Search Results (283)

The governing reaction–diffusion model for carbohydrate oxidation catalyzed by an immobilized bienzyme system glucose dehydrogenase and laccase within a spherical porous microreactor is adapted from Baronas et al. and extended here to the non-steady-state regime. The model consists of coupled non-linear partial differential equations based on non-Michaelis–Menten kinetics. The principal novelty of this work lies in the derivation of closed-form semi-analytical expressions for transient and steady-state concentrations of the carbohydrate substrate, oxygen, and product, as well as for the effectiveness factor, using the Laplace Homotopy Perturbation Method (LHPM). The LHPM solutions are validated against MATLAB R2026a numerical simulations (maximum error $<0.009\%$) and demonstrate superior accuracy compared to previously reported Adomian Decomposition Method (ADM) and Taylor Series Method (TSM) solutions. Parametric analysis reveals that the Thiele modulus, saturation parameters, and dimensionless time strongly influence the internal concentration profiles and reactor effectiveness. These analytical results provide rapid, closed-form predictive tools for optimizing catalyst particle size, enzyme loading, and operating conditions in immobilized enzyme microreactor systems. Full article

Generalization of the ordinary hyperbolic functions called hyperbolic Ateb-functions is considered. They are the inverse of incomplete Beta-function. These functions are solutions of differential equations, which describe the aperiodic vibration motion. It is shown that hyperbolic Ateb-functions have different dependence levels on their parameters. Investigation of the domain of hyperbolic Ateb-functions is conducted. It is shown that the minimum value of the domain can be expressed in terms of the lemniscate constant. The formulas for derivatives of hyperbolic Ateb-functions are proved and the structure of the higher-order derivatives is obtained. Some other properties connected with symmetry are considered. Taylor expansions of Ateb-cosine and Ateb-sine are taken out. Based on the mathematical induction principle, the corresponding theorems are proven. Examples of Taylor expansions of Ateb-cosine and Ateb-sine for different parameters are presented. The comparison of Ateb-function calculation using Taylor expansion and numerical methods shows the advantage of the Taylor series approach. Full article

This study presents a numerical and analytical investigation of the nonlinear smooth-discontinuous (SD) oscillator. The nonlinear restoring force is approximated by a fifth-order polynomial using a Taylor series expansion to simplify the governing equation while preserving the essential nonlinear characteristics of the system. To analyze the oscillator dynamics, two analytical approaches are applied, namely the spreading residue harmonic balance method (SRHPM) and the Multiple Scales Method (MSM). The obtained analytical solutions are validated through comparison with numerical simulations carried out using the classical fourth-order Runge–Kutta scheme. The results reveal a strong agreement between the analytical and numerical solutions, confirming the capability of both SRHPM and MSM to accurately describe the nonlinear oscillatory response of the SD system. Full article

This paper presents a hybrid numerical approach, termed the Radial Basis Function–Finite Difference Matrix Operator (RBF–FDMO) method, to enhance the accuracy and flexibility of the conventional FDMO technique for three-dimensional (3D) electromagnetic field analysis governed by the Laplace–Poisson equation. Conventional numerical methods often face challenges related to computational complexity and limited flexibility when handling non-uniform discretization and complex geometries. In the proposed method, spatial derivatives are approximated using RBF-based interpolation rather than finite difference schemes derived from Taylor series expansion. This formulation enables the construction of high-accuracy derivative operators on both uniform and non-uniform FD grids, thereby improving numerical robustness and adaptability to complex geometries. The performance of the proposed method is first compared with the FDMO in a 3D benchmark problem, with reductions of more than two orders of magnitude in both RMS and maximum errors. Furthermore, the RBF-FDMO approach is developed and, for the first time, applied to the analysis of grounding system (GS) configurations specified in IEEE Std. 80™, as well as a practical 110 kV substation GS in Vietnam. The obtained potential distributions, grounding resistances, and touch and step voltages confirm the effectiveness and reliability of the method. The results indicate that the proposed approach features a simple formulation and competitive computational efficiency, positioning it as a practical alternative to conventional methods like the finite element method (FEM) and the boundary element method (BEM) for GS analysis and design. Full article

Disturbances in dynamical systems pose a major challenge for parameter identification, particularly in the presence of unknown initial conditions and uncertain external influences. To address this issue, this paper proposes an algebraic parameter estimation methodology that incorporates fractional-order calculus in the Laplace domain for controlled linear engineering systems. The proposed approach eliminates the influence of unknown initial conditions and considers external disturbances that admit a local polynomial representation through Taylor series expansions over sufficiently small time intervals, while avoiding explicit numerical differentiation in the time domain. The manuscript includes analytical, numerical, and experimental validations to highlight the benefits of incorporating fractional-order differentiation in the derivation of algebraic estimators for online parameter estimation. The method is experimentally validated on two linear differentially flat electrical circuits, whose flat representations enable the proposed algebraic formulation under distinct disturbance signals. The results demonstrate that the fractional differentiation order acts as an additional tuning parameter, and that appropriately selected fractional orders can improve estimation accuracy, yielding parameter estimates consistently closer to their true values when compared with the conventional integer-order algebraic formulation. Full article

The convergence order of higher-order iterative methods for solving systems of nonlinear equations was analyzed using Taylor series expansion, which typically requires the computation of higher-order derivatives not inherently part of the method. This dependency limits the method’s applicability and increases the computational cost. The distinctiveness of our work lies in the development of improved convergence theorems that rely solely on first-order derivatives. The proposed approach offers a stronger framework than existing methods by incorporating details about the convergence region’s radius and providing precise error estimates. Furthermore, we explore semi-local convergence, which holds greater significance as it allows the identification of the specific domain where the iterative sequence remains valid. The theoretical findings are substantiated through suitable numerical illustrations. Full article

This paper introduces a new explicit integration method for second-order ordinary differential equations (ODEs) commonly encountered in engineering applications. Traditionally, these problems are solved either by reformulating them as first-order systems to apply one-step methods such as Runge–Kutta schemes, or by using direct second-order approaches widely adopted in linear dynamics, including the generalized-α, central difference, and Newmark methods. The proposed method is derived from a Taylor series expansion truncated at the third derivative, resulting in a fully explicit algorithm that requires only one function evaluation per time step. Similar to Newmark’s formulation, it includes adjustable parameters that allow the user to balance accuracy and stability. For a specific parameter choice, the method exhibits convergence and stability properties comparable to those of the central difference scheme. An important advantage is that it remains explicit even when nonlinearities depend on first-derivative terms. The paper presents a theoretical analysis covering stability, local truncation error, spectral properties, numerical damping, and period elongation. The method is validated through four test cases from multibody dynamics, including linear and nonlinear problems. Results demonstrate that the Explicit Integration Grade 3 (EIG-3) method achieves accuracy comparable to existing explicit second-order integrators while significantly reducing computational cost, particularly in nonlinear applications. Full article

This paper presents a comprehensive review of digital floating-point arithmetic algorithms that utilize Taylor series expansion in combination with mantissa-region division techniques, and it further demonstrates their generalization and applicability based on the findings of our research. While the discussion is broad in scope, this paper consolidates and systematizes the authors’ method within a broader contextual discussion, rather than presenting a fully systematic review of the entire state of the art in floating-point arithmetic algorithms. In many scientific computing applications, compact and low-power hardware implementations are essential. To address these requirements, this review presents algorithms specifically designed to operate under such constraints. The focus is placed on efficient floating-point operations—including division, inverse square root, square root, exponentiation, and logarithmic functions—all realized through Taylor series expansion with mantissa region division techniques. Furthermore, the trade-offs are examined in detail, covering factors such as the required numbers of additions, subtractions, and multiplications, along with the look-up table (LUT) size. The study further identifies the environments and application domains where the Taylor series expansion method combined with mantissa-region division is most effective, based on comparisons with various other floating-point computation algorithms and their corresponding hardware implementations. Overall, the review underscores the value of this unified framework in enabling efficient and adaptable floating-point computation across a wide range of hardware-constrained environments. Full article

Time series forecasting in power systems is crucial for power supply planning and exerts a direct impact on the electricity market. Accurate forecasting can effectively mitigate decision-making risks. This paper proposes a forecasting method based on a multi-dimensional Taylor network (MTN) and applies it to electricity price prediction. The time series is decomposed into one low-frequency signal and several high-frequency signals. The MTN model is constructed for each frequency sequence. The final forecast is obtained by aggregating the predictions from all frequency components. Using European electricity price data as a case study, experimental results demonstrate that the proposed method achieves high predictive accuracy. Full article

The existing methods for solving non-linear equations encounter convergence issues and computing constraints, especially when used in fractional-order or complex non-linear problems. This study develops a higher-order fractional technique for solving non-linear equations based on the Caputo fractional derivative. The proposed method uses a fractional framework to improve local convergence and stability while ensuring high efficiency in every iteration step. Local convergence analysis using generalized Taylor series expansion reveals that the order of the new fractional scheme for solving non-linear equations is $5¢+1$, where $¢\in$ (0,1] represents the Caputo fractional order, determining the memory depth of the Caputo fractional derivative. The performance of the method is further investigated using a variety of non-linear problems from engineering optimization and applied sciences, such as engineering control systems, computational chemistry, thermodynamics models, and operations research, such as inventory optimization. Analyzing the key performance metrics, such as dynamical analysis, percentage convergence, residual error, and computation time, confirms the advantages of the developed method over the state-of-the-art. This study provides a solid framework for higher-order fractional iterative approaches, paving the way for advanced applications of non-linear problems. Full article

The Taylor approximation theorem is a fundamental tool in numerical analysis, providing a local polynomial representation of smooth functions. In practical computations, a function f is approximated by a finite Taylor polynomial $P_n$, and controlling the resulting truncation error is of central importance. In this paper, we introduce two novel a posteriori error estimation techniques for Taylor polynomial approximations. The proposed estimators are fully computable and do not require prior bounds on the $(n+1)$st derivatives of f. We prove that the estimators converge to the exact error both pointwise and in the $L^2$-norm as $n\to \infty$, and we establish their asymptotic sharpness through effectivity analysis. Based on these results, we develop two adaptive algorithms that automatically determine the minimal degree n required to achieve a prescribed tolerance, either at a specific point or over a domain. We further extend the analysis to multivariate functions and show that analogous estimators and effectivity properties hold in higher dimensions. Numerical experiments are presented to validate the theoretical results and demonstrate the practical performance of the proposed methods. Full article

The applicability of a highly efficient sixth-order convergent method, originally proposed by Kansal et al., is extended in this study to a Banach space setting. The initial development of this method relied upon Taylor series expansions in ${\mathbb{R}}^n$ and the assumption that the nonlinear operator is sufficiently differentiable. This vague condition implies the existence of high-order derivatives that are not actually utilized by the algorithm. This study transcends these limitations by establishing convergence based solely on generalized continuity conditions of the first Fréchet derivative. By dispensing with these strong smoothness requirements, the domain of applicability is significantly widened. We derive computable radii for the ball of convergence and establish error bounds under local analysis. Furthermore, a rigorous semi-local convergence analysis is presented, a feature previously absent in the literature for this specific scheme, utilizing a majorizing sequence technique to guarantee the existence and uniqueness of the solution. The theoretical results are validated through numerical experiments, which demonstrate that the method converges even when the standard sufficiently differentiable conditions are violated. Full article

In this article, the problem of control of the kinematic model of an omnidirectional robot with time delay in the control input is tackled through an Active Disturbance Rejection Control (ADRC) with a disturbance predictor-based scheme, which consists in predicting the generalized forward disturbance input in order to cancel it and then using a feedforward linearization approach to control the system in trajectory tracking tasks. The novelties of the scheme are to demonstrate that using the proposed extended state disturbance estimation leads to a forward estimation following the Taylor series approximation, and, to avoid using additional pose predictions, a feedforward input as an exact linearization approach is used, in which the remaining dynamics can be lumped into the generalized disturbance input. Thus, the use of extended states in prediction improves the robustness of the predictor while increasing the prediction horizon for larger time delays. The stability of the proposal is demonstrated using the second method of Lyapunov, which shows the closed-loop estimation/tracking ultimate bound behavior. Additionally, numerical simulations and experimental tests validate the robustness of the approach in trajectory-tracking tasks. Full article

The description of the Earth’s gravitational field, governed by the fundamental potential equation (the Laplace equation), is conventionally expressed using spherical harmonics, yet the Cartesian formulation, using a Taylor series representation, offers significant algebraic advantages. This paper proposes a novel Direct Cartesian Method for generating spherical basis functions and coefficients directly within the Cartesian coordinate system, utilising the partial derivatives of the inverse distance ($1/R$) function. The present study investigates the structural correspondence between the Cartesian form of spherical basis functions and the high-order partial derivatives of $1/R$. The study reveals that spherical basis functions can be categorised into four distinct groups based on the parity of the degree n and order m. It is demonstrated that each spherical basis function is equivalent to a weighted summation of the partial derivatives of the inverse distance ($1/R$) with respect to Cartesian coordinates. Specifically, the basis functions are combined with those derivatives that share the same order of Z-differentiation and possess matching parities in their orders of differentiation with respect to X and Y. In order to facilitate the practical calculation of these high-degree derivatives, a recursive numerical algorithm has been developed. The method generates the polynomial coefficients for the numerator of the $1/R$ derivatives. A pivotal innovation is the implementation of a step-wise normalization scheme within the recursive relations. The integration of the recursive ratios of global normalization factors (including full Schmidt normalization) into each step of the algorithm effectively neutralises factorial growth, rendering the process immune to numerical overflow. The validity and numerical stability of the proposed method are demonstrated through a detailed step-by-step derivation of a sectorial basis function ($n=8,m=2$). Full article

This study presents a systematic comparative analysis of nine stator current discretization methods within the Model Predictive Current Control (MPCC) framework for Permanent Magnet Synchronous Motors (PMSMs). These methods have generally been examined individually or in limited combinations in previous research, and this holistic and comprehensive comparison constitutes the core contribution of this work by addressing a significant gap in the existing literature. The investigated MPCC methods—Forward Euler (FE), Backward Euler (BE), Midpoint Euler (ME), Fourth-Order Runge–Kutta (RK4), Runge–Kutta Ralston (RKR), Taylor Series (TS), Verlet Integration (VI), Crank–Nicolson (CN), and Adams–Bashforth (AB)—are comprehensively evaluated for their dynamic performance, including speed tracking, torque response, settling time, rise time, overshoot, and Total Harmonic Distortion (THD). Additionally, these analysis results are benchmarked against conventional Proportional–Integral–Derivative (PID) and Field-Oriented Control (FOC) methods. In terms of key performance indicators, the MPCC–RKR method proved optimal for speed tracking under no-load conditions, achieving the lowest overshoot, specifically ranging from 0.097% to 1.450%. Conversely, MPCC–ME and MPCC–CN demonstrated superior transient performance under sudden-load conditions (1.7 Nm), yielding the smallest torque deviations, fastest settling times. Specifically, MPCC-ME recorded the lowest overshoot (1.512%) at the 7 s load step, while MPCC-CN performed best at 9 s (1.220%) and 11 s (1.577%). Among the predictive schemes, the MPCC–RKR method achieved the highest current quality with a minimum THD of 3.69% at nominal speed. Finally, it has been confirmed through the applied statistical analysis techniques that the performance differences among the discretization methods are significant. The comparative analysis examines both the dynamic performance of the methods and the fundamental trade-off between accuracy and computational burden in MPCC design. Simple single-step explicit methods (FE, ME, RKR, VI, AB) offer low computational cost and are well suited for high–sampling-frequency real-time applications, especially with sufficiently small sampling times, whereas more complex multi-step or implicit methods (BE, RK4, TS, CN) may increase the processor load despite their potential gains in accuracy and stability. This study provides practical, evidence-based guidelines for selecting an optimal discretization method by balancing accuracy and dynamic performance requirements for PMSM applications. Full article