Search Results (48)

Recently, studies on delay-based controllers that consider the positive effects of time delay have been increasing in the literature. In this context, a proportional integral retarded (PIR) controller structure has been proposed. There are two important factors in the design of a delay-based controller: closed-loop stability and controller tuning. The motivation for this study is that there is no previous literature on stability region analysis on PIR controllers. To address this gap in the literature, this paper presents, for the first time, a comprehensive stability region analysis for PIR controllers. The study derives the stability boundary locus (SBL) equations for time-delay processes with PIR controllers, allowing stable regions to be defined analytically and graphically in the two-dimensional controller parameter space. In order to obtain the SBL, the stability boundary line equations are derived, and the variations in the case of different types of time-delay processes are presented with mathematical equations and proved with examples. Another novelty of this work is its systematic investigation of how the controller time delay affects the stability region when it is smaller, equal or larger than the process delay. The proposed method fills a critical gap in the design and tuning of PIR controllers by providing a reliable tool by which to visualize and understand stability properties. Full article

This paper investigates stability switches induced by Hopf bifurcation in a fractional three-neuron network that incorporates both neutral time delay and communication delay, as well as a general structure. Initially, we simplified the characteristic equation by eliminating trigonometric terms associated with purely imaginary roots, enabling us to derive the Hopf bifurcation conditions for communication delay while treating the neutral time delay as a constant. The results reveal that communication delay can drive a stable equilibrium into instability once it exceeds the Hopf bifurcation threshold. Furthermore, we performed a sensitivity analysis to identify the fractional order and neutral delay as the two most sensitive parameters influencing the bifurcation value for the illustrative example. Notably, in contrast to neural networks with only retarded delays, our numerical observations show that the Hopf bifurcation curve is non-monotonic, highlighting that the neural network with a fixed communication delay can exhibit stability switches and eventually stabilize as the neutral delay increases. Full article

With the development of nuclear energy in China, the geological disposal of high-level radioactive waste (HLW) is increasingly receiving national attention. Among them, the study of nuclide migration is an important and complex technical system, which requires continuous in-depth research. Under the decay heat, radiation, and groundwater effects of HLW, buffer materials generate complex coupled thermo-hydro-mechanical-chemical (THMC) processes. The migration and diffusion of nuclides in buffer materials are controlled by the coupling effect of THMC. It is of great significance for the long-term safety of a HLW repository to study the long-term retarding effect of buffer material on nuclide strontium under the coupling effect of multiple factors. This study leverages the solving advantages of COMSOL Multiphysics 5.6, using a combination of the self-developed Mock-up experimental device as a geometric model and numerical simulations to study the multi-field coupling performance and radionuclide migration evolution characteristics of THMC buffer materials, which overcomes the difficulties due to the limitations of the experimental time and spatial scale. The simulation results can predict the migration range and distance variation of strontium in buffer materials at different time scales. In the initial stage, the migration and diffusion of nuclide in buffer materials are relatively slow, and the migration distance increases by about 0.03 m with time. In the mid-to-late stage, the migration distance increases by about 0.05 m over time; to ensure that, in 1000 years, core strontium does not penetrate the buffer material and migrate into the surrounding rock groundwater of the disposal facility, a buffer material thickness of 0.3 m needs to be set. The construction of THMC control equations for the migration and diffusion of nuclides in buffer materials under multi-field coupling conditions has been revealed, providing an important reference for a deeper understanding of the risk analysis of radionuclide contamination in disposal environments. Full article

An exact non-perturbative model of a gravitational wave with pure radiation is constructed. It is shown that the presence of dust matter in this model contradicts Einstein’s field equations. The exact solution to Einstein’s equations for gravitational wave and pure radiation is obtained. The trajectories of propagation and the characteristics of radiation are found. For the considered exact model of a gravitational wave, a retarded time equation for radiation is obtained. The obtained results are used to construct an exact model of gravitational wave and pure radiation for the Bianchi type IV universe. Full article

We discuss local magnetic field quenches using perturbative methods of finite time path field theory (FTPFT) in the following spin chains: Ising and XY in a transverse magnetic field. Their common characteristics are: (i) they are integrable via mapping to a second quantized noninteracting fermion problem; and (ii) when the ground state is nondegenerate (true for finite chains except in special cases), it can be represented as a vacuum of Bogoliubov fermions. By switching on a local magnetic field perturbation at finite time, the problem becomes nonintegrable and must be approached via numeric or perturbative methods. Using the formalism of FTPFT based on Wigner transforms (WTs) of projected functions, we show how to: (i) calculate the basic “bubble” diagram in the Loschmidt echo (LE) of a quenched chain to any order in the perturbation; and (ii) resum the generalized Schwinger–Dyson equation for the fermion two-point retarded functions in the “bubble” diagram, hence achieving the resummation of perturbative expansion of LE for a wide range of perturbation strengths under certain analyticity assumptions. Limitations of the assumptions and possible generalizations beyond it and also for other spin chains are further discussed. Full article

This study focuses on deriving and presenting an infinite series as the analytical solution for transient electroosmotic and pressure-driven flows in microtubes. Such a mathematical presentation of fluid dynamics under simultaneous electric field and pressure gradients leverages governing equations derived from the generalized continuity and momentum equations simplified for laminar and axisymmetric flow. Velocity profile developments, apparent slip-induced flow rates, and shear stress distributions were analyzed by varying values of the ratio of microtube radius to Debye length and the electroosmotic slip velocity. Additionally, the “retarded time” in terms of hydraulic diameter, kinematic viscosity, and slip-induced flow rate was derived. A simpler polynomial series approximation for steady electroosmotic flow is also proposed for engineering convenience. The analytical solutions obtained in this study not only enhance the fundamental understanding of the electroosmotic flow characteristics within microtubes, emphasizing the interplay between electroosmotic and pressure-driven mechanisms, but also serve as a benchmark for validating computational fluid dynamics models for electroosmotic flow simulations in more complex flow domains. Moreover, the analytical approach aids in the parametric analysis, providing deeper insights into the impact of physical parameters on electroosmotic and pressure-driven flow behavior, which is critical for optimizing device performance in practical applications. These findings also offer insightful implications for diagnostic and therapeutic strategies in healthcare, particularly enhancing the capabilities of lab-on-a-chip technologies and paving the way for future research in the development and optimization of microfluidic systems. Full article

The purpose of this article is to present some new nonlinear retarded integral inequalities which can be utilized to study the existence, stability, boundedness, uniqueness, and asymptotic behavior of solutions of nonlinear retarded integro-differential equations, and these inequalities can be used in the symmetrical properties of functions. These inequalities also generalize some former famous inequalities in the literature. Two examples as applications will be provided to demonstrate the strength of our inequalities in estimating the boundedness and global existence of the solution to initial value problems for nonlinear integro-differential equations and differential equations which can be seen in graphs. This research work will ensure opening new opportunities for studying nonlinear dynamic inequalities on a time-scale structure of a varying nature. Full article

In this paper we develop an approach for obtaining the solutions to systems of linear retarded and neutral delay differential equations. Our analytical approach is based on the Laplace transform, inverse Laplace transform and the Cauchy residue theorem. The obtained solutions have the form of infinite non-harmonic Fourier series. The main advantage of the proposed approach is the closed-form of the solutions, which are capable of accurately evaluating the solution at any time. Moreover, it allows one to study the asymptotic behavior of the solutions. A remarkable discovery, which to the best of our knowledge has never been presented in the literature, is that there are some particular linear systems of both retarded and neutral delay differential equations for which the solution asymptotically approaches a limit cycle. The well-known method of steps in many cases is unable to obtain the asymptotic behavior of the solution and would most likely fail to detect such cycles. Examples illustrating the Laplace transform method for linear systems of DDEs are presented and discussed. These examples are designed to facilitate a discussion on how the spectral properties of the matrices determine the manner in which one proceeds and how they impact the behavior of the solution. Comparisons with the exact solution provided by the method of steps are presented. Finally, we should mention that the solutions generated by the Laplace transform are, in most instances, extremely accurate even when the truncated series is limited to only a handful of terms and in many cases become more accurate as the independent variable increases. Full article

The aerodynamic noise of an open rotor is one of the critical challenges that must be considered in its design and application. FODNOPP, a program specifically programmed to predict the aerodynamic discrete noise of single- and counter-rotating open rotors (such as propellers, propfans, and rotorcraft rotors) at subsonic, transonic, and supersonic helical blade tip speeds, has recently been developed by the first author. This program is composed of four prediction codes, namely code a1, code a2, code b, and code c, each based on Farassat-derived formulations Formu 1-RTE, Formu 1A, Formu 1-Sph, and Formu 3, providing time-domain solutions to the Ffowcs Williams–Hawkings equation. Four verification examples for both propeller low-speed flight noise and counter-rotating propfan take-off noise are presented, along with an application case for transonic helical tip speed counter-rotating propfan cruise noise. The results demonstrate the accuracy of FODNOPP in calculating the noise for these verification cases. And in the counter-rotating propfan cruise noise case, the maximum harmonic sound pressure level of the rear propfan is 5.5 dB higher than that of the front propfan. FODNOPP can be referred to as a comprehensive design tool, and it offers valuable guidance for engineering design focused on rotor-related noise reduction. Full article

In this article, we establish certain time-scale-retarded dynamic inequalities that contain nonlinear retarded integral equations on various time scales. These inequalities extend and generalize some significant inequalities existing in the literature to their more general forms. The qualitative and quantitative characteristics of solutions to various dynamic equations on time scales involving retarded integrals can be studied using these inequalities. The results presented in this manuscript furnish a powerful tool to analyze the boundedness of nonlinear integral equations with retarded integrals on several time scales. In the end, we also include numerical illustrations to signify the applicability of these results to power nonlinear retarded integral equations on real and quantum time scales. Full article

In this work, we review and extend a version of the old attempt made by Louis de Broglie for interpreting quantum mechanics in realistic terms, namely, the double solution. In this theory, quantum particles are localized waves, i.e., solitons, that are solutions of relativistic nonlinear field equations. The theory that we present here is the natural extension of this old work and relies on a strong time-symmetry requiring the presence of advanced and retarded waves converging on particles. Using this method, we are able to justify wave–particle duality and to explain the violations of Bell’s inequalities. Moreover, the theory recovers the predictions of the pilot-wave theory of de Broglie and Bohm, often known as Bohmian mechanics. As a direct consequence, we reinterpret the nonlocal action-at-a-distance in the pilot-wave theory. In the double solution developed here, there is fundamentally no action-at-a-distance but the theory requires a form of superdeterminism driven by time-symmetry. Full article

This study contains a two-dimensional mathematical model of solute transport in a river with temporally and spatially dependent flow, explicitly focusing on pulse-type input point sources with a fractional approach. This model is analyzed by assuming an initial concentration function as a declining exponential function in both the longitudinal and transverse directions. The governing equation is a time-fractional two-dimensional advection–dispersion equation with a variable form of dispersion coefficients, velocities, decay constant of the first order, production rate coefficient for the solute at the zero-order level, and retardation factor. The solution of the present problem is obtained by the fractional reduced differential transform method (FRDTM). The analysis of the initial retardation factor has been carried out via plots. Also, the influence of initial longitudinal and transverse dispersion coefficients and velocities has been examined by graphical analysis. The impact of fractional parameters on pollution levels is also analyzed numerically and graphically. The study of convergence for the FRDTM technique has been conducted to assess its efficacy and accuracy. Full article

The Jeffreys-type heat conduction equation with flux precedence describes the temperature of diffusive hot electrons during the electron–phonon interaction process in metals. In this paper, the deformation resulting from ultrafast surface heating on a “nanoscale” plate is considered. The focus is on the anomalous heat transfer mechanisms that result from anomalous diffusion of hot electrons and are characterized by retarded thermal conduction, accelerated thermal conduction, or transition from super-thermal conductivity in the short-time response to sub-thermal conductivity in the long-time response and described by the fractional Jeffreys equation with three fractional parameters. The recent double-strip problem, Awad et al., Eur. Phy. J. Plus 2022, allowing the overlap between two propagating thermal waves, is generalized from the semi-infinite heat conductor case to thermoelastic case in the finite domain. The elastic response in the material is not simultaneous (i.e., not Hookean), rather it is assumed to be of the Kelvin–Voigt type, i.e., $\sigma =E\left(\epsilon +{\tau }_{\epsilon }\dot{\epsilon }\right)$, where $\sigma$ refers to the stress, $\epsilon$ is the strain, $E$ is the Young modulus, and ${\tau }_{\epsilon }$ refers to the strain relaxation time. The delayed strain response of the Kelvin–Voigt model eliminates the discontinuity of stresses, a hallmark of the Hookean solid. The immobilization of thermal conduction described by the ordinary Jeffreys equation of heat conduction is salient in metals when the heat flux precedence is considered. The absence of the finite speed thermal waves in the Kelvin–Voigt model results in a smooth stress surface during the heating process. The temperature contours and the displacement vector chart show that the anomalous heat transfer characterized by retardation or crossover from super- to sub-thermal conduction may disrupt the ultrafast laser heating of metals. Full article

In a series of recent papers we developed a formulation of general relativity in which spacetime and the dynamics of matter evolve with a Poincaré invariant parameter $\tau$. In this paper, we apply the formalism to derive the metric induced by a ‘static’ event evolving uniformly along its t-axis at the spatial origin $\mathbf{x}=0$. The metric is shown to vary with t and $\tau$, as well as spatial distance r, taking its maximum value for a test particle at the retarded time $\tau =t-r/c$. In the resulting picture, an event localized in space and time produces a metric field similarly localized, where both evolve in $\tau$. We first derive this metric as a solution to the wave equation in linearized field theory, and discuss its limitations by studying the geodesic motion it produces for an evolving event. By then examining this solution in the 4+1 formalism, which poses an initial value problem for the metric under $\tau$-evolution, we clarify these limitations and indicate how they may be overcome in a solution to the full nonlinear field equations. Full article

The gelation time $t_{\mathrm{g}}$ necessary for a solution of functional (associating) molecules to reach its gel point after a temperature jump, or a sudden concentration change, is theoretically calculated on the basis of the kinetic equation for the stepwise cross-linking reaction as a function of the concentration, temperature, functionality f of the molecules, and multiplicity k of the cross-link junctions. It is shown that quite generally $t_{\mathrm{g}}$ can be decomposed into the product of the relaxation time $t_{\mathrm{R}}$ and a thermodynamic factor Q. They are functions of a single scaled concentration $x\equiv \lambda \left(T\right)\varphi$, where $\lambda \left(T\right)$ is the association constant and $\varphi$ is the concentration. Therefore, the superposition principle holds with $\lambda \left(T\right)$ as a shift factor of the concentration. Additionally, they all depend on the rate constants of the cross-link reaction, and hence it is possible to estimate these microscopic parameters from macroscopic measurements of $t_{\mathrm{g}}$. The thermodynamic factor Q is shown to depend on the quench depth. It generates a singularity of logarithmic divergence as the temperature (concentration) approaches the equilibrium gel point, while the relaxation time $t_{\mathrm{R}}$ changes continuously across it. Gelation time $t_{\mathrm{g}}$ obeys a power law $t_{\mathrm{g}}^{-1}\sim x^n$ in the high concentration region, whose power index n is related to the multiplicity of the cross-links. The retardation effect on the gelation time due to the reversibility of the cross-linking is explicitly calculated for some specific models of cross-linking to find the rate-controlling steps in order for the minimization of the gelation time to be easier in the gel processing. For a micellar cross-linking covering a wide range of the multiplicity, as seen in hydrophobically-modified water-soluble polymers, $t_{\mathrm{R}}$ is shown to obey a formula similar to the Aniansson–Wall law. Full article