B-spline Method Research Articles

Investigating novel optical soliton solutions for a generalized (3+1)-dimensional q-deformed equation

In this study, we investigate the (3+1) q-deformed tanh-Gordon equation due to its importance in the context of mathematical physics. It describes solitonic solutions in quantum field theory; it can sometimes be used in condensed matter physics to describe interactions between particles in magnetic materials or superconductors; it can model light propagation in nonlinear optical fibers or photonic crystals where the refractive index has a q-deformed structure; and it also can be applied in studying shock waves, turbulence and rogue waves where the deformation introduces corrections to classical wave phenomena. Utilizing the (G′ωG′+G+r)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$(\\frac{{\\mathfrak {G}}^{\\prime }}{\\omega {\\mathfrak {G}}^{\\prime }+{\\mathfrak {G}}+r})$$\\end{document}-expansion technique, we derive novel analytical solutions that enhance our understanding of the underlying dynamics. Additionally, we employ a finite element method (extended cubic B-spline method) to validate our analytical findings and explore the behavior of the q-deformed equation under different parameter regimes. Our results demonstrate the versatility of the q-deformed framework in generating rich optical phenomena, paving the way for future research in both theoretical and applied physics.

A Sixth-Order Cubic B-Spline Approach for Solving Linear Boundary Value Problems: An In-Depth Analysis and Comparative Study

This research presents an efficient and highly accurate cubic B-spline method (CBSM) for solving second-order linear boundary value problems (BVPs). The method achieves sixth-order convergence, supported by rigorous error analysis, ensuring rapid error reduction with mesh refinement. The effectiveness of the CBSM is validated through four numerical examples, showcasing its accuracy, reliability, and computational efficiency, making it well-suited for large-scale problems. A comparative analysis with existing methods confirms the superior performance of the CBSM, positioning it as a practical and powerful tool for solving second-order BVPs.

Comparative Analysis of NURBS and Finite Element Method in Computational Fluid Dynamics Applications: Case Study on NACA 2412 Airfoil Aerodynamics

In this research, an attempt was made to employ the Non-Uniform Rational B-Splines (NURBS) method for a challenging computational fluid dynamics (CFD) problem of aerodynamics around NACA 2412 airfoils. The comparison was carried out thoroughly by using the same boundary conditions and geometry, comparing NURBS to standard FEM implementations. Our study was interested in demonstrating the foreseeable functionalities of NURBS for solving complex CFD problems and conducting a comparative effectiveness performance evaluation between them with traditional FEM methodologies.

Extended B-Spline Collocation Method for Singularly Perturbed Time Delay Parabolic Differential-Difference Problems

The paper presents a robust collocation B-spline numerical scheme for solving singularly perturbed time lag parabolic differential–difference problems. The method uses Taylor’s series expansion for the approximation of retarded terms, the [Formula: see text]-method (when [Formula: see text]) for time discretization, and the extended third-degree B-spline collocation method for spatial discretization in a piecewise uniform Shishkin mesh. The advantage of using an extended cubic B-spline rather than the classical third-degree B-spline method is that it introduces one additional free parameter to control the global shape parameter. The stability and uniform convergence of the proposed scheme are investigated. The method is first-order accurate in [Formula: see text] and almost second-order accurate in [Formula: see text], surpassing existing literature methods.

Adaptive mesh extended cubic B-spline method for singularly perturbed delay Sobolev problems

The purpose of this paper is to develop a robust numerical scheme for a class of singularly perturbed delay Sobolev (pseudo-parabolic) problems that have wide application in various branches of mathematical physics and fluid mechanics.For the small perturbation parameter, the standard numerical schemes for the solution of these problems fail to resolve the boundary layer(s) and the oscillations occur near the boundary layer. Thus, in this paper to resolve the boundary layer(s), im-plicit Euler scheme for the time derivatives on uniform mesh and extended B-spline basis functions consisting of free parameter λ are presented for spatial variable on Bakhvalov type mesh. The stability and uniform convergence analyisis of the proposed method are established. The error estimation of the developed method is shown to be firts order accurate in time and second order accurate in space. Numerical exprementation is carried out to validate the applicability of the developednumerical method. The numerical results reveals that the computational result is in agreement with the theoretical estimations

Cubic B-spline based elastic and viscoelastic wave propagation method

The requirement for high computational efficiency in inversion imaging poses challenges to further enhancing the accuracy of imaging large elastic and viscoelastic models. Considering that forward modeling serves as a prerequisite for inversion imaging, it is essential to introduce a wave propagation simulation method that can enhance simulation accuracy without significantly compromising computational speed. In our paper, cubic B-spline is introduced into the forward modeling of the wave equation. The B-spline collocation method offers a linear complexity for computation and exhibits fourth-order convergence in terms of computational errors. We also find that it can perfectly adapt to the boundary conditions of the perfectly matched layer in the forward modeling solution of the wave simulation. Several typical numerical examples, including 2-D acoustic wave equation, 2-D elastic wave propagation, and 2-D viscoelastic propagation in homogeneous media, are provided to validate its convergence, accuracy, and computational efficiency.

A modified basis of cubic B-spline with free parameter for linear second order boundary value problems: Application to engineering problems

The traditional cubic B-spline method offers limited local control over the curve solution. Adjusting the position of a control point affects the entire curve, making it challenging to make localized changes, e.g., smoothness. Moreover, the basis functions vanish on one side by the cubic B-spline method near the end conditions where the initial and boundary conditions are applied. To address these limitations, this research proposes a new basis by including a free parameter γ with the purpose of modifying the weights of nearby control points. This free parameter γ can influence the curve’s behavior in specific regions as well as the entire curve. This modification of the cubic B-spline method was used to approximate the second-order derivative at each collocation point. The convergence test showed that the proposed method was second-order convergent. Numerical examples of ordinary differential equations were used with different step values to evaluate the accuracy of the proposed method. The findings persistently indicated that the proposed technique provided better error estimates as compared to the other methods discussed in the literatures.

Comparative analysis of parametric B-spline and Hermite cubic spline based methods for accurate ECG signal modeling

Analyzing Electrocardiogram (ECG) signals is imperative for diagnosing cardiovascular diseases. However, evaluating ECG analysis techniques faces challenges due to noise and artifacts in actual signals. Machine learning for automatic diagnosis encounters data acquisition hurdles due to medical data privacy constraints. Addressing these issues, ECG modeling assumes a crucial role in biomedical and parametric spline-based methods have garnered significant attention for their ability to accurately represent the complex temporal dynamics of ECG signals. This study conducts a comparative analysis of two parametric spline-based methods—B-spline and Hermite cubic spline—for ECG modeling, aiming to identify the most effective approach for accurate and reliable ECG representation. The Hermite cubic spline serves as one of the most effective interpolation methods, while B-spline is an approximation method. The comparative analysis includes both qualitative and quantitative evaluations. Qualitative assessment involves visually inspecting the generated spline-based models, comparing their resemblance to the original ECG signals, and employing power spectrum analysis. Quantitative analysis incorporates metrics such as root mean square error (RMSE), Percentage Root Mean Square Difference (PRD) and cross correlation, offering a more objective measure of the model's performance. Preliminary results indicate promising capabilities for both spline-based methods in representing ECG signals. However, the analysis unveils specific strengths and weaknesses for each method. The B-spline method offers greater flexibility and smoothness, while the cubic spline method demonstrates superior waveform capturing abilities with the preservation of control points, a critical aspect in the medical field. Presented research provides valuable insights for researchers and practitioners in selecting the most appropriate method for their specific ECG modeling requirements. Adjustments to control points and parameterization enable the generation of diverse ECG waveforms, enhancing the versatility of this modeling technique. This approach has the potential to extend its utility to other medical signals, presenting a promising avenue for advancing biomedical research.

A robust B-spline method for two parameter singularly perturbed parabolic differential equations with discontinuous initial condition

A robust B-spline method for two parameter singularly perturbed parabolic differential equations with discontinuous initial condition

Exponential B-Spline Method for Second Order Singularly Perturbed Boundary Value Problem with Negative Shift

Exponential B-Spline Method for Second Order Singularly Perturbed Boundary Value Problem with Negative Shift

Hawking radiation of 5D Ricci-flat black hole

In this paper, we investigate the properties of a five-dimensional Ricci-flat black hole within the framework of the Space–Time–Matter theory. By employing the Klein–Gordon equation, we derive the wave function equations for both the mass dimensional coordinate and the radial dimensional coordinate. We present the analytical solutions for the wave function on the mass dimensional coordinate under two distinct scenarios. Remarkably, we observe the absence of quantum effects on the mass dimensional coordinate, which aligns with the absence of a barrier in the wave function equation. For the radial component, we employ the B-spline method to numerically solve its wave function. Our findings reveal the existence of Hawking radiation between the event horizon and the cosmological horizon, with the transmission and reflection probabilities being determined by the presence of a barrier. Furthermore, we discover a significantly higher probability of particle detection near the cosmological horizon, while the probability near the event horizon approaches zero.

Optimal section design of surface piercing propeller using artificial neural networks algorithm

ABSTRACT Section optimisation of a surface piercing propeller (SPP) with the combination of artificial neural networks and NSGA II algorithm is investigated in this research. The blade section have a sharp leading edge and thick trailing edge. The geometry of this propeller is mostly designed experimentally, and comprehensive studies have not been performed to examine its parameters. In this regard, the propeller's parametric geometry is designed based on the cubic B-spline method. The SPP was first fabricated and tested, and the simulated data of the tested propeller were verified. Then, limited data were extracted according to a verified CFD model based on which the ANN was trained. The ANN results indicated the mean square error of the trained network is 3e-9, 1e-7, and 1e-7 for training, validation, and test data, respectively. comparing the optimisation and experimental results showed a relative error amounting to 3.5% in thrust and 4.24% in torque.

Stochastic occupancy modeling for spaces with irregular occupancy patterns using adaptive B-Spline-based inhomogeneous Markov Chains

This paper presents a discrete time, discrete state-space in-homogeneous Markov Chains model for stochastic occupancy modeling in spaces with irregular occupancy patterns. The goal of the model is to provide accurate predictions of occupancy numbers, enabling appropriate actions to be taken for HVAC system to maintain optimal indoor environment. The proposed Markov Chain model incorporates time in-homogeneity by coupling the time-varying model parameters using a Periodic B-Spline expansion with adaptive knots, which effectively captures patterns in occupancy activity. This method optimizes the distribution of knots based on specific occupancy characteristics observed in different types of rooms. To evaluate the effectiveness of the proposed method, six months of occupancy data collected from a meeting room are utilized. A comprehensive comparison is conducted between the proposed adaptive B-Spline method and other approaches, including the counting method and uniform B-Spline method. The comparison considers both model accuracy and complexity, using metrics such as the Akaike Information Criterion and Bayesian Information Criterion. Results indicate that the proposed model achieves more accurate predictions with fewer model parameters compared to other methods. These forecasts are particularly useful in optimizing the control of HVAC systems, where accurate predictions of future occupancy numbers are essential.

Numerical Solution of Heat Equation using Modified Cubic B-spline Collocation Method

In this paper, a collocation method is presented based on the Modified Cubic B-spline Method (MCBSM) for the numerical solution of the heat equation. The PDE is fully discretized by using the Modified Cubic B-spline basis collocation for spatial discretization and the finite difference method is used for the time discretization. A numerical example from PDE is used to evaluate the accuracy of the proposed method. The numerical results are evaluated in comparison to the exact solutions. The findings consistently indicate that the suggested technique provides good error estimates. We also discovered that our proposed method was unconditionally stable. Hence, based on the results and the efficiency of the method, the method is suitable for solving heat equation.

Quartic B-Spline Method for Non-Linear Second Order Singularly Perturbed Delay Differential Equations

Quartic B-Spline Method for Non-Linear Second Order Singularly Perturbed Delay Differential Equations

The use of polyurethane foam-sand mixtures in sandy embankment design- predicting seismic response using FEM, catastrophe theory, B-spline method, and artificial neural networks

High-permeability sand cannot control the water that is stored behind an embankment. In addition, if clay cannot be provided within a reasonable distance of the embankment construction site, an alternative method must be found. The study proposes using a polyurethane foam-sand mixture to construct an impermeable embankment. The main purpose of the paper was to predict the seismic stability of the embankment. The nonlinear finite element models (FEMs) are applied along with artificial neural networks (ANNs), and this research method applied was performed to investigate the main objectives of the research. Catastrophe theory was used to predict the mechanism of differential displacement in the Y direction at selected points of the embankment model. For model smooth functions, the basis spline (B-spline) method was applied to simulate the catastrophe progression index value. Results revealed that the suitability of the polyurethane foam-sand mixture controls the acceleration, displacement, strain, and stress of the model at points selected in different parts of the embankment. Moreover, it was found that the deformation pattern of the model was related to the polyurethane foam-sand mixture ratios. Furthermore, the main contribution was that the seismic response of the embankment model could be improved with the right percentage of polyurethane foam added to the sand. Results were validated by referencing those available in the literature.

HB-RRT:A path planning algorithm for mobile robots using Halton sequence-based rapidly-exploring random tree

Path planning remains crucial for efficient robot operation. A Halton Biased Rapidly-exploring Random Tree (HB-RRT) path planning algorithm is introduced in this study. The Halton sequence, known for its uniform distribution and low discrepancy, is employed for sampling. Issues arising from the pseudo-random sequence in the standard RRT algorithm, leading to uneven distribution of sampling points, are addressed. A mouse-inspired goal-oriented strategy and a candidate sampling pool strategy are incorporated to enhance the sampling point quality, thereby addressing the challenge of insufficient memory during node expansion. Path optimization is further achieved through a multi-level planning approach, which aims to minimize redundancy. A subsequent smoothing of the path is conducted using a cubic B-spline method. Comparisons with the RRT, Bionic Target Bias-RRT, and Informed-RRT* algorithms, through both numerical simulations and real-world testing, confirm the superiority of the HB-RRT algorithm in terms of planning time, path length, and overall path quality.

Numerical investigation of electroosmotic driven transport of MHD Casson fluid over an exponential stretching sheet using quartic B-spline method

The stretching surface flow phenomenon plays novel significance in the differential industrial sectors like manufacturing processes, food processing, polymers, petroleum engineering, glass fiber production, metal spinning, and many others. This research focuses on investigating the flow of electroosmotic-driven transport of magnetic Casson fluid over an exponential stretching sheet. The study employs mathematical modeling based on partial differential equations which are further truncated into ordinary system under the implementation of dimensionless variables. The quartic B-spline method is applied for obtaining solutions. Key findings indicate that electroosmotic and Helmholtz–Smoluchowski effects enhance fluid velocity, while magnetic forces reduce velocity. Additionally, an increase in the Hartmann parameter leads to a decrease in the skin friction coefficient.

Trends in Salmonella Infantis human illness incidence and chicken carcass prevalence in the United States; 1996-2019.

The incidence of human illness due to Salmonella Infantis reported to Foodborne Diseases Active Surveillance Network and the prevalence of Infantis on chicken carcasses reported by the United States Department of Agriculture Food Safety and Inspection Service have increased significantly in the past decade. However, the trends do not appear coincident, as would be expected if the increased prevalence in chicken led to the increase in the incidence of human illness. Salmonella Infantis incidence and prevalence trends are analyzed using penalized B-spline methods for generalized additive regression models. The association between the two time series is analyzed using time-lagged rank-order cross-correlation. Geographic variations in reported incidence and trends are also explored. The increase in human incidence of Salmonella Infantis began circa 2011. The increase in chicken carcass prevalence began circa 2015. A 4-year lag on chicken carcass prevalence maximizes the rank-order cross-correlation with the incidence of illness. While chicken consumption undoubtedly contributes to the incidence of human illness due to Salmonella Infantis, the initial increase in reported illness was likely due to one or more other transmission pathways. Other potential transmission pathways include non-chicken foodborne, waterborne, person-to-person, animal contact, and environmental.

Solve third order Lane–Emden–Fowler equation by Adomian decomposition method and quartic trigonometric B-spline method

In this work, we study the Lane–Emden–Fowler equation which has a wide range of applications in astrophysics, classical and quantum mechanics. Firstly, we convert this nonlinear differential equation into an equivalent integral equation. We employ the powerful Adomian decomposition method to derive an analytical solution that will also lead to numerical solutions. Moreover, we exhibit two theorems to ensure the uniqueness of the solution of the problem and to confirm the convergence of the proposed scheme. In addition, we use quartic trigonometric B-spline as a powerful numerical method to achieve numerical solutions. Finally we present a comparison between the proposed methods to show their accuracy performance. Proper graphs are also provided to illustrate the obtained solutions.