This paper is concerned with the existence and stability of traveling wave solutions for a class of non-local time-delayed equations with p-Laplacian diffusion. We first show the existence of traveling wave solutions via monotone iterative method. After providing the existence, uniqueness and regularity of original solution for the Cauchy problem by means of compactness technique, we prove the exponential stability of the non-critical waves. Here the method adapted is the approximated weighted energy method.

Our new approach to MWD mud pulse telemetry offers significant increases over conventional data rates. Using waveguide acoustics, we show that common drilling communications channels support carrier frequencies exceeding several hundred Hertz. Such signals are prone to attenuation over large drill pipe distances. Low power, self-spinning “turbosirens” are designed, providing high torque and rotation rate performance at all flow rates without electric or hydraulic motor drives. The sirens rotate, drawing only on the kinetic energy of the mud, like flapping flags oscillating in wind. Our turbosirens are minimally affected by LCM jamming. Turbine and siren rotor components ride freely along longitudinal trenches built into rotating shafts. Both magically float into the oncoming mud as it slows down, thus, widening all gap spaces and freeing any trapped debris even as both travel opposite to the mud flow. In addition, “turbosirens in series” signal superposition, drawing on constructive wave interference, offers strong pressure signals at high frequencies without mechanical complexity. Rapid modulations are possible, also without motor drive, relying on rapidly acting electro-magneto rheological brakes powered by turbosirens. To take advantage of hardware capabilities, “Intelligent <em>i</em> FSK” telemetry, or Frequency Shift Keying creates clean signals without the ambiguous wave reflections found with PSK and randomized time shifts. Frequency pairs are not selected arbitrarily, but chosen from “neighboring pressure peaks and valleys” in frequency space as determined from bottomhole assembly waveguide Fourier analysis. Finally, surface signal processing and reflection removal are facilitated with time delay and differential equation algorithms. For deep wells where multiple drill pipe reflections are unlikely, analytical solutions for both are obtained assuming single reflections which do not involve windowing challenges and complicated digital filter design. Software and hardware developments patented, published and validated over the years, are integrated, refined and readied for controlled field tests.

This paper proposes a new interval-based contractor for nonlinear equations which is minimal when dealing with narrow boxes. The method is based on the centered form classically used by interval algorithms combined with a Gauss Jordan band diagonalization preconditioning. As an illustration in stability analysis, we propose to compute the set of all parameters of a characteristic function of linear time-delayed equations which have at least one zero in the imaginary axis. Our approach is able compute a guaranteed and accurate enclosure of the solution set faster than existing approaches.

A non-fragile robust [formula omitted] controller for continuous-time delayed models

We study the disentanglement dynamics of two giant atoms coupled to a common one-dimensional waveguide. We focus on the non-Markovian retarded effect in the disentanglement of the two giant atoms by taking the photon transmission time into account. By solving the time-delayed equations of motion for the probability amplitudes, we obtain the evolution of the entanglement of the two giant atoms, which are initially in the maximally entangled states in the single-excitation space. It is found that the retardation-induced non-Markovianity leads to non-exponential decay and revivals of entanglement. Concretely, we consider separate-, braided-, and nested-coupling configurations, and find that the disentanglement dynamics in these configurations exhibits different features. We demonstrate that the steady-state entanglement depends on the time delay under certain conditions in these three coupling configurations. We also study the dependence of the disentanglement of the two giant atoms on both the detuning effect and the initial-state phase effect. In addition, we consider the disentanglement dynamics of the two giant atoms, which are initially in the state superposed by zero-excitation and two-excitation components. This work will pave the way for the generation of stationary entanglement between two giant atoms, which may have potential applications in the construction of large-scale quantum networks based on the giant-atom waveguide-QED systems.

This manuscript aims to investigate the existence, uniqueness, and stability of non-local random impulsiveneutral stochastic differential time delay equations (NRINSDEs) with Poisson jumps. First, we prove theexistence of mild solutions to this equation using the Banach fixed point theorem. Next, we prove thestability via continuous dependence initial value. Our study extends the work of Wang and Wu [15] wherethe time delay is addressed by the prescribed phase space B (defined in Section 3). An example is given toillustrate the theory.

In this paper, we study the application of iterative learning control to a class of nonlinear varying time-delay equations. The handled procedure is the distributed parameter system, hyperbolic type. The selected plan is based on PD-type iterative learning control with initial state learning. Initially, we present the equation and the control law utilised. Subsequently, we presented some dilemmas. Then, sufficient conditions for monotone convergence of the error under the proper assumption are found. Also, a detailed convergence analysis is given based on the once-offered lemmas and Gronwall inequality. Finally, we show the usefulness of the method utilising a numerical model. Abbreviations DPS: distributed parameter system; ILC: iterative learning control; PDE:partial differential equation; DPS: partial difference system; PD: proportional derivative

Integro-differential models of natural and anthropogenic processes and phenomena arise in many research problems ranging from physical and chemical processes to biophysics and life science. The theory of its theoretical description is closely connected with various areas of pure and applied mathematics including integro-differential equations, nonlinear dynamics, pattern formation, non-Markovian processes, nonlinear and anomalous transport, and time-delay equations. While working on the content of our special issue, we were focused on selecting the most original and high-quality contributions related to the mathematical theory of such processes and phenomena including the models, computational algorithms and analysis, and mathematical methods regarded as new and promising for understanding the problem that arises both in natural and anthropogenic conditions. Among the others, the contributions devoted to the real applications of integro-differential models in physical, physicochemical, and biophysical processes and natural phenomena were appreciated. We are now happy to announce that our work on special issue “Integro-differential models of natural and anthropogenic processes and phenomena” is successfully completed, so please enjoy the recent achievements in the field and feel free to contact the authors regarding your questions and hopefully new fruitful collaborations. We thank heartily all the authors who submitted their research papers and the reviewers who assisted in reviewing process. It is especially appreciated since we know how hard it is to manage work-life balance in pandemic times. We would also like to thank the editorial board of Mathematical Methods in the Applied Sciences for their great support, careful supervision, and outstanding team work. Open Access funding enabled and organized by Projekt DEAL. Irina Nizovtseva: Conceptualization; methodology; project administration; supervision; visualization. Dmitri Alexandrov: Conceptualization; formal analysis; investigation; methodology; supervision; validation.

FPGA implementation of nonlinear equations with delay

In this paper we present an approximate model for load frequency control system with time delay. The load frequency control is one of the conventional power system control problems. In order to secure the stability of the grid the frequency must remain within its limited range which is achieved through the load frequency control. The load frequency control signals experience time delay that could destabilize the power systems. The presence of the time delay complicates the analysis of the load frequency control system. In this paper we present a stability method based on the Direct Frequency Response approximation for the time delay. This approximation transforms the transcendental time delay equation into linear equation. This results in a simple stability criterion for the loadfrequency control system with time delay. A one-area load frequency control system is chosen as a case study. The effectiveness of the proposed approximation has been tested through simulation and comparison with the published research work. By tracking the eigenvalues or using Routh's criterion the maximum delay margin can be estimated. The proposed stability criterion has been compared with the most recent methods and showed it is merit. The range of the PI controller parameters for a given time delay can be determined which is very important in practice.

Representation of solutions of nonhomogeneous conformable fractional delay differential equations

This special issue “Nonlinear dynamics of phase transitions” is devoted to recent trends in the mathematical description of phase transitions, which arise in many research problems ranging from physical and chemical processes to biophysics and life science. The theory of these processes and phenomena presented by leading researchers is closely connected with various areas of pure and applied mathematics including nonlinear dynamics, pattern formation, non-Markovian processes, anomalous transport, time delay, and integral equations. The present special issue contains 29 original and high-quality contributions related to the mathematical theory of phase transitions including the models, computational algorithms, and analysis. Contributions devoted to real applications of nonlinear dynamics of phase transitions in physical, physicochemical, and biophysical processes and natural phenomena are included as well. We sincerely thank all the authors who submitted research articles and the reviewers who assisted in reviewing these articles. We would also like to thank the editorial board of Mathematical Methods in the Applied Sciences for their invaluable support of our special issue at all stages of its preparation.

Novel Control-Oriented Models for Liquid Transport in Falling Film Evaporator Tubes

This paper is devoted to solve a set of non-linear optimal control problems which are touched with time-delay Fredholm integro-differential equations. The serious objective of this work is to contribute an appropriate direct scheme for solving these problems. The technique used in this paper is based upon the Dickson polynomials and collocation points. Getting through the solutions, the states and controls variables can be approximated with Dickson polynomials. Therefore, the optimal control problem with time-delay integro-differential equation transforms into a system of algebraic equations that by solving it, we can obtain the unknown coefficients of the main problem. The residual error estimation of this technique is also investigated. Accuracy amount of the absolute errors have been studied for the performance of this method by solving several non-trivial examples.

We report on the design and modeling of vertical-cavity surface-emitting laser (VCSEL) integrated in the lateral direction with a cascade of multiple passive cavities. The scheme is proposed to increase the bandwidth of the VCSEL in the mm-waveband to generate ultra-high-frequency oscillations with low-frequency chirp. The model treats the transverse feedback induced by the cascaded transverse coupled cavities (TTCs) as time delay of the transverse slow light due to round trips in these cavities. The simulation is based on numerical integration of the modified time-delay rate equations of the intensity and phase of the electric field. The simulation results are used to optimize the TCCs parameters, including the coupling ratio and TCC length, for generating ultra-high-frequency signals with high intensity, low chirping and low distortion. The obtained results show that the proposed structure could achieve 300% enhancement of the modulation bandwidth of the C-VCSEL due to either extended carrier-photon resonance (CPR) frequencies of photon–photon resonance (PPR). Signals with frequencies as high as (40–49 GHz) with second-harmonic distortion lower than − 30 dB are presented.

Nonlinear singularly perturbed problem for time-delay evolution equation with two parameters is studied. Using the variables of the multiple scales method, homogeneous equilibrium method, and approximation expansion method from the singularly perturbed theories, the structure of the solution to the time-delay problem with two small parameters is discussed. Under suitable conditions, first, the outer solution to the time-delay initial boundary value problem is given. Second, the multiple scales variables are introduced to obtain the shock wave solution and boundary layer corrective terms for the solution. Then, the stretched variable is applied to get the initial layer correction terms. Finally, using the singularly perturbed theory and the fixed point theorem from functional analysis, the uniform validity of asymptotic expansion solution to the problem is proved. In addition, the proposed method possesses the advantages of being very convenient to use.

In this paper, a numerical method based on the Lagrangian piece-wise interpolation is proposed to solve variable-order fractal-fractional time delay equations with power law, exponential decay and Mittag-Leffler memories. These operators permit to describe physical phenomena with variable memory and fractal variable dimension. Numerical methods were applied to simulate the variable-order time delay Mackey–Glass and synaptically coupled FitzHugh–Nagumo models. Our numerical simulations display several new attractors.

This paper focuses on the moving target localization and velocity measurement in incoherent centralized multiple-input and multiple-output (MIMO) radar systems with widely separated antennas in 3D space. In this paper, we assume that parameters such as time-of-arrival (TOA), frequency-of-arrival (FOA), azimuth angles and elevation angles have already been measured. With these measurements, a final closed-form solution of the target position and velocity can be obtained via the weighted least squares (WLS) method. When the target is located in the far field, due to poor system observability, a first-order Taylor expansion based on the WLS solution is necessary to obtain a more accurate and unbiased solution. Unlike the preceding papers which were based on the two-stage weighted least squares (2SWLS) method [1]-[3], in this paper, the angle information is introduced into the time delay equations and the Doppler frequency equations, so that the intermediate variables in the estimator can be eliminated [4], [5]. Meanwhile, the time delay equations and the Doppler equations are transformed into linear equations only related to the position and the speed of the target. This method, unlike 2SWLS-based methods [1]-[3], does not introduce auxiliary variables, so it does not require the decorrelation procedure. Simulation results show that the root mean-square error (RMSE) of position and velocity can reach Cramer-Rao lower bound (CRLB) when the noise is at a moderate level before the thresholding effect occurs.

In this paper, a numerical method consists of combining Haar wavelet method and Tikhonov regularization method to determine unknown boundary condition and unknown nonlinear source term for the generalized time-delayed Burgers-Fisher equation using noisy data is presented. A stable numerical solution is determined for the problem. We also show that the rate of convergence of the method is as exponential $ \Bigl(O\left(\frac{1}{2^{J+1}}\right)\Bigr) $, where $ J $ is maximal level of resolution of wavelet. Some numerical results are reported to show the efficiency and robustness of the proposed approach for solving the inverse problems.

ABSTRACTLaser self-mixing interferometry (SMI) is a well-known measurement technique having been applied to the fields of geometrical quantities detection, medical treatment and industry manufacturing. Its detection sensitivity can be improved by frequency shifting towards the laser's relaxation oscillation frequency (ROF), which has always been implemented by a pair of acousto-optic modulators (AOM). This manuscript presents a novel method based on Doppler effect for frequency shifting, which is based on a rotating disk and has the advantages of convenient adjustment and low cost. The theoretical analysis in the form of Doppler shift and time-delayed rate equations is followed, and simulative and experimental results are included to prove its validity. To our knowledge, this technique has not been used in practice till now, and the proposed structure, with these advantages, can find significant applications in designing high performance SMI sensors, and can extend SMI measurement to more kinds of non-cooperative surfaces.