Search Results (199)

Regenerative chatter is an unfavorable phenomenon that severely affects machining efficiency and surface finish in milling operations. The prediction of chatter stability is an important way to obtain the stable cutting zone. Based on implicit multistep schemes, this paper presents the third-order and fourth-order implicit exponentially fitted methods (3rd IEM and 4th IEM) for milling stability prediction. To begin with, the delay differential equations (DDEs) with time-periodic coefficients are employed to describe the milling dynamics models, and the principal period of the coefficient matrix is firstly decomposed into two different subintervals according to the cutting state. Subsequently, the fourth-step and fifth-step implicit exponential fitting schemes are applied to more accurately estimate the state term. Two benchmark milling models are utilized to illustrate the effectiveness and advantages of the high-order implicit exponentially fitted methods by making comparisons with the three typical existing methods. Under different radial immersion conditions, the numerical results demonstrate that the 3rd IEM and the 4th IEM exhibit both faster convergence rates and higher prediction accuracy than the other three existing prediction methods, without much loss of computational efficiency. Finally, in order to verify the feasibility of the 3rd IEM and the 4th IEM, a series of experimental verifications are conducted using a computer numerical control machining center. It is clearly visible that the stability boundaries predicted by the 3rd IEM and the 4th IEM are mostly consistent with the cutting test results, which indicates that the proposed high-order exponentially fitted methods achieve significantly better prediction performance for actual milling processes. Full article

This article introduces two numerical methods based on the Theory of Functional Connections (TFC) for solving linear ordinary differential equations that involve step discontinuities in the forcing term. The novelty of the first proposed approach lies in the direct incorporation of discontinuities into the free function of the TFC framework, while the second proposed method resolves discontinuities through piecewise constrained expressions comprising particular weighted support functions systematically chosen to enforce continuity conditions. The accuracy of the proposed methods is validated for both a second-order initial value and boundary value problem. As a final demonstration, the methods are applied to a third-order differential equation with non-constant coefficients and multiple discontinuities, for which an analytical solution is known. The methods achieve error levels approaching machine precision, even in the case of equations involving functions whose Laplace transforms are not available. Full article

This work analyzes the aero-elastic oscillations of cantilever beams reinforced by carbon nanotubes (CNTs). Four different distributions of single-walled CNTs are assumed as the reinforcing phase, in the thickness direction of the polymeric matrix. A modified third-order piston theory is used as an accurate tool to model the supersonic air flow, rather than a first-order piston theory. The nonlinear dynamic equation governing the problem accounts for Von Kármán-type nonlinearities, and it is derived from Hamilton’s principle. Then, the Galerkin decomposition technique is adopted to discretize the nonlinear partial differential equation into a nonlinear ordinary differential equation. This is solved analytically according to a multiple time scale method. A comprehensive parametric analysis was conducted to assess the influence of CNT volume fraction, beam slenderness, Mach number, and thickness ratio on the fundamental frequency and lateral dynamic deflection. Results indicate that FG-X reinforcement yields the highest frequency response and lateral deflection, followed by UD and FG-A patterns, whereas FG-O consistently exhibits the lowest performance metrics. An increase in CNT volume fraction and a reduction in slenderness ratio enhance the system’s stiffness and frequency response up to a critical threshold, beyond which a damped beating phenomenon emerges. Moreover, higher Mach numbers and greater thickness ratios significantly amplify both frequency response and lateral deflections, although damping rates tend to decrease. These findings provide valuable insights into the optimization of CNTR composite structures for advanced aeroelastic applications under supersonic conditions, as useful for many engineering applications. Full article

The oscillatory behavior of a class of third-order hybrid-type delay differential equations—used to model various real-world phenomena in fluid dynamics, control systems, biology, and beam deflection—is investigated in this study. A novel method is proposed, whereby these complex trinomial equations are reduced to a simpler binomial form by employing solutions of the corresponding linear differential equations. Through the use of comparison techniques and integral averaging methods, new oscillation criteria are derived to ensure that all solutions exhibit oscillatory behavior. These results are shown to extend and enhance existing theories in the oscillation analysis of functional differential equations. The effectiveness and originality of the proposed approach are illustrated by means of two representative examples. Full article

The aim of this study is to investigate the oscillatory behaviour of a new class of third-order advanced differential equations in their noncanonical form. By employing newly refined transformations, the noncanonical equation is converted into its canonical form. This transformation reduces the number of possible nonoscillatory solution categories from four to two. The present study is based on a thorough and comprehensive review of existing literature and introduces new oscillation criteria by the comparison principle and other analytical techniques. These criteria provide sufficient conditions for the oscillation of solutions without imposing additional restrictive assumptions. The validity and significance of the obtained results are demonstrated through illustrative examples. Full article

This study explores the (1+1)-dimensional Klein–Fock–Gordon equation, a distinct third-order nonlinear differential equation of significant theoretical interest. The Klein–Fock–Gordon equation (KFGE) plays a pivotal role in theoretical physics, modeling high-energy particles and providing a fundamental framework for simulating relativistic wave phenomena. To find the exact solution of the proposed model, for this purpose, we utilized two effective techniques, including the sine-Gordon equation method and a new extended direct algebraic method. The novelty of these approaches lies in the form of different solutions such as hyperbolic, trigonometric, and rational functions, and their graphical representations demonstrate the different form of solitons like kink solitons, bright solitons, dark solitons, and periodic waves. To illustrate the characteristics of these solutions, we provide two-dimensional, three-dimensional, and contour plots that visualize the magnitude of the (1+1)-dimensional Klein–Fock–Gordon equation. By selecting suitable values for physical parameters, we demonstrate the diversity of soliton structures and their behaviors. The results highlighted the effectiveness and versatility of the sine-Gordon equation method and a new extended direct algebraic method, providing analytical solutions that deepen our insight into the dynamics of nonlinear models. These results contribute to the advancement of soliton theory in nonlinear optics and mathematical physics. Full article

This study addresses the standardization of Singular Distributed Parameter Systems (SDPSs). It focuses on classifying and simplifying first- and second-order linear SDPSs using characteristic matrix theory. First, the study classifies first-order linear SDPSs into three canonical forms based on characteristic curve theory, with an example illustrating the standardization process for parabolic SDPSs. Second, under regular conditions, first-order SDPSs can be decomposed into fast and slow subsystems, where the fast subsystem reduces to an Ordinary Differential Equation (ODE) system, while the slow subsystem retains the spatiotemporal characteristics of the original system. Third, the standardization and classification of second-order SDPSs is proposed using three reversible transformations that achieve structural equivalence. Finally, an illustrative example of a building temperature control is built with SDPSs. The simulation results show the importance of system standardization in real-world applications. This research provides a theoretical foundation for SDPS standardization and offers insights into the practical implementation of distributed temperature systems. Full article

The Navier equations are reformulated to be third-order partial differential equations. New anti-Cauchy-Riemann equations can express a general solution in 2D space for incompressible materials. Based on the third-order solutions in 3D space and the Boussinesq–Galerkin method, a third-order method of fundamental solutions (MFS) is developed. For the 3D Navier equation in linear elasticity, we present three new general solutions, which have appeared in the literature for the first time, to signify the theoretical contributions of the present paper. The first one is in terms of a biharmonic function and a harmonic function. The completeness of the proposed general solution is proven by using the solvability conditions of the equations obtained by equating the proposed general solution to the Boussinesq–Galerkin solution. The second general solution is expressed in terms of a harmonic vector, which is simpler than the Slobodianskii general solution, and the traditional MFS. The main achievement is that the general solution is complete, and the number of harmonic functions, three, is minimal. The third general solution is presented by a harmonic vector and a biharmonic vector, which are subjected to a constraint equation. We derive a specific solution by setting the two vectors in the third general solution as the vectorizations of a single harmonic potential. Hence, we have a simple approach to the Slobodianskii general solution. The applications of the new solutions are demonstrated. Owing to the minimality of the harmonic functions, the resulting bases generated from the new general solution are complete and linearly independent. Numerical instability can be avoided by using the new bases. To explore the efficiency and accuracy of the proposed MFS variant methods, some examples are tested. Full article

This work presents a one-dimensional (1D) model for simulating the behavior of an FCC riser reactor processing bio-oil. The FCC riser is modeled as a plug-flow reactor, where the bio-oil feed undergoes vaporization followed by catalytic cracking reactions. The bio-oil droplets are represented using a Lagrangian framework, which accounts for their movement and evaporation within the gas-solid flow field, enabling the assessment of droplet size impact on reactor performance. The cracking reactions are modeled using a four-lumped kinetic scheme, representing the conversion of bio-oil into gasoline, kerosene, gas, and coke. The resulting set of ordinary differential equations is solved using a stiff, second- to third-order solver. The simulation results are validated against experimental data from a full-scale FCC unit, demonstrating good agreement in terms of product yields. The findings indicate that heat exchange by radiation is negligible and that the Buchanan correlation best represents the heat transfer between the droplets and the catalyst particles/gas phase. Another significant observation is that droplet size, across a wide range, does not significantly affect conversion rates due to the bio-oil’s high vaporization heat. The proposed reduced-order model provides valuable insights into optimizing FCC riser reactors for bio-oil processing while avoiding the high computational costs of 3D CFD simulations. The model can be applied across multiple applications, provided the chemical reaction mechanism is known. Compared to full models such as CFD, this approach can reduce computational costs by thousands of computing hours. Full article

River barriers constitute a key factor that is degrading river connectivity and represent a critical research focus in riverine ecosystem conservation. Management authorities and river restoration agencies globally have increasingly employed barrier removal or modification for connectivity restoration projects in recent years, practices that are widely discussed and empirically supported in academia. However, existing research predominantly focuses on large dams in primary rivers, overlooking the more severe fragmentation caused by low-head barriers within low-order streams. This study targets the Yanjing River (total length: 70 km), a third-order tributary of the Yangtze River basin, implementing culvert modification and complete removal measures, respectively, for two river barriers distributed within its terminal 9 km reach. Using differential analysis, principal component analysis (PCA), cluster analysis, Mantel tests, and structural equation modeling (SEM), we systematically examined the mechanisms by which connectivity restoration projects influences aquatic habitat and fish diversity, the evolution of reach heterogeneity, and intrinsic relationships between aquatic environmental factors and diversity metrics. Results indicate that (1) the post-restoration aquatic habitat significantly improved with marked increases in fish diversity metrics, where hydrochemical factors and species diversity exhibited the highest sensitivity to connectivity changes; (2) following restoration, the initially barrier-fragmented river segments (upstream, middle, downstream) exhibited significantly decreased differences in aquatic habitat and fish diversity, demonstrating progressive homogenization across reaches; (3) hydrological factors exerted stronger positive effects on fish diversity than hydrochemical factors did, particularly enhancing species diversity, with a significant positive synergistic effect observed between species diversity and functional diversity. These studies demonstrate that “culvert modification and barrier removal” represent effective project measures for promoting connectivity restoration in low-order streams and eliciting positive ecological effects, though they may reduce the spatial heterogeneity of short-reach rivers in the short term. It is noteworthy that connectivity restoration projects should prioritize the appropriate improvement of hydrological factors such as flow velocity, water depth, and water surface width. Full article

This paper mainly investigates a class of third-order semilinear delay differential equations with a nonhomogeneous term ${\left({\left[x^{″}\left(t\right)\right]}^{\alpha }\right)}^{\prime }+q\left(t\right)x^{\alpha }\left(\sigma \left(t\right)\right)+f\left(t\right)=0,t\ge t_0.$ Under the oscillation criteria, we propose a sufficient condition to ensure that all solutions for the equation exhibit oscillatory behavior when $\alpha$ is the quotient of two positive odd integers, supported by concrete examples to verify the accuracy of these conditions. Furthermore, for the case $\alpha =1,$ a sufficient condition is established to guarantee that the solutions either oscillate or asymptotically converge to zero. Moreover, under these criteria, we demonstrate that the global oscillatory behavior of solutions remains unaffected by time-delay functions, nonhomogeneous terms, or nonlinear perturbations when $\alpha =1.$ Finally, numerical simulations are provided to validate the effectiveness of the derived conclusions. Full article

This paper concerns accurate spectral collocation solutions, more precisely Chebyshev collocation (ChC), to some third-order nonlinear and singular boundary value problems on unbounded domains. The problems model some draining or coating fluid flows. We use exclusively ChC, in the form of Chebfun, avoid any obsolete shooting-type method, and provide reliable information about the convergence and accuracy of the method, including the order of Newton’s method involved in solving the nonlinear algebraic systems. As a complete novelty, we combine a graphical representation of the convergence of the Newton method with a numerical estimate of its order of convergence for a more realistic value. We treat five challenging examples, some of which have only been solved by approximate methods. The found numerical results are judged in the context of existing ones; at least from a qualitative point of view, they look reasonable. Full article

In this paper, we introduce null Cartan normal helices in Minkowski space ${\mathbb{E}}_1^3$. We obtain explicit expressions for their torsions by considering the cases when the C-constant vector field is orthogonal to their axis or not orthogonal to it. We find that the tangent vector field of a null Cartan normal helix satisfies the third-order linear homogeneous differential equation and obtain its general solution in a special case. We prove that null Cartan helices are the only normal helices having two axes and, in a particular case, three axes. Finally, we provide the necessary and sufficient conditions for null Cartan normal helices lying on a timelike surface to be isophotic curves, silhouettes, normal isophotic curves and normal silhouettes with respect to the same axis and provide some examples. Full article

This study aims to establish new oscillation criteria for solutions of a specific class of functional differential equations. Our findings extend and refine the recently developed criteria for this type of equation by various authors and also encompass classical criteria for related problems. Our approach relies on the Riccati technique to derive conditions that preclude the possibility of non-oscillatory solutions. The inherent symmetry of these solutions plays a key role in formulating the new criteria presented here. By applying techniques from the theory of symmetric differential equations and leveraging symmetric functions, we are able to establish precise conditions for oscillation. To enhance practical applicability, we propose multiple distinct criteria while minimizing the constraints typically imposed. Several examples are provided to illustrate the accuracy, applicability, and versatility of the new criteria. Full article

A new kind of graph-based contraction in a metric space is introduced in this article. We investigate results concerning the best proximity points and fixed points for these contractions, supported by illustrated examples. The practical applicability of our results is demonstrated through particular instances in the setting of integral equations and differential equations. We also describe how a class of third-order boundary value problems satisfying the present contraction can be solved iteratively. To support our findings, we conduct a series of numerical experiments with various third-order boundary value problems. Full article