Search Results (19)

In this paper, we demonstrate that neutral fractional evolution equations with finite delay possess a stable mild solution. Our model incorporates a mixed fractional derivative that combines the Riemann–Liouville and Caputo fractional derivatives with orders $0<\alpha <1$ and $1<\beta <2$. We identify the infinitesimal generator of the cosine family and analyze the stability of the mild solution using both Hyers–Ulam–Rassias and Hyers–Ulam stability methodologies, ensuring robust and reliable results for fractional dynamic systems with delay. In order to guarantee that the features of invariance under transformations, such as rotations or reflections, result in the presence of fixed points that remain unchanging and represent the consistency and balance of the underlying system, fixed-point theorems employ the symmetry idea. Lastly, the results obtained are applied to a fractional order nonlinear wave equation with finite delay with respect to time. Full article

This paper aims to investigate the nonlinear transition to turbulence in generalized 3D Kolmogorov flow. The difference between this and classical Kolmogorov flow is that the forcing term in the x direction $sin\left(y\right)$ is replaced with $sin\left(y\right)cos\left(z\right)$. This drastically complicates the problem. First, a stability analysis is performed by deriving the analog of the Orr–Sommerfeld equation. It is shown that for infinite stretching, the flow is stable, contrary to classical forcing. Next, a neutral curve is constructed, and the stability of the main solution is analyzed. It is shown that for the cubic domain, the main solution is linearly stable, at least for $0<R\le 100$. Next, we turn our attention to the numerical investigation of the solutions in the cubic domain. The main feature of this problem is that it is spatially periodic, allowing one to apply a relatively simple pseudo-spectral numerical method for its investigation. We apply the method of deflation to find distinct solutions in the discrete system and the method of arc length continuation to trace the bifurcation solution branches. Such solutions are called disconnected solutions if these are solutions not connected to the branch of the main solution. We investigate the influence of disconnected solutions on the dynamics of the system. It is demonstrated that when disconnected solutions are formed, the nonlinear transition to turbulence is possible, and dangerous initial conditions are these disconnected solutions. Full article

In the sense of an arbitrary time scale, some new sufficient conditions on oscillation are presented in this paper for a class of nonlinear third-order delay dynamic equations involving a local fractional derivative with a super-linear neutral term. The established oscillation results include known Kamenev and Philos-type oscillation criteria and are new oscillation results so far in the literature. Some inequalities, the Riccati transformation, the integral technique, and the theory of time scale are used in the establishment of these oscillation criteria. The proposed results unify continuous and discrete analysis, and the process of deduction is further extended to another class of nonlinear third-order delay dynamic equations involving a local fractional derivative with a super-linear neutral term and a damping term. As applications for the established oscillation criteria, some examples are given. Full article

Interior Permanent Magnet Synchronous Motor (IPMSM) motion-sensorless speed control necessitates precise knowledge of rotor flux, speed, and position. Due to numerous non-ideal aspects, such as converter nonlinearities, detection errors, integral initial value, and parameter mismatches, the conventional first-order integrator’s estimated rotor flux experiences a DC offset (Doff). Low-pass filters (LPF) with a constant cut-off frequency yield accurate estimates only in the medium- and high-speed range; however, at the low-speed area, both magnitude and phase estimates are inaccurate. The presented technique resolves the aforementioned issue for a broad speed range. In order to achieve precise flux estimation, this article presents an improved technique of flux estimator with two distinct drift mitigation strategies for the motion-sensorless field-oriented control (FOC) system of IPMSM. Using the orthogonality of the $\mathrm{\alpha }\text{-}$ and $\mathrm{\beta }$-axes, the proposed drift elimination system can estimate drift in different situations while maintaining a high level of dynamic performance. The stator flux linkage (SFL) computation in the synchronous coordinate is established from the estimation of the rotating shaft’s permanent magnetic flux linkage orientation and the statistical equations model of the SFL. By comparing the calculated SFL vector to the SFL vector derived from the stator winding voltage and currents integral model with a drift PI compensation loop, a feedback loop is formed to neutralize integral drift, and the rotational speed and position of an IPMSM is estimated utilizing the vector product of the two flux linkages in a phase-locked loop. Theoretical interpretation is presented, and Matlab Simulink simulations, as well as experimental outcomes, consistently demonstrate that the suggested estimation techniques can eliminate the phenomenon of flux drift. Full article

pH neutralization reaction process plays a crucial role in Waste Water Treatment Process (WWTP). Traditional PID Proportion Integral Differential, (or even advanced PID control) algorithms have poor performance on WWTP due to the strong non-linearity, large time lag, and large inertia characteristics of pH neutralization. Therefore, finding a superior control method to maintain the pH value of wastewater within the normal range will greatly help to improve the efficiency and effectiveness of wastewater treatment. The chemical reaction mechanism of pH neutralization reaction process is first analyzed, and a mechanistic model of pH neutralization reaction process is developed based on the reaction of ions during acid-alkali neutralization and the electric balance equation. Then, combining the characteristics of generalized predictive control and Model-Free Adaptive Control (MFAC), a Model-Free Adaptive Predictive Control (MFAPC) method based on compact format dynamic linearization is introduced. An Improved Model Free Adaptive PI Predictive Control algorithm (IMFAPC) with proportional (P) and integral (I) algorithms is proposed to further improve the control performance. IMFAPC is proposed on the basis of MFAPC, combining the advantages of generalized predictive control, introducing a PI module consisting of error and error sum, and predicting the PI module, making it possible to produce more accurate constraints on the control inputs, avoiding increasing errors, and improving the control effect of delayed systems at the same time. pH neutralization process simulation experimental results show that compared with the ordinary Model-Free Adaptive Control (MFAC) and MFAPC, the IMFAPC control algorithms has the best performance in terms of accuracy, overshoot, and the robustness. Full article

Mean wind profiles within a unit-aspect-ratio street canyon have been estimated by solving the three-dimensional Poisson equation for a set of discrete vortex sheets. The validity of this approach, which assumes inviscid vortex dynamics away from boundaries and a small nonlinear contribution to the growth of turbulent fluctuations, is tested for a series of idealised and realistic flows. In this paper, the effects of urban geometry on accuracy are examined with neutral flow over shallow, deep, asymmetric and realistic canyons, while thermal effects are investigated for a single street canyon and both bottom cooling and heating. The estimated mean profiles of the streamwise and spanwise velocity components show good agreement with reference profiles obtained from the large-eddy simulation: the canyon-averaged errors (e.g., normalised absolute errors around 1%) are of the same order of magnitude as those for the unit-aspect-ratio street canyon. It is argued that the approach generalises to more realistic flows because strong spatial localisation of the vorticity field is preserved. This work may be applied to high-resolution modelling of winds and pollutants, for which mean wind profiles are required, and fast statistical modelling, for which physically-based estimates can serve as initial guesses or substitutes for analytical models. Full article

In this work, a damped modified Kawahara equation (mKE) with cubic nonlinearity and two dispersion terms including the third- and fifth-order derivatives is analyzed. We employ an effective semi-analytical method to achieve the goal set for this study. For this purpose, the ansatz method is implemented to find some approximate solutions to the damped mKE. Based on the proposed method, two different formulas for the analytical symmetric approximations are formally obtained. The derived formulas could be utilized for studying all traveling waves described by the damped mKE, such as symmetric solitary waves (SWs), shock waves, cnoidal waves, etc. Moreover, the energy of the damped dressed solitons is derived. Furthermore, the obtained approximations are used for studying the dynamics of the dissipative dressed (modified Kawahara (mK)) dust-ion acoustic (DIA) solitons in an unmagnetized collisional superthermal plasma consisting of inertia-less superthermal electrons and inertial cold ions as well as immobile negative dust grains. Numerically, the impact of the collisional parameter that arises as a result of taking the ion-neutral collisions into account and the electron spectral index on the profile of the dissipative structures are examined. Finally, the analytical and numerical approximations using the finite difference method (FDM) are compared in order to confirm the high accuracy of the obtained approximations. The achieved results contribute to explaining the mystery of several nonlinear phenomena that arise in different plasma physics, nonlinear optics, shallow water waves, oceans, and seas, and so on. Full article

This paper presents for the first time a closed-form solution of the dynamic response of sigmoid bidirectional functionally graded (SBDFG) microbeams under moving harmonic load and thermal environmental conditions. The formulation is established in the context of the modified couple stress theory to integrate the effects of microstructure. On the basis of the elasticity theory, nonclassical governing equations are derived by using Hamilton’s principle in combination with the parabolic higher-order shear deformation theory considering the physical neutral plane concept. Sigmoid distribution functions are used to describe the temperature-dependent thermomechanical material of bulk continuums of the beam in both the axial and thickness directions, and the gradation of the material length scale parameter is also considered. Linear and nonlinear temperature profiles are considered to present the environmental thermal loads. The Laplace transform is exploited for the first time to evaluate the closed-form solution of the proposed model for a simply supported (SS) boundary condition. The solution is verified by comparing the predicted fundamental frequency and dynamic response with the previously published results. A parametric study is conducted to explore the impacts of gradient indices in both directions, graded material length scale parameters, thermal loads, and moving speed of the acted load on the dynamic response of microbeams. The results can serve as a principle for evaluating the multi-functional and optimal design of microbeams acted upon by a moving load. Full article

The purpose of this paper is to study the existence of periodic solutions for the first-order nonlinear neutral differential equation on time scales. Burton–Krasnoselskii’s fixed point theorem will be sufficiently general for application to the considered equation. An example has been carried out to show our results. It should be pointed out that the problem of periodic solutions is one of the current hot topics in the study of dynamic equations, which contains rich symmetry ideas and methods. Full article

As one of the ways to achieve carbon neutralization, shrub biomass plays an important role for natural resource management decision making in arid regions. To investigate biomass dynamic variations of Caragana korshinskii, a typical shrub found in the arid desert area of Ningxia, northwest China, we combined a nonlinear simultaneous (NLS) equation system with theoretical growth (TG) and allometric growth (AG) equations. On the basis of a large biomass survey dataset and analytical data of shrub stems, four methods (NOLS, NSUR, 2SLS, and 3SLS) of the NLS equations system were combined with the TG and AG equations. A model was subsequently established to predict the AGB growth of C. korshinskii. The absolute mean residual (AMR), root mean system error (RMSE), and adjusted determination coefficient (adj-R²) were used to evaluate the performance of the equations. Results revealed that the NSUR method of the NLS equations had better performance than other methods and the independent equations for BD and H growth and AGB. Additionally, the NSUR method exhibited extremely significant differences (p < 0.0001) when compared with the equations without heteroscedasticity on the basis of the likelihood ratio (LR) test, which used the power function (PF) as the variance function. The NSUR method of the NLS equations was an efficient method for predicting the dynamic growth of AGB by combining the TG and AG equations and could estimate the carbon storage for shrubs accurately, which was important for stand productivity and carbon sequestration capacity. Full article

This paper mainly studies the exponential stability of the highly nonlinear hybrid neutral stochastic differential equations (NSDEs) with multiple unbounded time-dependent delays and different structures. We prove the existence and uniqueness of the exact global solution of the new stochastic system, and then give several criteria of the exponential stability, including the $q_1$th moment and almost surely exponential stability. Additionally, some numerical examples are given to illustrate the main results. Such systems are widely applied in physics and other fields. For example, a specific case is pantograph dynamics, in which the delay term is a proportional function. These are widely used to determine the motion of a pantograph head on an electric locomotive collecting current from an overhead trolley wire. Compared with the existing works, our results extend the single constant delay of coefficients to multiple unbounded time-dependent delays, which is more general and applicable. Full article

In this paper, we are concerned with the oscillation of solutions to a class of third-order nonlinear neutral dynamic equations on time scales. New oscillation criteria are presented by employing the Riccati transformation and integral averaging technique. Two examples are shown to illustrate the conclusions. Full article

Under a couple of canonical and mixed canonical-noncanonical conditions, we investigate the oscillation and asymptotic behavior of solutions to a class of third-order nonlinear neutral dynamic equations with mixed deviating arguments on time scales. By means of the double Riccati transformation and the inequality technique, new oscillation criteria are established, which improve and generalize related results in the literature. Several examples are given to illustrate the main results. Full article

In the paper, we provide sufficient conditions for the oscillatory and asymptotic behavior of a new type of third-order nonlinear dynamic equations with mixed nonlinear neutral terms. Our theorems not only improve and extend existing theorems in the literature but also provide a new approach as far as the nonlinear neutral terms are concerned. The main results are illustrated by some particular examples. Full article

In this paper, we established a continuous-time agency model in which an ambiguity-averse venture capitalist (VC) employs an ambiguity-neutral entrepreneur (EN) to manage an innovative project. We analyzed the connection between ambiguity sharing and incentives under double moral hazard. Applying a stochastic dynamic programming approach, we solved the VC’s maximization problem and obtained the Hamilton–Jacobi–Bellman (HJB) equation under a special form of the value function. We showed that the optimal pay-performance sensitivity was a fixed point of a nonlinear equation. The model ambiguity on the probability measure induced a tradeoff between ambiguity sharing and the incentive compensation that improved the EN’s pay-performance sensitivity level. Besides, we simulated the model and showed that when two efforts were complementary, the VC’s effort did not monotonically decrease with respect to the pay-performance sensitivity, while the EN’s effort did not monotonically increase in the pay-performance sensitivity level. More importantly, we found that as efforts tended to be more complementary, the optimal pay-performance sensitivity tended to approach those that maximized the efforts exerted by the EN and the VC. Full article