Second-order Nonlinear Equations Research Articles

About tautochronic movements

It is shown that a material point, under the influence of an attractive linear force and a repulsive force inversely proportional to the cube of the distance from the center of attraction, performs a periodic motion, the period of which does not depend on the initial data (tautochronic motion). The problem is reduced to a nonlinear autonomous second-order equation, the general solution of which is expressed in terms of elementary functions. It has also been proven that for other power laws of repulsive force, except for degrees 0, 1 and –3, the movement of a material point is not tautochronous.

Physical Reservoir Computing Using Tellurium-Based Gate-Tunable Artificial Photonic Synapses.

We report tellurium (Te) thin-film-based artificial photonic synapses and their application to physical reservoir computing (PRC). The Te-based artificial photonic synapses were fabricated by using sputtered Te thin films and spray-coated MXene (Ti3C2) electrodes. A thorough investigation of the field-dependent persistent photoconductivity (PPC) of the Te channel revealed that the relaxation speed of the transient photocurrent depended on the gate bias. Utilizing the PPC property, the Te device served as an excellent photonic synapse under light pulse stimulus, exhibiting multiple synaptic characteristics such as excitatory postsynaptic current and paired-pulse facilitation, as well as highly linear potentiation-depression characteristics; a simulation-based study further confirmed the effectiveness of the device. Most importantly, by exploiting the nonlinear and fading memory characteristics of the Te photonic synapse, we demonstrate two advanced examples of PRC. In classifying handwritten digits, our system carried out successful digit recognition without binarization or another simplification process with reduced computational cost compared to conventional systems. To solve second-order nonlinear equations, we introduce the strategy of utilizing historical nodes. The combination of historical nodes and the gate-tunable responses of the photonic synapses, which provide an enriched reservoir state, yielded excellent prediction accuracy. Overall, this work will offer an understanding of Te-based optoelectronic devices and their synergetic integration with neuromorphic devices and PRC.

A Near-Infrared Retinomorphic Device with High Dimensionality Reservoir Expression.

Physical reservoir-based reservoir computing (RC) systems for intelligent perception have recently gained attention because they require fewer computing resources. However, the system remains limited in infrared (IR) machine vision, including materials and physical reservoir expression power. Inspired by biological visual perception systems, the study proposes a near-infrared (NIR) retinomorphic device that simultaneously perceives and encodes narrow IR spectral information (at ≈980nm). The proposed device, featuring core-shell upconversion nanoparticle/poly (3-hexylthiophene) (P3HT) nanocomposite channels, enables the absorption and conversion of NIR into high-energy photons to excite more photo carriers in P3HT. The photon-electron-coupled dynamics under the synergy of photovoltaic and photogating effects influence the nonlinearity and high dimensionality of the RC system under narrow-band NIR irradiation. The device also exhibits multilevel data storage capability (≥8 levels), excellent stability (≥2000 s), and durability (≥100 cycles). The system accurately identifies NIR static and dynamic handwritten digit images, achieving recognition accuracies of 91.13% and 90.07%, respectively. Thus, the device tackles intricate computations like solving second-order nonlinear dynamic equations with minimal errors (normalized mean squared errorof 1.06 × 10⁻3 during prediction).

Solutions of Second-Order Nonlinear Implicit ψ-Conformable Fractional Integro-Differential Equations with Nonlocal Fractional Integral Boundary Conditions in Banach Algebra

In this paper, we introduce and thoroughly examine new generalized ψ-conformable fractional integral and derivative operators associated with the auxiliary function ψ(t). We rigorously analyze and confirm the essential properties of these operators, including their semigroup behavior, linearity, boundedness, and specific symmetry characteristics, particularly their invariance under time reversal. These operators not only encompass the well-established Riemann–Liouville and Hadamard operators but also extend their applicability. Our primary focus is on addressing complex fractional boundary value problems, specifically second-order nonlinear implicit ψ-conformable fractional integro-differential equations with nonlocal fractional integral boundary conditions within Banach algebra. We assess the effectiveness of these operators in solving such problems and investigate the existence, uniqueness, and Ulam–Hyers stability of their solutions. A numerical example is presented to demonstrate the theoretical advancements and practical implications of our approach. Through this work, we aim to contribute to the development of fractional calculus methodologies and their applications.

Dynamics of chains of a large number of fully coupled oscillators, as well as oscillators with one-way and two-way delay couplings

This articles considers chains consisting of identical nonlinear equations of second order with couplings in linear elements. It is assumed that there is a large delay in the coupling elements. Moreover, we assume that the chain possesses a large number of elements. Therefore, instead of the original system with many elements, we can study a second order nonlinear integro-differential equation with periodic boundary conditions. The main study is focused on the local dynamics of chains with one-sided, two-sided couplings, as well as fully coupled chains. Also, we study the stability of equilibrium states and identify the critical cases. The condition of a sufficiently large delay helps to determine the parameters for the implementation of critical cases explicitly. Our methodology is based on the infinite-dimensional normalisation method proposed by the author, namely the method of quasinormal forms. We construct quasinormal forms for the chains under consideration that determine the asymptotics of the leading terms of the asymptotic expansions of the solutions to the original system.

Linear and Nonlinear Approach on Kinetic Studies in the Adsorption of Ciprofloxacin onto Banana Peels-base Carbon from Aqueous Solution

The adsorption process investigates the adsorption kinetic of ciprofloxacin onto banana peels-based activated carbon (BPAC) with a batch process. The pyrolysis method prepared the banana peel-based activated carbon (BPAC). XRD investigated the crystallinity of the BPAC. The effect of shaking time, pH, concentration of BPAC, concentration of ciprofloxacin in aqueous solution, and time duration. The maximum removal percentage of ciprofloxacin from its aqueous solution was 98.7 %. In the kinetic study, the pseudo-first and second-order linear and non-linear equations were fitted with experimental data using the origin LAB program. The pseudo-first-order linear equation R2 values have an R2 of 0.9103. In contrast, the R2 values of the linear equations about pseudo-second-order of types I, II, III, and IV are 0.9982, 0.9968, 0.9986, and 0.9886, respectively. Considering the R2 values, linear pseudo-second-order types III & IV are preferable for ciprofloxacin elimination from its aqueous solution.

On the Existence of a Positive Solution of a Boundary Value Problem for a Nonlinear Second-Order Functional-Differential Equation with Integral Boundary Conditions

On the Existence of a Positive Solution of a Boundary Value Problem for a Nonlinear Second-Order Functional-Differential Equation with Integral Boundary Conditions

Fast physical reservoir computing, achieved with nonlinear interfered spin waves

Reservoir computing is a promising approach to implementing high-performance artificial intelligence that can process input data at lower computational costs than conventional artificial neural networks. Although reservoir computing enables real-time processing of input time-series data on artificial intelligence mounted on terminal devices, few physical devices are capable of high-speed operation for real-time processing. In this study, we introduce spin wave interference with a stepped input method to reduce the operating time of the physical reservoir, and second-order nonlinear equation task and second-order nonlinear autoregressive mean averaging, which are well-known benchmark tasks, were carried out to evaluate the operating speed and prediction accuracy of said physical reservoir. The demonstrated reservoir device operates at the shortest operating time of 13 ms/5000-time steps, compared to other compact reservoir devices, even though its performance is higher than or comparable to such physical reservoirs. This study is a stepping stone toward realizing an artificial intelligence device capable of real-time processing on terminal devices.

Schauder and Calderón–Zygmund type estimates for fully nonlinear parabolic equations under “small ellipticity aperture” and applications

In this manuscript, we derive some Schauder estimates for viscosity solutions to non-convex fully nonlinear second-order parabolic equations of the form: ∂tu−F(x,t,D2u)=f(x,t)inQ1=B1×(−1,0],provided that the source f and the coefficients of F are Hólder continuous functions, and F enjoys a small ellipticity aperture. Furthermore, for problems with merely bounded data, we prove that such solutions are C1,Log-Lip-regular. We also obtain Calderón-Zygmund estimates for such a class of non-convex operators. Finally, we connect our results and recent estimates for fully nonlinear models in certain solution classes.

Opto-magnonic reservoir computing coupling nonlinear interfered spin wave and visible light switching

Physical reservoir computing is a promising approach to realize high-performance artificial intelligence systems utilizing physical devices. Recently, it has been experimentally found that nonlinear interfered spin wave multi-detection shows excellent performance for processing nonlinear time-series data due to its outstanding features: nonlinearity, short-term memory, and the ability to map in high dimensional space. However, said performance is considerably inferior to reservoir computing utilizing an optical circuit with a large volume. Herein, we develop reservoir computing with nonlinear interfered spin wave coupled with light switching, namely opto-magnonic reservoir computing. The spin wave was modulated through a crystal field transition that occurred in two different Fe3+ sites of Y3Fe5O12 by visible light switching, and it was found that the spin wave modulated by visible light switching dramatically reduced normalized mean square errors to 4.96 × 10−3, 0.163, and 3.66 × 10−5 for NARMA2, NARMA10, and second-order nonlinear dynamical equation tasks. Said excellent performance results from the strong nonlinearity caused by chaos and large memory capacity induced by reservoir states diversified by visible light switching.

Lagrangian, game theoretic, and PDE methods for averaging G-equations in turbulent combustion: existence and beyond

G-equations are popular level set Hamilton–Jacobi nonlinear partial differential equations (PDEs) of first or second order arising in turbulent combustion. Characterizing the effective burning velocity (also known as the turbulent burning velocity) is a fundamental problem there. We review relevant studies of the G-equation models with a focus on both the existence of effective burning velocity (homogenization), and its dependence on physical and geometric parameters (flow intensity and curvature effect) through representative examples. The corresponding physical background is also presented to provide motivations for mathematical problems of interest. The lack of coercivity of Hamiltonian is a hallmark of G-equations. When either the curvature of the level set or the strain effect of fluid flows is accounted for, the Hamiltonian becomes highly nonconvex and nonlinear. In the absence of coercivity and convexity, the PDE (Eulerian) approach suffers from insufficient compactness to establish averaging (homogenization). We review and illustrate a suite of Lagrangian tools, most notably min-max (max-min) game representations of curvature and strain G-equations, working in tandem with analysis of streamline structures of fluid flows and PDEs. We discuss open problems for future development in this emerging area of dynamic game analysis for averaging noncoercive, nonconvex, and nonlinear PDEs such as geometric (curvature-dependent) PDEs with advection.

The Presence of Chaos in a Viscoelastic Harmonically Forced Von Mises Truss

Abstract This work investigates how viscoelasticity affects the dynamic behavior of a lumped-parameter model of a bistable von Mises truss. The system is controlled by a linear first-order equation and a second-order nonlinear Duffing equation with a quadratic nonlinearity that governs mechanical behavior. The second-order equation controls mechanical oscillations, while the linear first-order equation controls viscoelastic force evolution. Combined, the two equations form a third-order jerk equation that controls system dynamics. Viscoelasticity adds time scales and degrees-of-freedom to material behavior, distinguishing it from viscosity-only systems. Due to harmonic excitation, the system exhibits varied dynamic responses, from periodic to quasi-periodic to chaotic. We explore the dynamics of a harmonically forced von Mises truss with viscous damping to address this purpose. We demonstrate this system's rich dynamic behavior due to driving amplitude changes. This helps explain viscoelastic system behavior. A viscoelastic unit replaces the viscous damper, and we show that, although viscous damping merely changes how fast the trajectory decays to an attractor, viscoelasticity modifies both the energy landscape and the rate of decay. In a conventional linear solid model, three viscoelastic parameters control the system's behavior instead of one, as in pure viscous damping. This adds degrees-of-freedom that affect system dynamics. We present the parameter space for chaotic behavior and the shift from regular to irregular motion. Finally, Melnikov's criteria identify the regular-chaotic threshold. The system's viscous and elastic components affect the chaotic threshold amplitude

Effects of electron-to-ion mass ratio in driving magnetic oscillations of magnetohydrodynamic plasmas and self-organized criticality

Evolutions of nonlinear magnetic fields have been shown to be governed by a set of coupled nonlinear equations of second order in magnetohydrodynamic (MHD) plasmas by Lee and Parks [Geophys. Res. Lett. 19, 637–640 (1992)]. We have considered the same set of coupled nonlinear equations for further analysis in this work by neglecting the presence of external forcing term in it. Different modes of oscillations of magnetic field have been found to exist in special limiting cases of this set of undriven second order coupled nonlinear equations having frequencies that are multiples of lower hybrid frequency. Numerical solutions of these coupled equations have been analysed revealing a quasi-periodic route to chaotic oscillations of the nonlinear magnetic fields as electron-to-ion mass ratio signifying presence of linear coupling effects is increased. Some signatures of the phenomenon of self-organized criticality (SOC) in typical quasi-periodic oscillations of magnetic field have also been noticed using Fourier analysis. The presence of long range correlations has been witnessed in quasi-periodic oscillations whereas both long range correlations and anticorrelations are found in chaotic oscillations using rescaled range analysis. Concluding remarks are provided in addition to various results and discussions.

Magnetization Vector Rotation Reservoir Computing Operated by Redox Mechanism.

Physical reservoir computing is a promising way to develop efficient artificial intelligence using physical devices exhibiting nonlinear dynamics. Although magnetic materials have advantages in miniaturization, the need for a magnetic field and large electric current results in high electric power consumption and a complex device structure. To resolve these issues, we propose a redox-based physical reservoir utilizing the planar Hall effect and anisotropic magnetoresistance, which are phenomena described by different nonlinear functions of the magnetization vector that do not need a magnetic field to be applied. The expressive power of this reservoir based on a compact all-solid-state redox transistor is higher than the previous physical reservoir. The normalized mean square error of the reservoir on a second-order nonlinear equation task was 1.69 × 10-3, which is lower than that of a memristor array (3.13 × 10-3) even though the number of reservoir nodes was fewer than half that of the memristor array.

Challenges in Numerical Solutions of Higher-Dimensional Differential Equations

Differential equations constitute a fundamental tool in modeling various natural phenomena across scientific disciplines such as physics, engineering, and finance. We provide an overview of fractional differential equations, focusing on the computational requirements associated with their numerical solutions from a computer science perspective. We analyze the computational intricacies concerning First-Order Linear ODE, First-Order Nonlinear ODE, Second-Order Linear ODE, Second-Order Nonlinear ODE, Heat Equation (PDE), and Wave Equation (PDE). This comparative assessment delves into the computational demands of solving these equations using differential equation methodologies. While analytical solutions provide deep insights, obtaining numerical solutions, particularly in higher dimensions, remains a persistent challenge. Finite difference methods commonly employed for numerical solutions, In higher-dimensional problems, traditional numerical methods face challenges stemming from an exponential surge in grid points and the consequent demand for substantially decreased time step sizes. This paper explores the challenges posed by higher-dimensional differential equations in numerical solutions. It highlights the infeasibility of finite difference methods in such scenarios and emphasizes the need for innovative numerical techniques capable of efficiently handling the complexities of higher-dimensional differential equations. Overcoming these challenges is crucial for advancing our understanding and modeling capabilities in complex real-world systems governed by differential equations. Continued research efforts strive to develop novel numerical methodologies capable of addressing these challenges, aiming to broaden the scope of solvable higher-dimensional differential equations and expand their application across diverse scientific domains.

Eigenvalue intervals for iterative systems of second-order nonlinear equations with impulses and m-point boundary conditions

Eigenvalue intervals for iterative systems of second-order nonlinear equations with impulses and m-point boundary conditions

Inverted input method for computing performance enhancement of the ion-gating reservoir

Physical reservoir computing (PRC) is useful for edge computing, although the challenge is to improve computational performance. In this study, we developed an inverted input method, the inverted input is additionally applied to a physical reservoir together with the original input, to improve the performance of the ion-gating reservoir. The error in the second-order nonlinear equation task was 7.3 × 10−5, the lowest error in reported PRC to date. Improvement of high dimensionality by the method was confirmed to be the origin of the performance enhancement. This inverted input method is versatile enough to enhance the performance of any other PRC.

Multidimensional Diffusion-Wave-Type Solutions to the Second-Order Evolutionary Equation

The paper concerns a nonlinear second-order parabolic evolution equation, one of the well-known objects of mathematical physics, which describes the processes of high-temperature thermal conductivity, nonlinear diffusion, filtration of liquid in a porous medium and some other processes in continuum mechanics. A particular case of it is the well-known porous medium equation. Unlike previous studies, we consider the case of several spatial variables. We construct and study solutions that describe disturbances propagating over a zero background with a finite speed, usually called ‘diffusion-wave-type solutions’. Such effects are atypical for parabolic equations and appear since the equation degenerates on manifolds where the desired function vanishes. The paper pays special attention to exact solutions of the required type, which can be expressed as either explicit or implicit formulas, as well as a reduction of the partial differential equation to an ordinary differential equation that cannot be integrated in quadratures. In this connection, Cauchy problems for second-order ordinary differential equations arise, inheriting the singularities of the original formulation. We prove the existence of continuously differentiable solutions for them. A new example, an analog of the classic example by S.V. Kovalevskaya for the considered case, is constructed. We also proved a new existence and uniqueness theorem of heat-wave-type solutions in the class of piece-wise analytic functions, generalizing previous ones. During the proof, we transit to the hodograph plane, which allows us to overcome the analytical difficulties.

Asymptotic behavior of solutions of a second-order nonlinear discrete equation of Emden-Fowler type

Abstract The article investigates a second-order nonlinear difference equation of Emden-Fowler type Δ 2 u ( k ) ± k α u m ( k ) = 0 , {\Delta }^{2}u\left(k)\pm {k}^{\alpha }{u}^{m}\left(k)=0, where k k is the independent variable with values k = k 0 , k 0 + 1 , … k={k}_{0},{k}_{0}+1,\ldots \hspace{0.33em} , u : { k 0 , k 0 + 1 , … } → R u:\left\{{k}_{0},{k}_{0}+1,\ldots \hspace{0.33em}\right\}\to {\mathbb{R}} is the dependent variable, k 0 {k}_{0} is a fixed integer, and Δ 2 u ( k ) {\Delta }^{2}u\left(k) is its second-order forward difference. New conditions with respect to parameters α ∈ R \alpha \in {\mathbb{R}} and m ∈ R m\in {\mathbb{R}} , m ≠ 1 m\ne 1 , are found such that the equation admits a solution asymptotically represented by a power function that is asymptotically equivalent to the exact solution of the nonlinear second-order differential Emden-Fowler equation y ″ ( x ) ± x α y m ( x ) = 0 . {y}^{^{\prime\prime} }\left(x)\pm {x}^{\alpha }{y}^{m}\left(x)=0. Two-term asymptotic representations are given not only for the solution itself but also for its first- and second-order forward differences as well. Previously known results are discussed, and illustrative examples are considered.

Theoretical understanding of reaction and kinetics in the reduction of the bromate anion to bromine on a rotating disk electrode

The reduction of the bromate anion to bromine on a rotating disk electrode (RDE) at steady-state conditions is discussed. This model is based on a system of nonlinear equations. The most crucial aspect of this investigation is finding the analytical expression of the concentration of reagents by solving the coupled second-order nonlinear equation using hyperbolic function methods. The effects of the kinetic parameters on concentration and current are explored in the unitizing graph and tables. The analytical results are validated in numerical results using MATLAB software. The effects of the parameters such as diffusion layer thickness, rotation rate, rate constants and the ratio of concentration of bromate anion and volume of concentration of bromine ions on current are discussed.