Nonlinear Terms Research Articles

Simulations of nerve signal propagation in axons

PurposeIn the literature, soliton solutions of the Heimburg–Jackson model have been proposed by Drab et al. (2022), but for the considered models, i.e. Eq.(1) and Eq.(2), the existence of solitons, the dispersion analysis and the pseudospectral method have been studied (Engelbrecht et al., 2006, 2018, 2020; Tamm et al., 2017, 2022; Peets et al., 2013). Therefore, the gap should be filled by this work.Design/methodology/approachWhen nonlinear terms, dissipative terms and forcing terms are ignored, the system (Eq.(2)) reduces to a single, sixth-order partial differential equation (Tamm et al., 2022). In this work, our aim is to propose analytical solutions in the explicit form via ansatz-based method. Therefore, the parameter effects in wave profile will be proposed clearly in figures.FindingsWhile progress has been made in signal propagation in nerves, thanks to many experimental studies and theoretical predictions over the last two centuries, the results obtained in this study may answer new questions that arise.Originality/valueIn the literature, the existence of solitons, dispersion analysis and pseudospectral method have been investigated for the Heimburg-Jackson model (Engelbrecht et al., 2006, 2018, 2020; Tamm et al., 2017, 2022; Peets et al., 2013), and this study fills the gap in soliton solutions. Additionally, when nonlinear terms, dissipative term and forcing terms are ignored, the system (Eq.(2)) reduced to a single equation that is sixth-order partial differential equation (Tamm et al., 2022). In this work, our aim is to propose analytical solutions in the explicit form via ansatz-based method. Therefore, the parameter effects in wave profile will be proposed clearly in figures.

Comparing methods of measuring interest fit: A large prediction study with career choice satisfaction

AbstractVocational interest inventories are widely used in both research and practice to help match people to well‐fitting work environments. However, because there are many different methods to operationalize interest fit, a debate remains regarding the best ways to do so. To empirically inform this debate, our study compared the predictive power of four widely used interest fit indices (i.e., matching scale scores, profile deviance scores, profile correlations, and polynomial regression scores) for predicting career choice satisfaction. Using a large and diverse U.S. sample (N = 257,320), results indicated that among the three single‐term interest fit measures, profile correlations (R2 = .04) explained more variance in career choice satisfaction than matching scale scores (R2 = .02) and profile deviance scores (R2 = .00). By comparison, the full 30‐term polynomial regression model explained the most variance in career choice satisfaction (R2 = .09); in this case, however, the nonlinear terms that capture fit effects only accounted for about 22% (R2 = .02) of the total variance explained by the model. Overall, these results indicate that researchers and practitioners should be cautious of the greater criterion‐related validity of polynomial regression models as fit information may not be a substantial contributor to their predictive capacities. In addition, our findings support the use of profile correlations as a predictive, single‐term measure of interest fit.

A Nonlinear Approach in the Quantification of Numerical Uncertainty by High-Order Methods for Compressible Turbulence with Shocks

This is a comprehensive overview on our research work to link interdisciplinary modeling and simulation techniques to improve the predictability and reliability simulations (PARs) of compressible turbulence with shock waves for general audiences who are not familiar with our nonlinear approach. This focused nonlinear approach is to integrate our “nonlinear dynamical approach” with our “newly developed high order entropy-conserving, momentum-conserving and kinetic energy-preserving methods” in the quantification of numerical uncertainty in highly nonlinear flow simulations. The central issue is that the solution space of discrete genuinely nonlinear systems is much larger than that of the corresponding genuinely nonlinear continuous systems, thus obtaining numerical solutions that might not be solutions of the continuous systems. Traditional uncertainty quantification (UQ) approaches in numerical simulations commonly employ linearized analysis that might not provide the true behavior of genuinely nonlinear physical fluid flows. Due to the rapid development of high-performance computing, the last two decades have been an era when computation is ahead of analysis and when very large-scale practical computations are increasingly used in poorly understood multiscale data-limited complex nonlinear physical problems and non-traditional fields. This is compounded by the fact that the numerical schemes used in production computational fluid dynamics (CFD) computer codes often do not take into consideration the genuinely nonlinear behavior of numerical methods for more realistic modeling and simulations. Often, the numerical methods used might have been developed for weakly nonlinear flow or different flow types other than the flow being investigated. In addition, some of these methods are not discretely physics-preserving (structure-preserving); this includes but is not limited to entropy-conserving, momentum-conserving and kinetic energy-preserving methods. Employing theories of nonlinear dynamics to guide the construction of more appropriate, stable and accurate numerical methods could help, e.g., (a) delineate solutions of the discretized counterparts but not solutions of the governing equations; (b) prevent numerical chaos or numerical “turbulence” leading to FALSE predication of transition to turbulence; (c) provide more reliable numerical simulations of nonlinear fluid dynamical systems, especially by direct numerical simulations (DNS), large eddy simulations (LES) and implicit large eddy simulations (ILES) simulations; and (d) prevent incorrect computed shock speeds for problems containing stiff nonlinear source terms, if present. For computation intensive turbulent flows, the desirable methods should also be efficient and exhibit scalable parallelism for current high-performance computing. Selected numerical examples to illustrate the genuinely nonlinear behavior of numerical methods and our integrated approach to improve PARs are included.

The uniqueness of minimizers for L2$$ {L}^2 $$‐subcritical inhomogeneous variational problems with a spatially decaying nonlinearity

By constructing various Pohozaev identities, we study the uniqueness of minimizers for ‐subcritical inhomogeneous variational problems with spatially decaying nonlinear terms, which contains as a singular point.

On the blow-up for a Kuramoto–Velarde type equation

It is known that the Kuramoto-Velarde equation is globally well-posed on Sobolev spaces in the case when the parameters γ1 and γ2 involved in the non-linear terms verify γ2=γ12 or γ2=0. In the complementary case of these parameters, the global existence or blow-up of solutions is a completely open (and hard) problem. Motivated by this fact, in this work we consider a non-local version of the Kuramoto-Velarde equation. This equation allows us to apply a Fourier-based method and, within the framework γ2≠γ12 and γ2≠0, we show that large values of these parameters yield a blow-up in finite time of solutions in the Sobolev norm. As a complement to it, we address an alternative result on the finite-time blow-up of smooth solutions by considering a virial-type estimate.

Long sequence Hopfield memory*

Abstract Sequence memory is an essential attribute of natural and artificial intelligence that enables agents to encode, store, and retrieve complex sequences of stimuli and actions. Computational models of sequence memory have been proposed where recurrent Hopfield-like neural networks are trained with temporally asymmetric Hebbian rules. However, these networks suffer from limited sequence capacity (maximal length of the stored sequence) due to interference between the memories. Inspired by recent work on Dense Associative Memories, we expand the sequence capacity of these models by introducing a nonlinear interaction term, enhancing separation between the patterns. We derive novel scaling laws for sequence capacity with respect to network size, significantly outperforming existing scaling laws for models based on traditional Hopfield networks, and verify these theoretical results with numerical simulation. Moreover, we introduce a generalized pseudoinverse rule to recall sequences of highly correlated patterns. Finally, we extend this model to store sequences with variable timing between states’ transitions and describe a biologically-plausible implementation, with connections to motor neuroscience.

A Sub-1 ppm/°C Reference Voltage Source with a Wide Input Range

With the continuous advancement of electronic technology, the application of high-voltage integrated circuits is becoming increasingly prevalent in fields such as power systems, medical devices, and industrial automation. The reference circuit within high-voltage integrated circuits must not only exhibit insensitivity to temperature variations but also maintain stability across a broad voltage supply. This paper presents a bandgap reference (BGR) source capable of operating over a wide input range. This BGR employs a high-order curvature compensation method to eliminate nonlinear voltage terms, resulting in minimal temperature drift. The circuit achieves an impressive temperature coefficient (TC) of 0.88 ppm/°C over a temperature range from −40 °C to 130 °C. To ensure stable operation within a 4–40 V range, the design incorporates a pre-regulation circuit that stabilizes the supply voltage of the BGR core at a fixed value, thereby enhancing the ability to withstand variations in power supply voltage.

Fractional Caputo Operator and Takagi–Sugeno Fuzzy Modeling to Diabetes Analysis

Diabetes is becoming more and more dangerous, and the effects continue to grow due to the population’s ignorance of the seriousness of this phenomenon. The studies that have been carried out have not been able to follow the phenomenon more precisely, which has led to the use of the fractional derivative tool, which has a very great capability to study real problems and phenomena but is somewhat limited on nonlinear models. In this work, we will develop a new fractional derivative model of a diabetic population, the Takagi–Sugeno fractional fuzzy model, which will enable us to study the phenomenon with these nonlinear terms in order to obtain greater precision in the results. We will study the existence and uniqueness of the solution using the Lipschizian theorem and then turn to the new fuzzy model, which leads us to four dynamical systems. The interpretation results show the quality of fuzzy membership in tracking the malleable phenomena of nonlinear terms existing in the system.

Distributed leader-following bipartite consensus for one-sided Lipschitz multi-agent systems via dual-terminal event-triggered mechanism

This article analyses leader-following bipartite consensus for one-sided Lipschitz multi-agent systems by dual-terminal event-triggered output feedback control approach. A distributed observer is designed to estimate unknown system states by employing relative output information at triggering time instants, and then an event-triggered output feedback controller is proposed. Dual-terminal dynamic event-triggered mechanisms are proposed in sensor–observer channel and controller–actuator channel, which can save communication resources to a great extent, and the Zeno behavior is ruled out. A new generalized one-sided Lipschitz condition is proposed to handle the nonlinear term and achieve bipartite consensus. Some stability conditions are presented to guarantee leader-following bipartite consensus. Finally, one-link robot manipulator systems are introduced to demonstrate the availability of the designed scheme. The results demonstrate that the agents of the robot manipulators can track the reference trajectories bi-directionally, and effectively reduce communication resources by 61.22% and 68.04% at the sensor–observer and controller–actuator channels, respectively.

Study of a non-linear Volterra integro-differential equation with a non-linear unknown source term

In this paper, we propose a new class of integro-differential nonlinear Volterra equations with a non-linear unknown source term. Our study focuses on both theoretical analysis and numerical approximation of solutions to these equations. In the theoretical analysis partition, we define the specific assumptions on the integral kernel's characteristics, on the source term function and also on the derivative of the solutions of equations, under which the Krasnoselskii's fixed point theorem can be applied and in order to ensures the existence and uniqueness of solutions to the proposed integro-differential equations. In the numerical approximation partition, we use a well-known numerical technique for solving integral equations called Nyström method. This method allows us to get an approximate solutions of the proposed equations. Furthermore, we provide some illustrative examples and according to the numerical method described in this paper we approach them, where the comparison results between the exact and approximate solutions of these examples confirm the effectiveness of the numerical Nyström method for approaching the solutions of this class of integro-differential equations.

Stationary scattering for the nonlinear Schrödinger equation with point-like obstacles: exact solutions

AbstractWe solve the Nonlinear Schrödinger Equation (NLSE) in 1D in presence of one, two and several Dirac delta potentials. With the help of an equivalent central force problem we obtain the analytical solutions in terms of a biparametric family containing the Jacobi functions. Elliptic Jacobi functions are already reported in the literature but they have not been used in the context of a scattering problem under causal boundary conditions. In the simplest examples of one or two Dirac deltas we analyze how the nonlinear term of the equation affects the modulus and phase profiles of the wave function. We also study the transmission curves under the nonlinear modification of the tunneling behavior for the first time. For a Fabri-Perot configuration made of two deltas, we obtain the effect of nonlinear coupling in the positions of the local maxima (resonances). We lay the foundations for nonlinear Anderson localization of 1D BECs in a speckle field. Upon redefinition of parameters these novel results describe the dynamics of a stationary Higgs field in 1D. Finally, we discuss the conditions for soliton formation under the influence of a Dirac comb potential, giving rise to fully correlated locations and intensities of the defects.

Ultra-low-complexity weight-sharing trigonometric nonlinear equalizer for beyond net-200-Gb/s/λ short-reach optical interconnects.

An ultra-low-complexity third-order weight-sharing trigonometric nonlinear equalizer (WS-TNLE) is proposed to eliminate nonlinear signal distortions in short-reach optical interconnects exceeding net 200 Gb/s/λ. By replacing the second- and third-order nonlinear terms in a third-order weight-sharing diagonally pruned Volterra nonlinear equalizer (WS-DP-VNLE) with cosine and sine terms, respectively, the required number of real-valued multiplications per symbol of the proposed third-order WS-TNLE is significantly reduced to the same value as the number of weight-sharing kernels. When transmitting probabilistically shaped 16-level pulse amplitude modulation (PS-PAM-16) signals at net rates ranging from 200.4 Gb/s to 300.4 Gb/s over a 1-km standard single-mode fiber (SSMF), the proposed third-order WS-TNLE requires the lowest number of real-valued multiplications per symbol, ranging from 10 to 44, which reduces the computational complexity by up to 96.2% and 52.4% compared to the third-order WS-DP-VNLE and WS-DP-absolute-term nonlinear equalizer (WS-DP-ATNLE), respectively.

Experimentally-efficient data-driven rational feedforward control for multivariable systems

Feedforward control with task flexibility for MIMO systems is essential to meet the growing demands on throughput and accuracy of high-tech systems. The aim of this paper is to develop an experimentally efficient framework for data-driven tuning of rational feedforward controllers for general non-commutative MIMO systems. In the developed approach, the nonlinear terms in the non-convex optimisation problem of iterative learning control (ILC) are iteratively circumvented via approximation. This leads to a series of convex optimisation problems that can be solved offline to obtain the parameters of the rational feedforward controller. In addition, by limiting the number of offline iterations an experimentally intensive algorithm is derived, which could be beneficial in the case of severe model mismatch. A simulation study shows that the experimentally efficient approach converges fast through offline iterations and has improved convergence properties through the use of regularisation.

Prediction of wall geometry forcold-metal-transfer-based wire-arc additive manufacturing

Wire-arc additive manufacturing (WAAM) is an advanced technique for fabricating large metal components through layer-by-layer material deposition using arc welding methods. This study focused on optimizing the WAAM process by employing machine learning models to predict and control bead geometries, specifically bead height (BH) and bead width (BW), while ensuring consistent height increments in multibead walls. Based on CMT technology in cold metal transfer experiments, linear regression models achieved high accuracy in predicting BH and BW. Analysis of variance results highlighted the considerable influence of voltage (V) and travel speed (TS) on bead geometries. For multibead wall characteristics, polynomial regression models incorporating non-linear terms, such as travel speed (TS²) and dwell time (Dt²), were developed to predict height (H) and waviness (W). Various optimization metrics were employed to balance the trade-offs between H and W for identifying optimal welding conditions that achieved the target H while minimizing W. A notable innovation of this research is the optimization of dwell time (Dt) for each layer to achieve a linear incremental H profile, minimizing W and ensuring consistent layer quality.

Unconditional error analysis of weighted implicit-explicit virtual element method for nonlinear neutral delay-reaction–diffusion equation

In this paper, we develop a virtual element method in space for the nonlinear neutral delay-reaction–diffusion equation, while a weighted implicit-explicit scheme is utilized in time. The nonlinear term is adopted by using the Newton linearized method. The calculation efficiency is improved using an implicit scheme to analyze linear terms and an explicit scheme to deal with nonlinear terms. Then, the time–space error splitting technique and G-stability are creatively combined to rigorously analyze the unconditionally optimal convergence results of the numerical scheme without any restrictions on the mesh ratio. Finally, numerical examples on a set of polygonal meshes are provided to confirm the theoretical results. In particular, when the weighted parameter is taken θ=12 (or θ=1), the method degenerates into the Crank–Nicolson (or second-order backward differential formula (BDF2)) scheme.

Characteristics of solution to singular fractional differential equation with two Riemann-Stieltjesdintegral boundary value conditions

The purpose of this paper is to study unique solution and iterative sequence of approximate solution for uniformly approaching unique solution to a new class of singular fractional differential equations with two kinds of Riemann-Stieltjes integral boundary value conditions by using some fixed point theorems. Because of different properties of the nonlinear terms and complexity of the boundary conditions in equations, we first probe several fixed point theorems of sum-type operators which expand many existing works in this research area. It is essential to point out that some conditions in our works greatly simplify the proof process of fixed point theorems. By applying the operator conclusions obtained in this paper, some sufficient conditions that guarantee the existence and uniqueness of solutions to singular differential equations are obtained, two iterative schemes that uniformly converges to the unique solution are given which provide computational methods of approximating solutions. As applications, some examples are provided to illustrate our main results.

Fast heuristics for the time‐constrained immobile server problem

AbstractWe propose easy‐to‐implement heuristics for time‐constrained applications of a problem referred to in the literature as the facility location problem with immobile servers, stochastic demand, and congestion, the service system design problem, or the immobile server problem (ISP). The problem is typically posed as one of allocating capacity to a set of M/M/1 queues to which customers with stochastic demand are assigned with the objective of minimizing a cost function composed of a fixed capacity‐acquisition cost, a variable customer‐assignment cost, and an expected‐waiting‐time cost. The expected‐waiting‐time cost results in a nonlinear term in the objective function of the standard binary programming formulation of the problem. Thus, the solution approaches proposed in the literature are either sophisticated linearization or relaxation schemes, or metaheuristics. In this study, we demonstrate that an ensemble of straightforward, greedy heuristics can rapidly find high‐quality solutions. In addition to filling a gap in the literature on ISP heuristics, new stopping criteria for an existing cutting plane algorithm are proposed and tested, and a new mixed‐integer linear model requiring no iterating algorithm is developed. In many cases, our heuristic approach finds solutions of the same or better quality than those found by exact methods implemented with expensive, state‐of‐the‐art mathematical programming software, in particular a commercial nonlinear mixed‐integer linear programming solver, given a five‐minute time limit.

Mathematical Modelling of Spatially Inhomogeneous Non-Stationary Interaction of Pests with Transgenic and Non-Modified Crops Considering Taxis

Introduction. This paper addresses a unified spatially inhomogeneous, non-stationary model of interaction between genetically modified crop resources (corn) and the corn borer pest, which is also present on a relatively small section of non-modified corn. The model assumes that insect pests influence both types of crops and are capable of independent movement (taxis) towards the gradient of plant resources. It also considers diffusion processes in the dynamics of all components of the unified model, biomass growth, genetic characteristics of both types of plant resources, processes of crop consumption, phenomena of growth and degradation, diffusion, and mutation of pests. The model allows for predictive calculations aimed at reducing crop losses and increasing the resistance of transgenic crops to pests by slowing down the natural mutation rate of the pest. Materials and Methods. The mathematical model is an extension of Kostitsin’s model and is formulated as an initial-boundary value problem for a nonlinear system of convection-diffusion equations. These equations describe the spatiotemporal dynamics of biomass density changes in two types of crops — transgenic and non-modified — as well as the specific populations (densities) of three genotypes of pests (the corn borer) resulting from mutations. The authors linearized the convection-diffusion equations by applying a time-lag method on the time grid, with nonlinear terms from eachequation taken from the previous time layer. The terms describing taxis are presented in a symmetric form, ensuring the skew-symmetry of the corresponding continuous operator and, in the case of spatial grid approximation, the finite-difference operator. Results. A stable monotonic finite-difference scheme is developed, approximating the original problem with second-order accuracy on a uniform 2D spatial grid. Numerical solutions of model problems are provided, qualitatively corresponding to observed processes. Solutions are obtained for various ratios of modified and non-modified sections of the field. Discussion and Conclusion. The obtained results regarding pest behavior, depending on the type of taxis, could significantly extend the time for pests to acquire Bt resistance. The concentration dynamics of pests moving in the direction of the food gradient differs markedly from the concentration of pests moving towards a mate for reproduction.

Modeling of the Black Sea circulation using equations of heat and salt advection–diffusion having discrete nonlinear invariants

In this work the accuracy of reconstructing the Black Sea circulation by using new approximations of nonlinear terms in the transport equations, ensuring the conservation of temperature and salinity to a power greater than two, is analyzed based on the results of forecast calculations. Three numerical experiments with differences in the schemes for calculating temperature and salinity are carried out. In the first experiment – traditional schemes are used to conserve of temperature and salinity in the first and second degrees; in the second one – the temperature is conserved in the first and fifth degrees, salinity in the first and third; in the third one – the temperature in the first and third, salinity in the first and fifth degrees. Calculations are performed on the basis of the MHI model with a resolution of 1.6 km and taking into account realistic atmospheric forcing for 2016. Validation of the results is carried out based on comparison of model fields with in-situ and satellite measurements of temperature and salinity in 2016. Analysis of mean and root mean square errors showed that new schemes for the advection-diffusion equations of heat and salt, ensuring the conservation of predictive parameters to a power greater than two, improve the accuracy of reconstructing the salinity in the Black Sea upper 100m layer throughout the year compared with traditional approximation. The root mean square errors in the salinity field are reduced by 15–20%, the thickness of the upper mixed layer in winter and the depth of the upper boundary of the thermocline layer in summer in the central part of the sea are modeled approximately 10% more accurately. Based on the results of three experiments, the smallest deviations from observational data are obtained when using approximations that ensure the conservation of temperature to the third power and salinity to the fifth power.

Unconditionally superconvergence error analysis of an energy-stable finite element method for Schrödinger equation with cubic nonlinearity

In this paper, the unconditionally superconvergence analysis is studied for the cubic Schrödinger equation with an energy-stable finite element method. A different approach is proposed to obtain the unconditionally superclose error estimate in H1-norm firstly without using the time splitting technique required in the previous literature. The key to the analysis is to use a priori boundedness of the numerical solution in energy norm and control the nonlinear terms rigorously by two cases, i.e., τ≤h2 and τ≥h2, where τ denotes the temporal size and h is the spatial size. Subsequently, the global superconvergence error estimate in H1-norm is derived by an effective interpolation post-processing approach. Finally, some numerical experiments are carried out to confirm the theoretical findings.