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This paper introduces a numerical approach for the fractional-order Rössler chaotic systems and gives error analysis. The effectiveness of the present method is determined by comparing the numerical results of the high-precision difference scheme and predictor-correctors scheme. Some complex dynamical behaviors for the fractional-order Rössler chaotic systems are shown. [ABSTRACT FROM AUTHOR]

The numerical solution of a 1D fractional Laplacian boundary value problem is studied. Although the fractional Laplacian is one of the most important and prominent nonlocal operators, its numerical analysis is challenging, partly because the problem’s solution has in general a weak singularity at the boundary of the domain. To solve the problem numerically, we use piecewise linear collocation on a mesh that is graded to handle the boundary singularity. A rigorous analysis yields a bound on the maximum nodal error which shows how the order of convergence of the method depends on the grading of the mesh; hence, one can determine the optimal mesh grading. Numerical results are presented that confirm the sharpness of the error analysis. [ABSTRACT FROM AUTHOR]

The goal of this article is to develop a spectral collocation method for solving a distributed-order fractional Korteweg-de Vries equation using fractional Zigzag rising diagonal functions. To meet this target, we first introduce Zigzag and Jaiswal polynomials. Then, using a transformation, we find their fractional counterparts. As a linear combination of these functions, we seek a solution to the problem. We will generate operational matrices for fractional Zigzag raising diagonal functions and apply Simpson's rule to approximate the distributed fractional derivative. The resultant approximate equations are collocated to create a system of nonlinear equations. Error analysis of the numerical scheme is fully discussed. Numerical experiments have demonstrated the capability and efficiency of the method. We also demonstrate how the approach might be helpful for problems with non-smooth solutions. [ABSTRACT FROM AUTHOR]

In this article, we propose an efficient hybrid method to calculate the highly oscillatory Bessel integral ∫ 0 1 f (x) x - τ J m (ω x γ) d x with the Cauchy type singular point, where 0 < τ < 1 , m ≥ 0 , 2 γ ∈ N +. The hybrid method is established by combining the complex integration method with the Clenshaw– Curtis– Filon– type method. Based on the special transformation of the integrand and the additivity of the integration interval, we convert the integral into three integrals. The explicit formula of the first one is expressed in terms of the Meijer G function. The second is computed by using the complex integration method and the Gauss– Laguerre quadrature rule. For the third, we adopt the Clenshaw– Curtis– Filon– type method to obtain the quadrature formula. In particular, the important recursive relationship of the required modified moments is derived by utilizing the Bessel equation and the properties of Chebyshev polynomials. Importantly, the strict error analysis is performed by a large amount of theoretical analysis. Our proposed methods only require a few nodes and interpolation multiplicities to achieve very high accuracy. Finally, numerical examples are provided to verify the validity of our theoretical analysis and the accuracy of the proposed methods. [ABSTRACT FROM AUTHOR]

This work presents a new numerical method for solving Volterra-type integro functional equations with variable bounds and mixed delay. This paper applies discrete orthogonal Hahn polynomials and their properties numerically. A discrete scalar product is associated with discrete orthogonal polynomials. Several numerical experiments (including linear and nonlinear) for multiple test problems are provided to validate the accuracy of this method. [ABSTRACT FROM AUTHOR]

In this article, we derive accurate Horner methods in real and complex floating-point arithmetic. In particular, we show that these methods are as accurate as if computed in k-fold precision and then rounded into the working precision. When k is two, our methods are comparable or faster than the existing compensated Horner routines. When compared to multi-precision software, such as MPFR and MPC, our methods are significantly faster, up to k equal to eight, that is, up to 489 bits in the significand. [ABSTRACT FROM AUTHOR]

In this article, we present a flux‐based formulation of the hybridizable discontinuous Galerkin (HDG) method for steady‐state diffusion problems and propose a new method derived by letting a stabilization parameter tend to infinity. Assuming an inf‐sup condition, we prove its well‐posedness and error estimates of optimal order. We show that the inf‐sup condition is satisfied by some triangular elements. Numerical results are also provided to support our theoretical results. [ABSTRACT FROM AUTHOR]

This study introduces a novel fractional-order model to investigate the interplay between cancer and obesity and their treatment. Initially, we examine the solution’s existence and uniqueness for the proposed model. Additionally, we establish the boundedness of these solutions. Subsequently, we identify some potential equilibrium points of the cancer–obesity model and investigate their stability. To address the considered model, we propose fractional Euler’s and Adam’s methods. Theoretical and numerical analyses are conducted to assess the error estimates and performance of both methods with varying fractional-order derivatives. Moreover, we formulate an optimal control problem concerning cancer density and drug concentration. We delve into the existence of control and explore the first-order optimality conditions. We validate the analytical findings through numerical computations, demonstrating that administering drugs with control variables enhance immunity levels and reduce the burden of cancer. [ABSTRACT FROM AUTHOR]

In this paper, we present the general Bernstein functions for the first time. The properties of generalized Bernstein basis functions are given and demonstrated. The classical Bernstein polynomial bases are merely a subset of the general Bernstein functions. Based on the new Bernstein base functions and the collocation method, we present a numerical method for solving linear and nonlinear χ-fractional differential equations (χ-FDEs) with variable coefficients. The fractional derivative used in this work is the χ-Caputo fractional derivative sense (χ-CFD). Combining the Bernstein functions basis and the collocation methods yields the approximation solution of nonlinear differential equations. These base functions can be used to solve many problems in applied mathematics, including calculus of variations, differential equations, optimal control, and integral equations. Furthermore, the convergence of the method is rigorously justified and supported by numerical experiments. [ABSTRACT FROM AUTHOR]

The obsidian dating method converts the quantity of diffused molecular water within a near‐surface hydration layer to elapsed time using an experimentally derived diffusion coefficient predicted from the structural water content of the glass. Infrared spectroscopic transmission measurements on transparent archaeological samples record vibrational responses of water bands in the near‐infrared region, permitting determination of structural water content (OH), and the amount of diffused ambient water (H2O). In this application, the H2O water band at 5200 cm−1 is measured directly. The accuracy of the approach is assessed by an evaluation of the precision of each contributing variable. The new protocol is evaluated using obsidian artifacts from radiocarbon‐dated deposits at Salamanca Cave in Argentina. [ABSTRACT FROM AUTHOR]

In this paper, we analyze the time‐dependent flow of an incompressible micropolar fluid in a multiple‐pipe system. Motivated by the applications, we assume that the pipes have circular cross‐section and that the ratio between pipes' thickness and its length is small, denoted by the parameter ε$$ \varepsilon $$. Far from the junction, the fluid exhibits different behavior depending on the magnitude of the viscosity coefficients with respect to the small parameter ε$$ \varepsilon $$. Focusing on the critical case described by the strong coupling between velocity and microrotation, the complete asymptotic expansion of the solution (up to an arbitrary order) is built. To improve the accuracy of the asymptotic approximation, we introduce the boundary layer correctors near the pipes' ends and take into account the interior layer correction in the vicinity of the junction as well. The convergence is also proved via error estimates, providing the rigorous justification of the proposed effective model. [ABSTRACT FROM AUTHOR]

Compared to traditional partial differential equation modeling methods, Markov switching models can accurately capture the abrupt changes or jumps that complex systems often experience in the real world. In this paper, we propose a novel wave equation model with Markovian switching to represent complex systems with state‐jumping phenomena better, and the well‐posedness of the model is proved. In addition, a numerical method with non‐uniform grids is also proposed for the proposed model to simulate the data in realistic situations, which is based on the use of finite element discretization in space and central difference discretization in time. Finally, we conduct several experiments to analyze the errors and stability of the proposed model and the traditional model. The results show that the Markov switching model proposed in this paper has a smaller error than the traditional models while ensuring stability and can more accurately simulate the state jump phenomena of real‐world systems. [ABSTRACT FROM AUTHOR]

The current study aims to utilize the homotopy perturbation method (HPM) to solve nonlinear dynamical models, with a particular focus on models related to predicting and controlling pandemics, such as the SIR model. Specifically, we apply this method to solve a six-compartment model for the novel coronavirus (COVID-19), which includes susceptible, exposed, asymptomatic infected, symptomatic infected, and recovered individuals, and the concentration of COVID-19 in the environment is indicated by S (t) , E (t) , A (t) , I (t) , R (t) , and B (t) , respectively. We present the series solution of this model by varying the controlling parameters and representing them graphically. Additionally, we verify the accuracy of the series solution (up to the (n − 1) t h -degree polynomial) that satisfies both the initial conditions and the model, with all coefficients correct at 18 decimal places. Furthermore, we have compared our results with the Runge–Kutta fourth-order method. Based on our findings, we conclude that the homotopy perturbation method is a promising approach to solve nonlinear dynamical models, particularly those associated with pandemics. This method provides valuable insight into how the control of various parameters can affect the model. We suggest that future studies can expand on our work by exploring additional models and assessing the applicability of other analytical methods. [ABSTRACT FROM AUTHOR]

In this paper, we provide a collocation spectral scheme for systems of nonlinear Caputo–Hadamard differential equations. Since the Caputo–Hadamard operators contain logarithmic kernels, their solutions can not be well approximated using the usual spectral methods that are classical polynomial-based schemes. Hence, we construct a non-polynomial spectral collocation scheme, describe its effective implementation, and derive its convergence analysis in both L 2 and L ∞ . In addition, we provide numerical results to support our theoretical analysis. [ABSTRACT FROM AUTHOR]

Accurate bearing‐time recordings play a crucial role in hydroacoustic target motion analysis and situation estimation. The target bearing estimated by vector hydrophone in the interference environment suffers from poor display and low signal‐to‐noise ratio. To solve this problem, a two‐stage bearing‐only motion analysis method is proposed. Firstly, the bearing peak information and bearing dispersion are extracted for each sampling moment of the bearing‐time recording. Custom constraint regions and adaptive parameters are then set based on this information. The order truncate average algorithm is improved to enhance the quality of the low‐quality bearing‐time recording map. Secondly, a bearing‐only target motion model is established based on the bearing‐time recording distribution features, and a bearing‐time recording trust interval is constructed. Then a modified pseudo‐linear estimation algorithm is proposed to solve the target motion situation information, including target abeam time, target abeam bearing, target course, velocity‐to‐initial‐distance ratio, and so forth. Finally, by comparing the solved values of the sea trial data with the automatic identification system calculation results, the enhancement effect of the target bearing‐time recording map is significant and the solving error of the target motion elements meets the needs of engineering applications. [ABSTRACT FROM AUTHOR]

In this paper, a new direct calculation method of frost ice shape on the blade surface of horizontal axis wind turbine is proposed. Using linear interpolation algorithm, the airfoil ice shape obtained by LEWICE 2D software or ice wind tunnel experiment was fitted with equidistant step length in the first and fourth quadrants and equidistant step length in the second and third quadrants. The key point coordinates of ice shapes on cross-sections along the span-wise were mapped into lagging and flapping surfaces through the mathematical dimension reduction, respectively. The polynomial fitting was used to deal with ice projection points of multiple sections in lagging and flapping surfaces, and then the blade's frost ice shape was obtained. By calculating the sum of squared residuals of the polar diameter at the same polar angle, the errors between experimental and airfoil frost shape fitting methods, experimental and FENSAP, and blade frost shape formation methods and FENSAP were analyzed. The results show that the new method is in good agreement with the ice shape of FENSAP simulation results and experimental results. The residual sum of squares is small. This method makes the analysis of frost ice morphology of wind turbine blades do not need to consider interdisciplinary. The calculation process is simple and reliable. [ABSTRACT FROM AUTHOR]

IntroductionMethodResultsConclusionsProspective memory (PM) deficits have been documented in multiple sclerosis (MS). This study aimed to explore the specific types of errors made by persons with MS (PwMS), including differences between PwMS and healthy controls (HC) and PwMS who do and do not have impairments in processing speed and/or verbal learning and memory.PwMS (n = 111) and HC (n = 75) completed the Memory for Intentions Test (MIST), an objective measure of PM that has five types of errors that can be coded (PM failure, task substitution, loss of content, loss of time, and random errors). The number and types of PM errors were calculated for the overall MIST and six subscales, which break down performance by types of delay (2-Minute and 15-Minute), cue (Time and Event), and response (Verbal and Action). Impairment was defined as performing < 1.5 SD on either the Symbol Digit Modalities Test (SDMT) or Rey Auditory Verbal Learning Test (RAVLT). Bivariate analyses were used to examine group differences, with post-hoc pairwise comparisons with Bonferroni corrections.Nearly 93% of PwMS made at least one PM error, compared to 76% of HC (V = .24, p = .001). The most commonly made PM error by PwMS was loss of content errors (45.0%). PwMS made significantly more task substitution errors (26.4% vs. 7.6%, p < .001) and fewer loss of time errors (9.5% vs. 21.2%, p < .001) than HC. Impaired PwMS made more errors than non-impaired PwMS, specifically PM failures on time-based tasks.PM errors are common in PwMS, particularly when there are longer delays and time-based cues. Not only do PwMS make more errors than demographically similar HC, but they exhibit different cognitive process failures. [ABSTRACT FROM AUTHOR]

The XY model is a simplified insulation model for transformers that aims to connect the frequency spectrum of major insulation, insulation paper, and oil. It enables the calculation of the paper's frequency spectrum, facilitating a quantitative evaluation of the transformer's insulation status. Given the scarcity of research addressing the accuracy of the XY model and its corresponding equivalent circuit, this paper investigates the accuracy of the XY model and conducts an in‐depth analysis of the error associated with its equivalent circuit. Firstly, the frequency spectrum of oil‐paper insulations based on the XY model with varying moisture content and structure configurations were measured, followed by the simulation of both the corresponding 2D‐coaxial and XY models. By comparing the results obtained from these simulations with experimental data on oil‐paper insulation, the accuracy of XY model is verified. Subsequently, two variants of the equivalent circuit of the XY model are derived by combining the oil, spacer, and paper differently, which denoted as Type I and II circuits. Finally, we analyzed and explained the errors present in these equivalent circuits compared with the XY model. Our findings show that the width‐thickness ratio of oil‐paper insulation is the most critical influencing factor, while moisture content and insulation structure have a less decisive impact. Notably, for general transformers, as the width‐thickness ratio is between 30 and 70, the relative error of the Type I circuit remains consistently below 0.3% across different frequencies. Consequently, the Type I circuit proves to be a more suitable representation of the equivalent circuit for the XY model in the context of general transformers. © 2024 Institute of Electrical Engineers of Japan. Published by Wiley Periodicals LLC. [ABSTRACT FROM AUTHOR]

In solid state physics, electronic properties of crystalline materials are often inferred from the spectrum of periodic Schrödinger operators. As a consequence of Bloch's theorem, the numerical computation of electronic quantities of interest involves computing derivatives or integrals over the Brillouin zone of so-called energy bands, which are piecewise smooth, Lipschitz continuous periodic functions obtained by solving a parametrized elliptic eigenvalue problem on a Hilbert space of periodic functions. Classical discretization strategies for resolving these eigenvalue problems produce approximate energy bands that are either non-periodic or discontinuous, both of which cause difficulty when computing numerical derivatives or employing numerical quadrature. In this article, we study an alternative discretization strategy based on an ad hoc operator modification approach. While specific instances of this approach have been proposed in the physics literature, we introduce here a systematic formulation of this operator modification approach. We derive a priori error estimates for the resulting energy bands and we show that these bands are periodic and can be made arbitrarily smooth (away from band crossings) by adjusting suitable parameters in the operator modification approach. Numerical experiments involving a toy model in 1D, graphene in 2D, and silicon in 3D validate our theoretical results and showcase the efficiency of the operator modification approach. [ABSTRACT FROM AUTHOR]

Stochastic differential equations (SDEs) are very useful in modeling many problems in biology, economic data, turbulence, and medicine. Fractional Brownian motion (fBm) and variable‐order fractional Brownian motion (vofBm) are suitable alternatives to standard Brownian motion (sBm) for describing and modeling many phenomena, since the increments of these processes are dependent of the past and for H>12$$ \mathcal{H}>\frac{1}{2} $$ these increments have the property of long‐term dependence. Classical mathematical techniques such as Ito's calculus do not work for stochastic computations on fBm and vofBm due to they are not semi‐Martingale for H(ξ)≠12$$ \mathcal{H}\left(\xi \right)\ne \frac{1}{2} $$. Therefore, solving these equations is much more difficult than solving SDEs with sBm. On the other hand, these equations do not have an analytical solution, so we have to use numerical methods to find their solution. In this paper, a computational approach based on hybrid of block‐pulse and parabolic functions (HBPFs) has been introduced for simulating vofBm and solving a modern class of SDEs. The mechanism of this approach is based on stochastic and fractional integration operational matrices, which transform the intended problem to a nonlinear system of algebraic equations. Thus, the complexity of solving the mentioned problem is reduced significantly. Also, convergence analysis of the expressed method has been theoretically examined. Finally, the accuracy and efficiency of the proposed algorithm have been experimentally investigated through some test problems and comparison of obtained results with results of previous papers. High accurate numerical results are obtained by using a small number of basic functions. Therefore, this method deals with smaller matrices and vectors, which is one of the most important advantage of our suggested method. Also, presenting an applicable procedure to construct vofBm is another innovation of this work. To gain this aim, at first, discretized sBm is generated via fundamental features of this process, and afterward, block‐pulse functions (BPFs) and HBPFs are utilized for simulating discretized vofBm. Finally, spline interpolation method has been employed to provide a continuous path of vofBm. [ABSTRACT FROM AUTHOR]

Among the problems identified at Technical and Vocational Education and Training (TVET) colleges, low achievement in mathematical subjects is the most prominent one. This paper documents a qualitative case study undertaken at TVET College in Gauteng with the purpose of exploring the National Certificate Vocational (NC(V)) Level 4 students' errors and associated misconceptions when answering optimisation questions in differential calculus. The participants were 60 students who were registered for a course in mathematics. Data were generated from the written student responses to two non-routine test items followed by interviews. Using the Newman error hierarchical model to analyse the data, it was discovered that students errors were conceptual and procedural as well as systematic and non-systematic. The literature also alluded to instructional approaches as some of the causes of students' misconceptions and the errors and that interventions should target students as well as the instructors. [ABSTRACT FROM AUTHOR]

In view of the problems of high-cost and accumulation of errors in the existing attitude measurement methods of roadheader, an attitude measurement algorithm based on the principle of double-vector attitude determination is proposed. By constructing and sensing the gravity-vector and light-vector respectively in the roadway excavation environment, using inertial inclination measurement and binocular vision measurement techniques, the attitude of the roadheader carrier coordinate frame relative to the roadway navigation coordinate frame can be realized based on the mathematical expression of vector elements in the navigation coordinate frame and the roadheader carrier coordinate frame. A measuring device that consists of an inclinometer and a binocular camera measures the indicated laser and gravity-vectors, and then the attitude of the roadheader can be solved using the double-vector attitude determination algorithm. The static repeatability measurement experiment is designed, and the experimental results show that the repeatability measurement precision of the attitude angles is 0.066 2°. The simulation analysis of the errors that may be introduced using the Monte Carlo method, which shows that the effect of the errors on the yaw, pitch and roll angles are 0.786 4°, 0.454 8° and 0.476 5°, respectively. [ABSTRACT FROM AUTHOR]

In this study, we propose a moving least squares approximation with shifted Chebyshev polynomials to solve linear and nonlinear third-kind Volterra delay integral equations (VDIEs). The suggested approach does not use meshing and does not rely on the geometry of the domain; therefore, we may consider it as a meshless method. This method approximates the solution using the collocation method based on the moving least squares approximation. The formulation of the technique for the suggested equations is described, and its convergence is analysed. Numerical results are presented to demonstrate the high resolution of the proposed approach and confirm its capability to provide accurate and efficient computations for Volterra delay integral equations of the third kind. [ABSTRACT FROM AUTHOR]

We consider the nonlinear Schrödinger equation with a defocusing nonlinearity which is mass-(super)critical and energy-subcritical. We prove uniform in time error estimates for the Lie–Trotter time splitting discretization. This uniformity in time is obtained thanks to a vectorfield which provides time decay estimates for the exact and numerical solutions. This vectorfield is classical in scattering theory and requires several technical modifications compared to previous error estimates for splitting methods. [ABSTRACT FROM AUTHOR]

In the present work, India's primary energy use is analysed in terms of four socio-economic variables, including Gross Domestic Product, population, and the amounts of exports and imports. Historical data were obtained from the World Bank database for 44 years as annual values (1971–2014). Energy use is analysed as an optimisation problem, where a unique ensemble of two metaheuristic algorithms, Grammatical Evolution (GE), and Differential Evolution (DE), is applied. The energy optimisation problem has been investigated in two ways: estimation and a year-ahead prediction. Models are compared using RMSE (objective function) and further ranked using the Global Performance Index (GPI). For the estimation problem, RMSE values are found to be as low as 0.0078 and 0.0103 on training and test datasets, respectively. The average estimated energy use is found in good agreement with the data (RMSE = 6.3749 kgoe/capita), and the best model (E10) has an RMSE of 5.8183 kgoe/capita, with a GPI of 1.7249. For the prediction problem, RMSE is found to be 0.0096 and 0.0122 on training and test datasets, respectively. The average predicted energy use has RMSE of 7.8857 (kgoe/capita), while Model P20 has the best value of RMSE (7.9201 kgoe/capita) and a GPI of 1.8836. [ABSTRACT FROM AUTHOR]

The effect of first language transfer on second language acquisition and learning has been a major theoretical concept in second language research and pedagogy since the 1950s. In order to give a historical perspective, the authors offer a brief presentation of some of the major topics from the broad spectrum of issues that have been examined by the applied linguistics research community during the past six decades and some significant developments in neuroimaging and cognitive science that have allowed researchers to investigate the role that first language plays in transfer to second language during neural activity and cognitive processing. The paper concludes with some pedagogical implications for second language instructors. [ABSTRACT FROM AUTHOR]

New low-order H (div) -conforming finite elements for symmetric tensors are constructed in arbitrary dimension. The space of shape functions is defined by enriching the symmetric quadratic polynomial space with the (d + 1) -order normal-normal face bubble space. The reduced counterpart has only d (d + 1) 2 degrees of freedom. Basis functions are explicitly given in terms of barycentric coordinates. Low-order conforming finite element elasticity complexes starting from the Bell element, are developed in two dimensions. These finite elements for symmetric tensors are applied to devise robust mixed finite element methods for the linear elasticity problem, which possess the uniform error estimates with respect to the Lamé coefficient λ , and superconvergence for the displacement. Numerical results are provided to verify the theoretical convergence rates. [ABSTRACT FROM AUTHOR]

In this article, we analyse the fractional Cattaneo heat equation for studying the conduction of heat in porous medium. This equation is also used in studying extended irreversible thermodynamics, material, plasma, cosmological model, computational biology, and diffusion theory in crystalline solids. The Sumudu adomian decomposition technique, which is combination of Sumudu transform and a numerical technique, is applied for getting numerical solution. The existence and uniqueness is analysed by using the fixed point theorem and the highest error of the designed technique is also analysed. Finally, the accuracy of the designed numerical method is presented by solving two examples and the findings are compared with the existing method. [ABSTRACT FROM AUTHOR]

Accurate modeling of photovoltaic (PV) modules/cells is crucial for evaluating the efficiency of solar PV systems. However, the lack of specific parameters of solar cells, which are not included in the manufacturer's datasheet, often results in flawed cell modeling. Quick and convenient parameter extraction techniques are required to overcome this challenge and create a robust solar PV cell model. These models are useful in optimization, simulation, and enhanced energy harvesting from PV-based renewable energy systems. This paper uses a recent optimizer called Rat Search Algorithm Particle Swarm Optimization (RSAPSO), for extracting the parameters of single diode and double diode models of PV cells. RSAPSO combines the exploration and exploitation advantages of both algorithms. The RSAPSO's parameter optimization results were compared against five other techniques, and the superiority of the suggested algorithm was confirmed through ranking tests, statistical error analyses, and temperature variation sensitivity analyses. [ABSTRACT FROM AUTHOR]

Consider Dirichlet problems of Laplace's equation in a bounded simply‐connected domain S$$ S $$, and use the null field equation (NFE) of Green's representation formulation, where the source nodes Q$$ Q $$ are located on a pseudo‐boundary ΓR$$ {\Gamma}_R $$ outside S$$ S $$ but not close to its boundary Γ(=∂S)$$ \Gamma \kern0.3em \left(=\partial S\right) $$. Simple algorithms are proposed in this article by using the central rule for the NFE, and the normal derivatives uν$$ {u}_{\nu } $$ of the solutions on the boundary Γ(=∂S)$$ \Gamma \kern0.3em \left(=\partial S\right) $$ can be easily obtained. These algorithms are called the discrete null field equation method (DNFEM) because the collocation equations are, indeed, the direct discrete form of the NFE. The bounds of the condition number are like those by the method of fundamental solutions (MFS) yielding the exponential growth as the number of unknowns increases. One trouble of the DNFEM is the near singularity of integrations for the solutions in boundary layers in Green's representation formulation. The traditional BEM also suffers from the same trouble. To deal with the near singularity, quadrature by expansions and the sinh transformation are often used. To handle this trouble, however, we develop two kinds of new techniques: (I) the interpolation techniques by Taylor's formulas with piecewise q$$ q $$‐degree polynomials and the Fourier series, and (II) the mini‐rules of integrals, such as the mini‐Simpson's and the mini‐Gaussian rules. Error analysis is made for technique I to achieve optimal convergence rates. Numerical experiments are carried out for disk domains to support the theoretical analysis made. The numerical performance of the DNFEM is excellent for disk domains to compete with the MFS. The errors with O(10−4)$$ O\left(1{0}^{-4}\right) $$ can be obtained by combination algorithms, which are satisfactory for most engineering problems. In summary, the new simple DNFEM is based on the NFE, which is different from the boundary element method (BEM). The theoretical basis in error and stability has been established in this article. One trouble in seeking the numerical solutions in boundary layers has been handled well; this is also an important contribution to the BEM. Besides, numerical experiments are encouraging. Hence the DNFEM is promising, and it may become a new boundary method for scientific/engineering computing. [ABSTRACT FROM AUTHOR]

In this paper, we first introduce a new series of Legendre basis functions by using the matrix decomposition technique, which are simultaneously orthogonal in both L 2 - and H 1 -inner products. Then, we construct efficient space-time spectral method for linear Sobolev equation using spectral approximation in space and multi-domain collocation approximation in time, which can be implemented in a synchronous parallel fashion. Next, we propose a novel Legendre spectral element method for solving nonlinear Sobolev equation with Burgers' type term, which reduce the non-zero entries of linear systems and computational cost. Some rigorous error estimates are carried out for one-dimensional Sobolev equation. Numerical experiments illustrate the effectiveness and accuracy of the suggested approaches. [ABSTRACT FROM AUTHOR]

In this paper, we first investigate the well-posedness and regularity of mild solution to a stochastic active fluids model driven by the additive noise. A fully-discrete scheme is proposed for solving the given model, which is based on the finite element method for spatial discretization and the backward Euler method for temporal discretization. By overcoming the difficulty of error analysis caused by the discrete Laplacian operator, we obtain the convergence results of the developed scheme. Finally, some numerical examples are provided to validate the theoretical results and we also simulate the motion states of the active fluids. [ABSTRACT FROM AUTHOR]

This research article discusses a numerical solution of the radiative transfer equation based on the weak Galerkin finite element method. We discretize the angular variable by means of the discrete-ordinate method. Then the resulting semi-discrete hyperbolic system is approximated using the weak Galerkin method. The stability result for the proposed numerical method is devised. A priori error analysis is established under the suitable norm. In order to examine the theoretical results, numerical experiments are carried out. [ABSTRACT FROM AUTHOR]

This study concentrates on time fractional convection–diffusion equations (TFCDEs) with variable coefficients and their numerical solutions. Caputo derivative is used to calculate the time fractional order derivatives. In order to give an approximate solution to the TFCDE, an effective approach is proposed utilizing Fibonacci wavelet and block pulse functions. The Fibonacci wavelets operational matrices of fractional order integration are constructed. By combining the collocation technique, they are used to simplify the fractional model to a collection of algebraic equations. The suggested approach is quite practical for resolving issues of this nature. The comparison and analysis with other approaches demonstrate the effectiveness and precision of the suggested approach. [ABSTRACT FROM AUTHOR]

AbstractAccurate assessment of uncertainties in cross-section data is crucial for reliable nuclear reactor simulations and safety analyses. In this study, we focus on the interpolation procedure of the partial derivatives (PD) cross-section model used to evaluate nodal parameters from pregenerated multigroup libraries. Our primary objective is to develop a systematic methodology that enables prediction of the incurred errors in the cross-section model, leading to the development of optimal case matrices, more efficient cross-section models, and informed case matrix construction for reactor types lacking substantial data and experience. We make progress toward this objective through a rigorous analytic error analysis enabled by the derivation of error expressions and bounds for the PD model based on the discovery that the method is a form of Lagrange interpolation. Our investigations reveal distinct outcomes depending on the chosen cross-section functionalizations, achieved by identifying the sources of error. These error sources are found to include interpolation error, which is always present, and model form error, which is a property of the supplied case matrix. We show that simply increasing grid refinement through the addition of branches may not always lead to decreased cross-section errors, particularly in cases where the model form error predominantly contributes to the total error. We present numerical results and verification in a companion paper, showing these different error characteristics for various cross-section functionalizations. Although applied to current light water reactor environments, our methodology offers a means for advanced reactor analysts to develop case matrices with quantified error levels, advancing the goal of a general methodology for robust two-step reactor analysis. Future work includes exploring different lattice types and functionalizations, extending reactivity bounds to multilattice problems, and investigating historical effects within the macroscopic depletion model. [ABSTRACT FROM AUTHOR]

The Nyström method is a convenient strategy to obtain low-rank approximations to kernel matrices in nearly linear complexity. Existing studies typically use the method to approximate positive semidefinite matrices with low or modest accuracies. In this work, we propose a series of heuristic strategies to make the Nyström method reach high accuracies for nonsymmetric and/or rectangular matrices. The resulting methods (called high-accuracy Nyström methods) treat the Nyström method and a skinny rank-revealing factorization as a fast pivoting strategy in a progressive alternating direction refinement process. Two refinement mechanisms are used: alternating the row and column pivoting starting from a small set of randomly chosen columns, and adaptively increasing the number of samples until a desired rank or accuracy is reached. A fast subset update strategy based on the progressive sampling of Schur complements is further proposed to accelerate the refinement process. Efficient randomized accuracy control is also provided. Relevant accuracy and singular value analysis is given to support some of the heuristics. Extensive tests with various kernel functions and data sets show how the methods can quickly reach prespecified high accuracies in practice, sometimes with quality close to SVDs, using only small numbers of progressive sampling steps. [ABSTRACT FROM AUTHOR]

In this paper, the permeability of sandstone, mudstone, gangue, and limestone under the condition of initial elastic deformation is obtained by transient method test. Mercury intrusion test is carried out on the four kinds of rock samples to obtain the initial permeability of the rock samples. The results show that the magnitude of permeability obtained by mercury intrusion test and transient method test is completely consistent. The measurement error is analyzed. The relative error of permeability in this paper is less than 6.0%. It can be showed that the use of transient method to measure the permeability of rock samples is reasonable and reliable. [ABSTRACT FROM AUTHOR]

An outlier in the Internet of Things is an immediate change in data induced by a significant difference in the atmosphere (Event) or sensor malfunction (Error). Outliers in the data cause the decision-maker to make incorrect judgments about data analysis. Hence it is essential to detect outliers in any discipline. The detection of outliers becomes the most crucial task to improve sensor data quality and ensure accuracy, reliability and robustness. In this research, a novel outlier detection technique based on spatial, temporal correlations and attribute correlations is proposed to detect outliers (both Errors and Events). This research uses a correlation measure in the temporal correlation algorithm to determine outliers and the spatial correlation algorithm to classify the outliers, whether the outliers are events or errors. This research uses optimal clusters to improve network lifetime, and malicious nodes were also detected based on spatial–temporal correlations and attribute correlations in these clusters. The experimental results proved that the proposed method in this research outperforms some other models in terms of accuracy against the percentage of outliers infused and detection rate against the false alarm rate. [ABSTRACT FROM AUTHOR]

An error estimate for a canonical discretization of the harmonic map heat flow into spheres is derived. The numerical scheme uses standard finite elements with a nodal treatment of linearized unit-length constraints. The analysis is based on elementary approximation results and only uses the discrete weak formulation. [ABSTRACT FROM AUTHOR]

Geostationary Operational Environmental Satellites (GOES) Radar Estimation via Machine Learning to Inform NWP (GREMLIN) is a machine learning model that outputs composite reflectivity using GOES-R Series Advanced Baseline Imager (ABI) and Geostationary Lightning Mapper (GLM) input data. GREMLIN is useful for observing severe weather and initializing convection for short-term forecasts, especially over regions without ground-based radars. This study expands the evaluation of GREMLIN's accuracy against the Multi-Radar Multi-Sensor (MRMS) System to the entire contiguous United States (CONUS) for the entire annual cycle. Regional and temporal variation of validation metrics are examined over CONUS by season, day of year, and time of day. Since GREMLIN was trained with data in spring and summer, root-mean-square difference (RMSD) and bias are lowest in the order of summer, spring, fall, and winter. In summer, diurnal patterns of RMSD follow those of precipitation occurrence. Winter has the highest RMSD because of cold surfaces mistaken as precipitating clouds, but some of these errors can be removed by applying the ABI clear-sky mask product and correcting biases using a lookup table. In GREMLIN, strong echoes are closely related to the existence of lightning and corresponding low brightness temperatures, which result in different error distributions over different regions of CONUS. This leads to negative biases in cold seasons over Washington State, lower 30-dBZ critical success index caused by high misses over the Northeast, and higher false alarms over Florida that are due to higher frequency of lightning. [ABSTRACT FROM AUTHOR]

A collocation method for the numerical solution of Volterra integro-differential equations with weakly singular kernels, based on piecewise polynomials of fractional order, is constructed and analysed. Typical exact solutions of this class of problems have a weak singularity at the initial time t = 0 . A rigorous error analysis of our method shows that, with an appropriate choice of the fractional-order polynomials and a suitably graded mesh, one can attain optimal orders of convergence to the exact solution and its derivative, and certain superconvergence results are also derived. In particular, our analysis shows that on a uniform mesh our method attains a higher order of convergence than standard piecewise polynomial collocation. Numerical examples are presented to demonstrate the sharpness of our theoretical results. [ABSTRACT FROM AUTHOR]

It has been known that the traditional scaling argument cannot be directly applied to the error analysis of immersed finite elements (IFE) because, in general, the spaces on the reference element associated with the IFE spaces on different interface elements via the standard affine mapping are not the same. By analyzing a mapping from the involved Sobolev space to the IFE space, this article is able to extend the scaling argument framework to the error estimation for the approximation capability of a class of IFE spaces in one spatial dimension. As demonstrations of the versatility of this unified error analysis framework, the manuscript applies the proposed scaling argument to obtain optimal IFE error estimates for a typical first-order linear hyperbolic interface problem, a second-order elliptic interface problem, and the fourth-order Euler-Bernoulli beam interface problem, respectively. [ABSTRACT FROM AUTHOR]

AbstractAccurately predicting errors incurred in a cross-section model for two-step reactor analysis enables the development of optimal case matrices and more efficient cross-section models. In a companion paper, we developed a systematic methodology for the partial derivatives cross-section model through rigorous analytic error analysis. In this paper, we verify our methodology against the conventional “brute force” numerical approach using a typical pressurized water reactor (PWR) lattice. By successfully reproducing known results, we gain confidence in our methodology’s application to advanced reactor environments, where optimal case matrices are generally not known. Our error methodology relies on accurately estimating bounds on the derivatives of the cross-section functions, a task we achieve through an order of convergence study. We use these derivative bounds in derived error expressions to obtain pointwise and setwise cross-section error bounds and verify these results with reference solutions of various two-group cross sections. We then propagate the cross-section error bounds to reactivity error using first-order perturbation theory and analyze their combined effect. Application of this approach to our test problem corroborates our prior qualitative findings with quantitative evidence and reveals the relative magnitudes of interpolation and model form error sources across diverse PWR cross-section functionalizations. Our results suggest systematic pathways for improving case matrix construction to minimize the overall error. These findings also confirm what is well known to the light water reactor design community, which is that interpolation error of cross sections for standard interpolation procedures and case matrix structures is on the order of 10 pcm or less. Future work includes exploring different lattice types and functionalizations, extending reactivity bounds to multi-lattice problems, and investigating historical effects within the macroscopic depletion model. [ABSTRACT FROM AUTHOR]

In this study, banana peel powder (BPP) was used to remove carcinogenic crystal violet (CV) dye from aqueous solution. The BPP adsorbent was characterized by Fourier-Transform Infrared (FTIR) spectroscopy and Scanning Electron Microscopy (SEM). In batch sorption experiments, the maximum removal of CV was recorded 93% at optimum levels of operating parameters, i.e., pH 7.0, adsorbent dosage 0.1 g, contact time 10 min, initial adsorbate concentration 90 ppm and temperature 20 C o. The kinetics of the adsorption process was found best to follow the pseudo first order model with high value of R 2 (0.9999) and low values of error functions. The Langmuir isotherm model was the best fit with high R 2 (0.9552) low values of error models. The plots of Freundlich and D-R isotherms confirmed the feasibility ( n = 1.3109 L g − 1 ) and the physisorption nature ( E s = 0.1107 kJ mol − 1 ) of the process respectively. The negative values of thermodynamic parameters (Δ G & Δ H) revealed that the adsorption was spontaneous and exothermic in nature respectively. The results of the desorption study showed that 91% of the adsorbent was regenerated. The novelty of the present research is that no work has been reported till date to remove crystal violet dye by using banana peel specifically. The high adsorption and desorption efficiencies (>90%) suggest that BPP possesses characteristics to be used as an effective, fast and low-cost adsorbent for adsorption of CV dye from industrial effluents. Adsorption technique was employed to remove crystal violet dye by using banana peel powder. The hydroxyl ( OH − ) & carboxylate ( OH − ) groups on adsorbent surface provided excellent binding with adsorbate molecules. Equilibrium attainment in 10 min at neutral pH. 93% adsorption and 91% desorption. [ABSTRACT FROM AUTHOR]

The constitutive model and its material parameters largely determine the effectiveness and accuracy of the finite element (FE) simulation for metal cutting processes. However, even for the same workpiece material, there are multiple constitutive models with different predictive abilities that are applicable to the cutting simulation. Therefore, a method for evaluating the constitutive models based on the coupled Eulerian-Lagrangian (CEL) orthogonal cutting model incorporating the experimentally determined Coulomb friction coefficient is proposed. The evaluation method can effectively avoid the influences of chip separation criteria, damage models, and adaptive remeshing on the evaluation results with the ability to simulate material side flow. This paper evaluates Johnson-Cook (JC) and Zerilli-Armstrong (ZA) constitutive models, which are often applied in cutting simulations. Material parameters for these constitutive models are identified from the constitutive data of the primary or secondary shear zone in cutting tests. A series of FE simulations for orthogonal cutting of 42CrMo4 steel is carried out under various cutting conditions using different constitutive models. The simulated cutting forces, chip thickness, and average temperature over the tool-chip interface are compared with experimental results under the same cutting conditions. The conclusion is that no constitutive model is the most accurate for all predictions. Nevertheless, for 42CrMo4 steel, the ZA model with material parameters identified from constitutive data of the secondary shear zone has the best comprehensive predictive performance. The evaluation method contributes to selecting the most suitable constitutive model and conducting efficient cutting simulations. Furthermore, several feasible approaches for improving the constitutive models are presented through error analysis. [ABSTRACT FROM AUTHOR]

In this paper, we consider a nonlinear switched dynamical system (NSDS) with unknown system parameters in the context of uncertain experimental data. This system is employed to model the continuous fermentation for the production of 1,3-propanediol through glycerol bioconversion. The uncertain experimental data points are regarded as stochastic variables and only their first-order moment information of the probability distributions is available. The target of this paper is to optimize these system parameters under the environment of uncertain experimental data. Taking these factors into account, we propose a distributionally robust system identification (DRSI) problem (i.e., a bi-level system identification problem) governed by the NSDS. The objective functional comprises two level objectives: (i) the inner-level objective aims to maximize the expectation of the relative error between the solution of the NSDS and the uncertain experimental data with respect to their probability distributions at approximately stable time; and (ii) the outer-level objective is to minimize the expectation with respect to these system parameters. The DRSI problem is equivalently transformed into a single-level system identification (SLSI) problem with non-smooth term through the application of the duality theory in the probability space. A smoothing technique is employed to approximate the non-smooth term in the SLSI problem. Subsequently, an error analysis of the employed smoothing technique is derived. The gradients of the objective functional in the SLSI with respect to these system parameters are obtained. An optimization algorithm is designed to solve the SLSI problem. Finally, the paper concludes with simulation results. • The DRSI problem lies between RSI problem and SSI problem. • An optimization algorithm with the duality theory is proposed to solve the DRSI problem. • A smoothing technique is employed to approximate the non-smooth term in the objective functional. [ABSTRACT FROM AUTHOR]

This article deals with two different numerical approaches for solving singularly perturbed parabolic problems with time delay and interior layers. In both approaches, the implicit Euler scheme is used for the time scale. In the first approach, the upwind scheme is used to deal with the spatial derivatives whereas in the second approach a hybrid scheme is used, comprising the midpoint upwind scheme and the central difference scheme at appropriate domains. Both schemes are applied on two different layer resolving meshes, namely a Shishkin mesh and a Bakhvalov–Shishkin mesh. Stability and error analysis are provided for both schemes. The comparison is made in terms of the maximum absolute errors, rates of convergence, and the computational time required. Numerical outputs are presented in the form of tables and graphs to illustrate the theoretical findings. [ABSTRACT FROM AUTHOR]

AbstractPartial differential equations (PDEs) have become an essential tool for modeling complex physical systems. Such equations are typically solved numerically via mesh-based methods, such as finite element methods, with solutions over the spatial domain. However, obtaining these solutions are often prohibitively costly, limiting the feasibility of exploring parameters in PDEs. In this article, we propose an efficient emulator that simultaneously predicts the solutions over the spatial domain, with theoretical justification of its uncertainty quantification. The novelty of the proposed method lies in the incorporation of the mesh node coordinates into the statistical model. In particular, the proposed method segments the mesh nodes into multiple clusters via a Dirichlet process prior and fits Gaussian process models with the same hyperparameters in each of them. Most importantly, by revealing the underlying clustering structures, the proposed method can provide valuable insights into qualitative features of the resulting dynamics that can be used to guide further investigations. Real examples are demonstrated to show that our proposed method has smaller prediction errors than its main competitors, with competitive computation time, and identifies interesting clusters of mesh nodes that possess physical significance, such as satisfying boundary conditions. An R package for the proposed methodology is provided in an open repository. [ABSTRACT FROM AUTHOR]

X-ray computed tomography (XCT) has become a powerful technique for studying lithium-ion batteries, allowing non-destructive 3D imaging across multiple spatial scales. Image quality is particularly important for observing the internal structure of lithium-ion batteries. During multiple rotations, the existence of cumulative errors and random errors in the rotary table leads to errors in the projection angle, affecting the imaging quality of XCT. The accuracy of the projection angle is an important factor that directly affects imaging. However, the impact of the projection angle on XCT reconstruction imaging is difficult to quantify. Therefore, the required precision of the projection angle sensor cannot be determined explicitly. In this research, we selected a common 18650 cylindrical lithium-ion battery for experiments. By setting up an XCT scanning platform and installing an angle sensor to calibrate the projection angle, we proceeded with image reconstruction after introducing various angle errors. When comparing the results, we found that projection angle errors lead to the appearance of noise and many stripe artifacts in the image. This is particularly noticeable in the form of many irregular artifacts in the image background. The overall variation and residual projection error in detection indicators can effectively reflect the trend in image quality. This research analyzed the impact of projection angle errors on imaging and improved the quality of XCT imaging by installing angle sensors on a rotary table. [ABSTRACT FROM AUTHOR]

The establishment of a high-fidelity simulation model for heating systems is crucial for practical guidance, as it serves predictive and evaluative purposes, offering accurate references for the operational management of actual engineering projects, thereby minimizing resource and cost wastage. This study, utilizing the Transient System Simulation Program (TRNSYS) software and focusing on an electric boiler water storage heating system in an elementary school, has developed a high-fidelity simulation model. Multiple TRNSYS (Types) modules were independently programmed, and the model's stability and precision were verified through error analysis. The model is capable of dynamically simulating the changes in supply and return water temperatures on both the storage and usage sides of the system under varying external environmental conditions and hydraulic states, achieving synchronous computation of the thermal and hydraulic characteristics of the heating system. The comparison and analysis of the temperature measurements at various system points against actual collected data revealed an average relative error of less than 5% across these measurements, indicating the simulation model's high accuracy and operational stability. It accurately and reliably reflects the thermal dynamics of the heating system, providing a reliable reference for practical application. This model is applicable for guiding the operational management of actual engineering projects and offers a scientific basis for subsequent system design and optimization. [ABSTRACT FROM AUTHOR]