Search Results (32)

This study investigates the coupling mechanism between a parabolic notch and dislocations in one-dimensional (1D) hexagonal piezoelectric quasicrystals (PQCs) based on the theory of complex variable functions. By applying perturbation techniques and the Cauchy integral, analytical solutions for complex potentials are derived, yielding closed-form expressions for the phonon–phason stress field and electric displacement field. Numerical examples reveal several key findings: significant stress concentration occurs at the notch root, accompanied by suppression of electric displacement; interference patterns between dislocation cores and notch-induced stress singularities are identified; the J-integral quantifies distance-dependent forces, size effects, and angular force distributions reflecting notch symmetry; and the energy-driven dislocation slip toward free surfaces leads to the formation of dislocation-free zones. These results provide new insights into electromechanical fracture mechanisms in quasicrystals. Full article

For a massive scalar field in a fixed Schwarzschild background, the radial wave equation obeyed by Fourier modes is first studied. After reducing such a radial wave equation to its normal form, we first study approximate solutions in the neighborhood of the origin, horizon and point at infinity, and then we relate the radial with the Heun equation, obtaining local solutions at the regular singular points. Moreover, we obtain the full asymptotic expansion of the local solution in the neighborhood of the irregular singular point at infinity. We also obtain and study the associated integral representation of the massive scalar field. Eventually, the technique developed for the irregular singular point is applied to the homogeneous equation associated with the inhomogeneous Zerilli equation for gravitational perturbations in a Schwarzschild background. Full article

In this work, we present various novelty methods by employing the fractional differential quadrature technique to solve the time and space fractional nonlinear Benjamin–Bona–Mahony equation and the Benjamin–Bona–Mahony–Burger equation. The novelty of these methods is based on the generalized Caputo sense, classical differential quadrature method, and discrete singular convolution methods based on two different kernels. Also, the solution strategy is to apply perturbation analysis or an iterative method to reduce the problem to a series of linear initial boundary value problems. Consequently, we apply these suggested techniques to reduce the nonlinear fractional PDEs into ordinary differential equations. Hence, to validate the suggested techniques, a solution to this problem was obtained by designing a MATLAB code for each method. Also, we compare this solution with the exact ones. Furthermore, more figures and tables have been investigated to illustrate the high accuracy and rapid convergence of these novel techniques. From the obtained solutions, it was found that the suggested techniques are easily applicable and effective, which can help in the study of the other higher-D nonlinear fractional PDEs emerging in mathematical physics. Full article

In this paper, we deal with a strongly singular problem involving a non-local operator, a critical nonlinearity, and a subcritical perturbation. We apply techniques from non-smooth analysis to the energy functional, in combination with the study of the topological properties of the sublevels of its smooth part, to prove the existence of three weak solutions: two points of local minimum and a third one as a mountain pass critical point. Full article

In this paper, we study the existence of positive solutions for a changing-sign perturbation tempered fractional model with weak singularity which arises from the sub-diffusion study of anomalous diffusion in Brownian motion. By two-step substitution, we first transform the higher-order sub-diffusion model to a lower-order mixed integro-differential sub-diffusion model, and then introduce a power factor to the non-negative Green function such that the linear integral operator has a positive infimum. This innovative technique is introduced for the first time in the literature and it is critical for controlling the influence of changing-sign perturbation. Finally, an a priori estimate and Schauder’s fixed point theorem are applied to show that the sub-diffusion model has at least one positive solution whether the perturbation is positive, negative or changing-sign, and also the main nonlinear term is allowed to have singularity for some space variables. Full article

This paper presents an indirect adaptive neural network (NN) control algorithm tailored for flexible joint robots (FJRs), aimed at achieving desired transient and steady-state performance. To simplify the controller design process, the original higher-order system is decomposed into two lower-order subsystems using the singular perturbation technique (SPT). NNs are then employed to reconstruct the aggregated uncertainties. An adaptive prescribed performance control (PPC) strategy and a continuous terminal sliding mode control strategy are introduced for the reduced slow subsystem and fast subsystem, respectively, to guarantee a specified convergence speed and steady-state accuracy for the closed-loop system. Additionally, a composite-learning optimal bounded ellipsoid algorithm (OBE)-based identification scheme is proposed to update the NN weights, where the tracking errors of the reduced slow and fast subsystems are integrated into the learning algorithm to enhance the identification and tracking performance. The stability of the closed-loop system is rigorously established using the Lyapunov approach. Simulations demonstrate the effectiveness of the proposed identification and control schemes. Full article

We review the hadronic light-by-light (HLbL) contribution to the muon anomalous magnetic moment. Upcoming measurements will reduce the experimental uncertainty of this observable by a factor of four; therefore, the theoretical precision must improve accordingly to fully harness such an experimental breakthrough. With regards to the HLbL contribution, this implies a study of the high-energy intermediate states that are neglected in dispersive estimates. We focus on the maximally symmetric high-energy regime and in-quark loop approximation of perturbation theory, following the method of the OPE with background fields proposed by Bijnens et al. in 2019 and 2020. We confirm their results regarding the contributions to the muon $g-2$. For this, we use an alternative computational method based on a reduction in the full quark loop amplitude, instead of projecting on a supposedly complete system of tensor structures motivated by first principles. Concerning scalar coefficients, mass corrections have been obtained by hypergeometric representations of Mellin–Barnes integrals. By our technique, the completeness of such kinematic singularity/zero-free tensor decomposition of the HLbL amplitude is explicitly checked. Full article

Singular singularly-perturbed problems (SSPPs) are a powerful mathematical tool for modelling a variety of real phenomena, such as nuclear reactions, heat explosions, mechanics, and hydrodynamics. In this paper, the numerical solutions to fourth-order singular singularly-perturbed boundary and initial value problems are presented using a novel quintic B-spline (QBS) approximation approach. This method uses a quasi-linearization approach to solve SSPNL initial/boundary value problems. And the non-linear problems are transformed into a sequence of linear problems by applying the quasi-linearization approach. The QBS functions produce more accurate results when compared to other existing approaches because of their local support, symmetry, and partition of unity features. This method can be applied to immediately solve the SSPPs without reducing the order in which they are presented. It has been demonstrated that the suggested numerical approach converges uniformly over the whole domain. The proposed approach is implemented on a few problems to validate the scheme. The computational results are compared, and they illustrate that the proposed approach performs better. Full article

In this study, we present a general form of nonlinear two-time-scale systems, where singular perturbation analysis is used to separate the dynamics of the slow and fast subsystems. Machine learning techniques are utilized to approximate the dynamics of both subsystems. Specifically, a recurrent neural network (RNN) and a feedforward neural network (FNN) are used to predict the slow and fast state vectors, respectively. Moreover, we investigate the generalization error bounds for these machine learning models approximating the dynamics of two-time-scale systems. Next, under the assumption that the fast states are asymptotically stable, our focus shifts toward designing a Lyapunov-based model predictive control (LMPC) scheme that exclusively employs the RNN to predict the dynamics of the slow states. Additionally, we derive sufficient conditions to guarantee the closed-loop stability of the system under the sample-and-hold implementation of the controller. A nonlinear chemical process example is used to demonstrate the theory. In particular, two RNN models are constructed: one to model the full two-time-scale system and the other to predict solely the slow state vector. Both models are integrated within the LMPC scheme, and we compare their closed-loop performance while assessing the computational time required to execute the LMPC optimization problem. Full article

In this paper, we provide sufficient conditions for the existence, uniqueness and global behavior of a positive continuous solution to some nonlinear Riemann-Liouville fractional boundary value problems. The nonlinearity is allowed to be singular at the boundary. The proofs are based on perturbation techniques after reducing the considered problem to the equivalent Fredholm integral equation of the second kind. Some examples are given to illustrate our main results. Full article

Pump-controlled hydraulic actuators (PHAs) contain slow mechanical and fast hydraulic dynamics, and thus singular perturbation theory can be adopted in the control strategies of PHAs. In this article, we develop a singular perturbation theory-based composite control approach for a PHA with position tracking error constraint. Disturbance observers (DOBs) are used to estimate the matched and mismatched uncertainties for online compensation. A sliding surface-like error variable is proposed to transform the second-order mechanical subsystem into a first-order error subsystem. Consequently, the position tracking error constraint of the PHA is decomposed into the output constraint of the first-order error subsystem and the stabilizing of the first-order hydraulic subsystem. Slow and fast control laws can be easily designed without using the backstepping technique, thus simplifying the control design and reducing the computational burden to a large extent. Theoretical analysis verifies that desired stability properties can be achieved by an appropriate selection of the control parameters. Simulations and experiments are performed to confirm the efficacy and practicability of the proposed control strategy. Full article

Deep brain stimulation (DBS) is widely used as a treatment option for patients with movement disorders. In addition to its clinical impact, DBS has been utilized in the field of cognitive neuroscience, wherein the answers to several fundamental questions underpinning the mechanisms of neuromodulation in decision making rely on the ways in which a burst of DBS pulses, usually delivered at a clinical frequency, i.e., 130 Hz, perturb participants’ choices. It was observed that neural activities recorded during DBS were contaminated with large artifacts, which lasts for a few milliseconds, as well as a low-frequency (slow) signal (~1–2 Hz) that can persist for hundreds of milliseconds. While the focus of most of methods for removing DBS artifacts was on the former, the artifact removal capabilities of the slow signal have not been addressed. In this work, we propose a new method based on combining singular value decomposition (SVD) and normalized adaptive filtering to remove both large (fast) and slow artifacts in local field potentials, recorded during a cognitive task in which bursts of DBS were utilized. Using synthetic data, we show that our proposed algorithm outperforms four commonly used techniques in the literature, namely, (1) normalized least mean square adaptive filtering, (2) optimal FIR Wiener filtering, (3) Gaussian model matching, and (4) moving average. The algorithm’s capabilities are further demonstrated by its ability to effectively remove DBS artifacts in local field potentials recorded from the subthalamic nucleus during a verbal Stroop task, highlighting its utility in real-world applications. Full article

The investigation of dynamic behaviors of inertial neural networks depicted by second-order delayed differential equations has received considerable attention. Substantial research has been performed on the transformed first-order differential equations using traditional variable substitution. However, there are few studies on bifurcation dynamics using direct analysis. In this paper, a multi-delay Hopfield neural system with inertial couplings is considered. The perturbation scheme and non-reduced order technique are firstly combined into studying multi-delay induced Hopf–Hopf singularity. This combination avoids tedious computation and overcomes the disadvantages of the traditional variable-substitution reduced-order method. In the neighbor of Hopf–Hopf interaction points, interesting dynamics are found on the plane of self-connected delay and coupled delay. Multiple delays can induce the switching of stable periodic oscillation and periodic coexistence. The explicit expressions of periodic solutions are obtained. The validity of theoretical results is shown through consistency with numerical simulations. Full article

The study first proposes the difficult nonlinear convergent radius and convergent rate formulas and the complete derivations of a mathematical model for the nonlinear five-link human biped robot (FLHBR) system which has been a challenge for engineers in recent decades. The proposed theorem simultaneously has very distinctive superior advantages including the stringent almost disturbance decoupling feature that addresses the major deficiencies of the traditional singular perturbation approach without annoying “complete” conditions for the discriminant function and the global exponential stability feature without solving the impractical Hamilton–Jacobi equation for the traditional H-infinity technique. This article applies the feedback linearization technique to globally stabilize the FLHBR system that greatly improved those shortcomings of nonlinear function approximator and make the effective working range be global for whole state space, whereas the traditional Jacobian linearization technique is valid only for areas near the equilibrium point. In order to make some comparisons with traditional approaches, first example of the representative ones, that cannot be addressed well for the pioneer paper, is shown to demonstrate the fact that the effectiveness of the proposed main theorem is better than the traditional singular perturbation technique. Finally, we execute a second simulation example to compare the proposed approach with the traditional PID approach. The simulation results show that the transient behaviors of the proposed approach including the peak time, the rise time, the settling time and the maximum overshoot specifications are better than the traditional PID approach. Full article

The purpose of this article is to solve a nonlinear fractional Klein–Fock–Gordon equation that involves a recently created non-singular kernel fractional derivative by Caputo–Fabrizio. Motivated by some physical applications related to the fractional Klein–Fock–Gordon equation, we focus our study on this equation and some phenomena rated to it. The findings are crucial and essential for explaining a variety of physical processes. In order to find satisfactory approximations to the offered problems, this work takes into account a modern methodology and fractional operator in this context. We first take the Yang transform of the Caputo–Fabrizio fractional derivative and then implement it to solve fractional Klein–Fock–Gordon equations. We will consider three cases of the nonlinear fractional Klein–Fock–Gordon equation to ensure the applicability and effectiveness of the suggested technique. In order to determine an approximate solution to the fractional Klein–Fock–Gordon equation in the fast convergent series form, we can use the fractional homotopy perturbation transform approach. The numerical simulation is provided to demonstrate the effectiveness and dependability of the suggested method. Furthermore, several fractional orders will be used to describe the behavior of the given solutions. The results achieved demonstrate the high efficiency, ease of use, and applicability of this strategy for resolving other nonlinear issues. Full article