. [ABSTRACT FROM AUTHOR]

A dynamic iteration scheme for linear differential-algebraic port-Hamiltonian systems based on Lions–Mercier-type operator splitting methods is developed. The dynamic iteration is monotone in the sense that the error is decreasing and no stability conditions are required. The developed iteration scheme is even new for linear port-Hamiltonian systems governed by ODEs. The obtained algorithm is applied to a multibody system and an electrical network. [ABSTRACT FROM AUTHOR]

In this article, the problem for a differential-algebraic equation with a significant loads is studied. Unlike previously studied problems for differential equations with a significant loads, in the considered equation, there is a matrix in the left part with a derivative that is not invertible. Therefore, the system of equations includes both differential and algebraic equations. To solve the problem, we propose a modification of the Dzhumabaev’s parametrization method. The considered problem is reduced to a parametric problem for the differential-algebraic equation with significant loads. We apply the Weierstrass canonical form to this problem. We obtain parametric initial value problem for a differential equations and an algebraic equations with a significant loads. The solvability conditions for the considered problem are established.

Background In this paper, we focus on an efficient and easy method for solving the given system of differential-algebraic equations (DAEs) of second order. Methods The approximate solutions are computed rapidly and efficiently with the help of a semi-analytical method known as Adomian decomposition method (ADM). The logic of this method is simple and straightforward to understand. Results To demonstrate the proposed method, we presented several examples and the computations are compared with the exact solutions to show the efficient. One can employ this logic to different mathematical software tools such as Maple, SCILab, Mathematica, NCAlgebra, Matlab etc. for the problems in real life applications. Conclusions In this paper, we offered a method for solving the given system of secondorder nonlinear DAEs with aid of the ADM. We shown that the proposed method is simple and efficient, also one can obtain the approximate solutions quickly using this method. A couple of examples are discussed for illustrating this method and graphical and mathematical assessments are discussed with the analytical solutions of the given problems.

Background: In this paper, we focus on an efficient and easy method for solving the given system of differential-algebraic equations (DAEs) of second order. Methods: The approximate solutions are computed rapidly and efficiently with the help of a semi-analytical method known as Adomian decomposition method (ADM). The logic of this method is simple and straightforward to understand. Results: To demonstrate the proposed method, we presented several examples and the computations are compared with the exact solutions to show the efficient. One can employ this logic to different mathematical software tools such as Maple, SCILab, Mathematica, NCAlgebra, Matlab etc. for the problems in real life applications. Conclusions: In this paper, we offered a method for solving the given system of secondorder nonlinear DAEs with aid of the ADM. We shown that the proposed method is simple and efficient, also one can obtain the approximate solutions quickly using this method. A couple of examples are discussed for illustrating this method and graphical and mathematical assessments are discussed with the analytical solutions of the given problems.

This paper introduces semi-explicit multilinear modeling to ac power systems, including voltage source converters. Firstly, the novel semi-explicit multilinear modeling class is explained. Then this class is applied to a small example of PV inverter connected to an infinite bus. Some part of the nonlinear current reference calculation is expressed as a multilinear algebraic constraint. The formulation of this multilinear algebraic constraint allowed modeling the whole nonlinear inverter as a semi-explicit multilinear model. The novel modeling approach is compared with common nonlinear time-based simulation in Simulink. The results show a good approximation of the nonlinear model by its semi-explicit multilinear model where the relative error is below 10−3%. This low error motivates the further usage of the semi-explicit multilinear modeling class for power system analysis.

Different representations of linear dissipative Hamiltonian and port-Hamiltonian differential–algebraic equations (DAE) systems are presented and compared. Using global geometric and algebraic points of view, translations between different representations are presented. Characterizations are also derived when a general DAE system can be transformed into one of these structured representations. Approaches for computing the structural information and the described transformations are derived that can be directly implemented as numerical methods. The results are demonstrated with a large number of examples. [ABSTRACT FROM AUTHOR]