Search Results (204)

The paper deals with the results of a study of a third-order nonlinear differential equation with moving singular points and critical poles. So far, this type of equation cannot be solved in quadratures. The development of the author’s approach in proving the theorem of the existence of moving singular points and solutions in the vicinity of a critical pole, based on a modified Cauchy majorant method, is given. An analytical approximate solution in the vicinity of a moving singular point is obtained, and an expression for the a priori error estimate is presented. A numerical experiment confirming the obtained theoretical results is provided. Full article

The possibility of determining the phase and current errors of an existing or newly designed current instrument transformer on the basis of special characteristics of the core material is examined. One of the characteristics represents the dependence between the magnetic field intensity on the core sheet surface, measured at the instant when induction is at its peak, and the mean peak induction in the cross section of the sheet. The other characteristic represents the dependence between the field intensity value measured at the instant when induction passes through zero and the peak induction value. The characteristics must be determined for the sinusoidal shape of the induction curve. The secondary winding of the current instrument transformer should be uniformly distributed along the core. One must know the following: the number of turns in the primary and secondary winding, respectively, the resistance of the secondary winding and the resistance at the secondary winding output when the primary current is being converted. Indicated relations provide a clear formula for designing effective current transformers. The main contribution of this paper is to present the method for estimating the parameters of current transformer a priori, relying on characteristics of the core material. However, this formula combined with elements of artificial intelligence—nature-inspired optimization algorithms—results in a convenient tool for optimal core geometry design. The paper presents an extension of the method to a smart design approach with application of the Birch-inspired Optimization Algorithm (BiOA). Full article

Hyperparameters allow metaheuristics to be tuned to a wide range of problems. However, even though formalized tuning of metaheuristic parameters can affect the quality of the solution, it is rarely performed. The empirical selection method and the trial-and-error method are the primary conventional parameter selection techniques for optimization heuristics. Both require a priori knowledge of the problem and involve multiple experiments requiring significant time and effort, yet neither guarantees the attainment of optimum parameter values. Of the studies that perform formal parameter tuning, experimental design is the most commonly used method. Although experimental design is feasible for systematic experimentation, it is also time-consuming and requires extensive effort for large optimization problems. The computational effort in this study refers to the number of experimental runs required for hyperparameter tuning, not the computational time for each run. This study proposes a simpler, faster method based on an optimized Latin hypercube sampling (OLHS) technique augmented with response surface methodology for estimating the best hyperparameter settings for a hybrid simulated annealing algorithm. The method is applied to solve the aircraft landing problem with time windows (ALPTW), a combinatorial optimization problem that seeks to determine the optimal landing sequence within a predetermined time window while maintaining minimum separation criteria. The results showed that the proposed method improves sampling efficiency, providing better coverage and higher accuracy with 70% fewer sample points and only 30% of the total runs compared to full factorial design. Full article

Lithium-ion batteries have been extensively utilized as a high-power, rechargeable, and dischargeable energy storage medium. Accurate estimation of the battery state of charge (SOC) in the battery management system (BMS) is imperative for ensuring the safe and stable operation of electric vehicles. This paper proposes an SOC estimation method based on the equivalent circuit model as well as the ampere-time integration method with a physical informed neural network. The network enhances the estimation of SOC by introducing two mechanistic information sources: the equivalent circuit model (ECM) and the ampere-time integration method (Ah-I method). These are utilized as a priori knowledge to constrain the estimation of SOC. Initially, the Rint model is selected as the physical analysis model of the lithium-ion battery, and subsequently, the Ah-I method is chosen as the auxiliary model for SOC output estimation. A deep learning network is then employed to establish the mapping between the battery input parameters and the SOC output. Finally, the SOC is estimated by fusing the physical model and the data-driven model. The results demonstrate the efficacy of the method in accurately estimating the state of charge of lithium batteries, with a root mean square error within 1%. The validity of the research methodology was further validated through comparison with other approaches. Full article

The paper addresses the problem of attitude determination using Global Navigation Satellite System (GNSS) measurements from multiple antennas mounted on a navigation platform. To achieve attitude determination by GNSS with typical accuracy down to tenths of a degree for one-meter baselines, GNSS phase measurements are employed. A key challenge with phase measurements is the presence of unknown integer ambiguities. Consequently, the attitude determination problem traditionally reduces to a nonlinear, non-convex optimization problem with integer constraints. No closed-form solution to this problem is known, and its real-time calculation is computationally intensive. Given an a priori initial attitude approximation, we propose a new algorithm for attitude tracking based on the reduction of the nonlinear orthogonality-constrained attitude estimation problem to a linear integer least squares problem, for which numerical methods are well known and computationally much less demanding. Additionally, a simple a priori model for GNSS measurement error variance is introduced, grounded on the geometry of satellite signal propagation through vacuum and the Earth’s atmosphere, providing a clear physical interpretation. Applying the algorithm to a real dataset collected from a quasi-static multi-antenna, multi-GNSS system with sub-meter baselines, we obtain promising results. Full article

Airplane cabin temperature is a critical environmental factor governing the safety and reliability of airborne equipment. Compared with measuring temperature, predicting temperature is more cost- and time-saving and can cover an extreme flight envelope. Physics-informed neural networks (PINNs) offer a promising prediction solution whose performance hinges on the availability of precise governing differential equations. However, building governing differential equations between flight parameters and cabin temperature is a great challenge, as it is comprehensively influenced by aerodynamic heat, avionic heat, and internal flow. To solve this, a new PINN framework based on “a priori monotonicity” is proposed. Underlying physical trends (monotonicity) from flight data are extracted to construct the loss function as a data-driven constraint, thus eliminating the need for any governing equations. The new PINN is developed to estimate the seven cabin temperatures of an unmanned aerial vehicle. The model was trained on data from four flight sorties and validated on another four independent sorties. Results demonstrate that the proposed PINN achieves a mean absolute error of 1.9 and a root mean square error of 2.6, outperforming a conventional neural network by approximately 35%. The core value of this work is a new PINN framework that bypasses the development of complex governing equations, which enhances its practicality for engineering applications. Full article

The local convergence analysis of the $m+1$-step Newton-Jarratt composite scheme with order $2m+1$ has been shown previously. But the convergence order $2m+1$ is obtained using Taylor series and assumptions on the existence of at least the fifth derivative of the mapping involved, which is not present in the method. These assumptions limit the applicability of the method. A priori error estimates or the radius of convergence or uniqueness of the solution results have not been given either. These drawbacks are addressed in this paper. In particular, the convergence is based only on the operators on the method, which are the operator and its first derivative. Moreover, the radius of convergence is established, a priori estimates and the isolation of the solution is discussed using generalized continuity assumptions on the derivative. Furthermore, the more challenging semi-local convergence analysis, not previously studied, is presented using majorizing sequences. The convergence for both analyses depends on the generalized continuity of the Jacobian of the mapping involved, which is used to control it and sharpen the error distances. Numerical examples validate the sufficient convergence conditions presented in the theory. Full article

We present a fully discrete splitting-Galerkin scheme for second-order, non-autonomous abstract Cauchy problems with time-dependent perturbations. By reformulating the second-order equation as a first-order system in the product space, we apply a Galerkin semi-discretization in space of order $O\left(h^k\right)$ and a Strang splitting in time of order $O(\Delta t^2)$. An embedded Runge–Kutta controller provides adaptive time-stepping to handle rapid temporal variations in the perturbation operator $B\left(t\right)$. Under standard regularity and commutator assumptions on $A\left(t\right)$ and $B\left(t\right)$, we establish a priori error estimates $\underset{0\le t_n\le T}{max}{\parallel u\left(t_n\right)-u_n\parallel }_Z=O(h^k+\Delta t^2).$ Numerical experiments for a 1D perturbed wave equation confirm the theoretical convergence rates, illustrate stability thresholds in the unstable regime, and demonstrate up to 40% savings in computational cost via adaptivity. Full article

This work presents a stabilized Nitsche-type Cut Finite Element Method (CutFEM) for simulating two-phase flows with complex interfaces. The method addresses the challenges of capturing discontinuities in material properties and governing equations that arise from implicitly defined interfaces. By employing a Continuous Interior Penalty (CIP) method, Nitsche’s method for weak interface coupling, and Ghost Penalty (GP) terms for stability, the formulation enables an accurate representation of abrupt changes in physical properties across cut elements. A stability analysis and a priori error estimation, utilizing Oseen’s formulation, demonstrate the method’s robustness. At the same time, a numerical convergence study incorporating adaptivity and a best-fit quadratic level-set interpolation validates its accuracy. Finally, the method’s efficacy in mitigating spurious currents is confirmed through the Spurious Current Test, demonstrating its potential for reliable simulation of multi-phase flow phenomena. Full article

This paper addresses an attraction–repulsion chemotaxis system governed by Neumann boundary conditions within a bounded domain $\Omega \subset {\mathbb{R}}^3$ that has a smooth boundary. The primary focus of the study is the chemotactic response of a species (cell population) to two competing signals. We establish the existence and uniqueness of a weak solution to the system by analyzing the solvability of an approximate problem and utilizing the Leray–Schauder fixed-point theorem. By deriving appropriate a priori estimates, we demonstrate that the solution of the approximate problem converges to a weak solution of the original system. Additionally, we conduct computational studies of the model using the finite element method. The accuracy of our numerical implementation is evaluated through error analysis and numerical convergence, followed by various numerical simulations in a two-dimensional domain to illustrate the dynamics of the system and validate the theoretical findings. Full article

This paper is focused on developing a Galerkin finite element framework for the fractional Allen–Cahn equation with regularized logarithmic potential over the ${\mathbb{R}}^d$ ($d=1,2,3$) domain, where the regularization of the singular potential extends beyond the classical double-well formulation. A fully discrete finite element scheme is developed using a k-th-order finite element space for spatial approximation and a backward Euler scheme for the temporal discretization of a regularized system. The existence and uniqueness of numerical solutions are rigorously established by applying Brouwer’s fixed-point theorem. Moreover, the proposed numerical framework is shown to preserve the discrete energy dissipation law analytically, while a priori error estimates are derived. Finally, numerical experiments are conducted to verify the theoretical results and the inherent physical property, such as phase separation phenomenon and coarsening processes. The results show that the fractional Allen–Cahn model provides enhanced capability in capturing phase transition characteristics compared to its classical equation. Full article

Accurate and reliable multi-scene positioning remains a critical challenge in autonomous driving systems, as conventional fixed-noise fusion strategies struggle to handle the dynamic error characteristics of heterogeneous sensors in complex operational environments. This paper proposes a novel noise-adaptive fusion framework integrating Global Navigation Satellite System (GNSS) and Inertial Navigation System (INS) measurements. Our key innovation lies in developing a dual noise estimation model that synergizes priori weighting with posterior variance compensation. Specifically, we establish an a priori weighting model for satellite pseudorange errors based on elevation angles and signal-to-noise ratios (SNRs), complemented by a Helmert variance component estimation for posterior refinement. For INS error modeling, we derive a bias instability noise accumulation model through Allan variance analysis. These adaptive noise estimates dynamically update both process and observation noise covariance matrices in our Error-State Kalman Filter (ESKF) implementation, enabling real-time calibration of GNSS and INS contributions. Comprehensive field experiments demonstrate two key advantages: (1) The proposed noise estimation model achieves 37.7% higher accuracy in quantifying GNSS single-point positioning uncertainties compared to conventional elevation-based weighting; (2) in unstructured environments with intermittent signal outages, the fusion system maintains an average absolute trajectory error (ATE) of less than 0.6 m, outperforming state-of-the-art fixed-weight fusion methods by 36.71% in positioning consistency. These results validate the framework’s capability to autonomously balance sensor reliability under dynamic environmental conditions, significantly enhancing positioning robustness for autonomous vehicles. Full article

The local convergence analysis of a two-step vectorial method of accelerators with order five has been shown previously. But the convergence order five is obtained using Taylor series and assumptions on the existence of at least the fifth derivative of the mapping involved, which is not present in the method. These assumptions limit the applicability of the method. Moreover, a priori error estimates or the radius of convergence or uniqueness of the solution results have not been given. All these concerns are addressed in this paper. Furthermore, the more challenging semi-local convergence analysis, not previously studied, is presented using majorizing sequences. The convergence for both analyses depends on the generalized continuity of the Jacobian of the mapping involved, which is used to control it and sharpen the error distances. Numerical examples validate the sufficient convergence conditions presented in the theory. Full article

In this paper, the inverse problem of identifying the source term of the time fractional diffusion-wave equation is studied. This problem is ill-posed, i.e., the solution (if it exists) does not depend on the measurable data. Under the priori bound condition, the condition stable result and the optimal error bound are all obtained. The fractional Landweber iterative regularization method is used to solve this inverse problem. Based on the priori regularization parameter selection rule and the posteriori regularization parameter selection rule, the error estimation between the regularization solution and the exact solution is obtained. Moreover, the error estimations are all order optimal. At the end, three numerical examples are given to prove the effectiveness and stability of this regularization method. Full article

Multistep methods typically use Taylor series to attain their convergence order, which necessitates the existence of derivatives not naturally present in the iterative functions. Other issues are the absence of a priori error estimates, information about the radius of convergence or the uniqueness of the solution. These restrictions impose constraints on the use of such methods, especially since these methods may converge. Consequently, local convergence analysis emerges as a more effective approach, as it relies on criteria involving only the operators of the methods. This expands the applicability of such methods, including in non-Euclidean space scenarios. Furthermore, this work uses majorizing sequences to address the more challenging semi-local convergence analysis, which was not explored in earlier research. We adopted generalized continuity constraints to control the derivatives and obtain sharper error estimates. The sufficient convergence criteria are demonstrated through examples. Full article