Oscillation Of Differential Equations Research Articles

Numerical integrator for highly oscillatory differential equations based on the Neumann series

AbstractWe propose a third-order numerical integrator based on the Neumann series and the Filon quadrature, designed mainly for highly oscillatory partial differential equations. The method can be applied to equations that exhibit small or moderate oscillations; however, counter-intuitively, large oscillations increase the accuracy of the scheme. With the proposed approach, the convergence order of the method can be easily improved. Error analysis of the method is also performed. We consider linear evolution equations involving first- and second-time derivatives that feature elliptic differential operators, such as the heat equation or the wave equation. Numerical experiments consider the case in which the space dimension is greater than one and confirm the theoretical study.

An Efficient New Scheme for Direct Simulation of Higher Order Third and Fourth Oscillatory Differential Equations

In this research, we have examined the general block approach for solving higher-order oscillatory differential equations using the linear block approach (LBA). The basic properties of the new method, such as order, error constant, zero-stability, consistency, convergence, linear stability, and region of absolute stability, were also analyzed and satisfied. Some distinct fourth-order oscillatory problems were directly applied to the new method in order to overcome the setbacks of the reduction method. The results obtained were compared with those in the literature, and the new method takes away the burden of solving fourth-order oscillatory differential equations. Therefore, from the results, the new method has shown better accuracy and faster convergence graphically. One of the advantages of the new method is that it does not require much computational burden and is also self-starting. Convergence, Consistency, Direct simulation, Linear stability, Oscillatory differential equations

Simulating the Dynamics of Oscillating Differential Equations of Mass in Motion

This research explores the practical implementation and simulation of oscillatory differential equations concerning objects in motion. The methodology incorporates power series polynomials, ensuring adherence to the fundamental properties of these functions. The novel approach is applied to various oscillatory differential equations, encompassing harmonic motion, spring motion, dynamic mass motion, Betiss and Stiefel equations, and nonlinear differential equations. The results demonstrate computational reliability, showcasing enhanced accuracy and quicker convergence compared to currently examined methods. Dynamic motion, Genesio, Harmonic motion, Mass, Spring of motion

On a Duffing-type oscillator differential equation on the transition to chaos with fractional q-derivatives

In this paper, by applying fractional quantum calculus, we study a nonlinear Duffing-type equation with three sequential fractional q-derivatives. We prove the existence and uniqueness results by using standard fixed-point theorems (Banach and Schaefer fixed-point theorems). We also discuss the Ulam–Hyers and the Ulam–Hyers–Rassias stabilities of the mentioned Duffing problem. Finally, we present an illustrative example and nice application; a Duffing-type oscillator equation with regard to our outcomes.

On the Numerical Approximation of Higher Order Differential Equation

This research examines the general K - step block approach for solving higher order oscillatory differential equations using Linear Block Approach (LBA). The basic properties of the new method such as order, error constant, zero-stability, consistency, convergence, linear stability and region of absolute stability were also analyzed and satisfied. Some distinct fourth order oscillatory differential equation were directly applied on the new method in order to overcome the setbacks in reduction method, where the step size varies. The results obtained were compared with those in literature and the new method takes away the burden of solving fourth order oscillatory differential equations. The accuracy of the new method proved to be better as it outperformed those of existing methods. Therefore, from the results, the new method has shown better accuracy and faster convergence graphically. One of the advantage of the new method is that it does not require much computational burden and it is also self-starting.

Analytical technique for solving strongly nonlinear oscillator differential equations

The main motivation of this paper is to apply general formulas of the gamma function to find the analytical solutions of some types of nonlinear differential equations with recognizable simplicity and less calculation compared to other existing analytical methods in the literature. The simplicity and accuracy of the Gamma function method (GFM) in solving strongly nonlinear differential equations can be displayed through three practical examples arising in many different fields of engineering and science. The results obtained analytically are compared with the well-known analytical methods already published in the literature, such as Akbari-Ganji's method (AGM), homotopy perturbation method (HPM), and exact numerical solutions to confirm the efficiency of GFM. Excellent agreement is found between the analytical solutions and the exact numerical ones. Accordingly, the GFM is more reliable for exploring approximate solutions for strongly nonlinear oscillators. Moreover, the GFM can be easily extended to other nonlinear problems and is, therefore, very applicable in engineering and applied sciences.

Supervised machine learning for jamming transition in traffic flow with fluctuations in acceleration and braking

Jamming transition in traffic flow refers to the sudden transition from a free-flowing state to a jammed state as the traffic density increases. This transition is of great interest to traffic engineers and physicists, as it can have significant implications for traffic safety, efficiency, traffic management, and urban planning. Homogeneous car following models is a popular framework used to study the jamming transition phenomenon. The mathematical structure of the problem is governed by the classical Lorenz system to consider the fluctuational effects. The analytical solution of such nonlinear oscillatory differential equations does not exist. Therefore, this study aims to utilize the machine learning approach with the optimization technique that could be used to fine-tune the weights/ parameters of a neural network model to predict the accurate and reliable solutions for the jamming transition in traffic flow. The headway deviations have been studied by considering the multiple scenarios based on the acceleration and braking of the vehicle.

On the Simulations of Second-Order Oscillatory Problems with Applications to Physical Systems

Second-order oscillatory problems have been found to be applicable in studying various phenomena in science and engineering; this is because these problems have the capabilities of replicating different aspects of the real world. In this research, a new hybrid method shall be formulated for the simulations of second-order oscillatory problems with applications to physical systems. The proposed method shall be formulated using the procedure of interpolation and collocation by adopting power series as basis function. In formulating the method, off-step points were introduced within the interval of integration in order to bypass the Dahlquist barrier, improve the accuracy of the method and also upgrade the order of consistence of the method. The paper further validated the some properties of the hybrid method derived and from the results obtained; the new method was found to be consistent, convergent and stable. The simulation results generated as a result of the application of the new method on some second-order oscillatory differential equations also showed that the new hybrid method is computationally reliable.

Analyses of solutions of Riemann‐Liouville fractional oscillatory differential equations with pure delay

This paper first finishes off the exact solutions of Riemann‐Liouville fractional differential time‐delay oscillatory system of order by using two newly defined delayed perturbations of Mittag‐Leffler matrix functions and constant variation method. In the light of the exact solutions, we explore the finite time stability of the nonhomogeneous fractional oscillatory differential equations with pure delay. Ultimately, an example is cited to verify the rationality of the results. Through our method, the public problems left by Mahmudov in 2022 were partially solved.

Synthesis of EEG signals modeled using non-linear oscillator based on speech data with EKF

In the existing literature, one of the major issue in regression technique is the noisy training data. In this research work, the mathematical model for parameter estimation of neurological signals is developed for investigation of Brain dynamics. This proposed model is developed by using non-linear dynamics of oscillator models. It is assumed that state of the brain of a given person is characterized by an unknown parameter vector ‘θ’ that parameterizes a coupled nonlinear oscillator differential equation model which generates a harmonic process serving as a model for the EEG data collected on the brain surface. It is assumed that the EEG data and speech data of the same person are correlated, so that the speech data obeys another differential equation with a parameter ‘ϕ’ that is a fixed function of the EEG a parameter ‘θ’. ‘θ’ and ‘ϕ’ are estimated by an EKF based on training process based on joint the EEG and speech data. This enables us to get accurate estimates of ‘θ’ and ‘ϕ’ for each person. This training process is repeated for ‘N’ persons and accordingly a parameter vector is obtained for each person. This completes the training process. The validation of this model involves choosing a fresh person not belonging to the training set and estimating the EEG parameter vector for this person based only on his speech data. For this, a function ‘ψ’ which correlates the speech parameter ‘ϕ’ with the EEG parameter ‘θ’ as ‘ϕ=ψ(θ̂)’ is estimated by an affine linear relationship which is substituted into the speech model. The EEG model is then used to generate this person’s EEG data and we compare this with the true EEG data of his brain. This validation scheme can also be used to classify the fresh person’s brain by comparing his parameters with those obtained from the training set.

Exponentially fitted two-derivative DIRK methods for oscillatory differential equations

In this work, we construct and derive a new class of exponentially fitted two-derivative diagonally implicit Runge–Kutta (EFTDDIRK) methods for the numerical solution of differential equations with oscillatory solutions. First, a general format of so-called modified two-derivative diagonally implicit Runge–Kutta methods (TDDIRK) is proposed. Their order conditions up to order six are derived by introducing a set of bi-colored rooted trees and deriving new elementary weights. Next, we build exponential fitting conditions in order for these modified TDDIRK methods to treat oscillatory solutions, leading to EFTDDIRK methods. In particular, a family of 2-stage fourth-order, a fifth-order, and a 3-stage sixth-order EFTDDIRK schemes are derived. These can be considered as superconvergent methods. The stability and phase-lag analysis of the new methods are also investigated, leading to optimized fourth-order schemes, which turn out to be much more accurate and efficient than their non-optimized versions. Finally, we carry out numerical experiments on some oscillatory test problems. Our numerical results clearly demonstrate the accuracy and efficiency of the newly derived methods when compared with existing trigonometically/exponentially fitted implicit Runge–Kutta methods and two-derivative Runge–Kutta methods of the same order in the literature.

Uniformly accurate nested Picard integrators for a system of oscillatory ordinary differential equations

A uniformly second order accurate nested Picard iterative integrator (NPI) is proposed to solve a system of oscillatory ordinary equations involving a parameter $$0<\varepsilon \le 1$$ . The solution of this system propagates wave with $$O(\varepsilon ^2)$$ wavelength and $$O(\varepsilon ^4)/O(\varepsilon ^2)$$ amplitude for well-prepared/ill-prepared initial data. Based on the NPI and the exponential integrator, a uniformly accurate (w.r.t $$\varepsilon $$ ) second order numerical scheme with $$O(\tau ^2)$$ error has been developed. The method can be generalized to higher order. Numerical tests are presented to confirm our error estimates.

Application of the Homotopy Perturbation method to solve nonlinear oscillatory differential equations

Application of the Homotopy Perturbation method to solve nonlinear oscillatory differential equations

Study on oscillatory ordinary differential equation based on linear multistep algorithm

Study on oscillatory ordinary differential equation based on linear multistep algorithm

Kalman-based Spectro-Temporal ECG Analysis using Deep Convolutional Networks for Atrial Fibrillation Detection

In this article, we propose a novel ECG classification framework for atrial fibrillation (AF) detection using spectro-temporal representation (i.e., time varying spectrum) and deep convolutional networks. In the first step we use a Bayesian spectro-temporal representation based on the estimation of time-varying coefficients of Fourier series using Kalman filter and smoother. Next, we derive an alternative model based on a stochastic oscillator differential equation to accelerate the estimation of the spectro-temporal representation in lengthy signals. Finally, after comparative evaluations of different convolutional architectures, we propose an efficient deep convolutional neural network to classify the 2D spectro-temporal ECG data. The ECG spectro-temporal data are classified into four different classes: AF, non-AF normal rhythm (Normal), non-AF abnormal rhythm (Other), and noisy segments (Noisy). The performance of the proposed methods is evaluated and scored with the PhysioNet/Computing in Cardiology (CinC) 2017 dataset. The experimental results show that the proposed method achieves the overall F1 score of 80.2%, which is in line with the state-of-the-art algorithms.

Oscillation-preserving algorithms for efficiently solving highly oscillatory second-order ODEs

In the last few decades, Runge-Kutta-Nystrom (RKN) methods have made significant progress and the study of RKN-type methods for solving highly oscillatory differential equations has received a great deal of attention. In this paper, from the point of view of geometric integration, the oscillation-preserving behaviour of the existing RKN-type methods in the literature are analysed in detail. To this end, it is convenient to introduce the concept of oscillation preservation for RKN-type methods. It turns out that if both the internal stages and the updates of an RKN-type method respect the qualitative and global features of the highly oscillatory solution, then the method is oscillation preserving. Since the internal stages of standard RKN and adapted RKN (ARKN) methods are inimical to the oscillation-preserving structure, neither ARKN methods nor the symplectic and symmetric RKN methods, and standard RKN methods are oscillation preserving. Other concerns relating to oscillation preservation are also considered. In particular, we are concerned with the computational issues for efficiently solving semi-discrete wave equations such as semi-discrete Klein-Gordon (KG) equations and damped sine-Gordon equations. The results of numerical experiments show the importance of the oscillation-preserving property for an RKN-type method and the remarkable superiority of oscillation-preserving integrators when applied to nonlinear multi-frequency highly oscillatory systems.

Harmonic-Enriched Reproducing Kernel Approximation for Highly Oscillatory Differential Equations

AbstractThe harmonic-enriched reproducing kernel (HRK) approximation together with collocation method is introduced to circumvent the discretization restriction for highly oscillatory partial diffe...

Transverse Vibration of Functionally Graded Tapered Double Nanobeams Resting on Elastic Foundation

The natural vibration behavior of axially functionally graded (AFG) double nanobeams is studied based on the Euler–Bernoulli beam and Eringen’s non-local elasticity theory. The double nanobeams are continuously connected by a layer of linear springs. The oscillatory differential equation of motion is established using the Hamilton’s principle and the constitutive relations. The Chebyshev spectral collocation method (CSCM) is used to transform the coupled governing differential equations of motion into algebraic equations. The discretized boundary conditions are used to modify the Chebyshev differentiation matrices, and the system of equations is put in the matrix-vector form. Then, the dimensionless transverse frequencies and the mode shapes are obtained by solving the standard eigenvalue problem. The effects of the coupling springs, Winkler stiffness, the shear stiffness parameter, the breadth and taper ratios, the small-scale parameter, and the boundary conditions on the natural transverse frequencies are carried out. Several numerical examples were conducted, and the authors believe that the results may be interesting in designing and analyzing double and multiple one-dimensional nano structures.

Oscillation of delay difference equations with finite non-monotone arguments

In this paper, the oscillation of difference equations with multiple non-monotone delay arguments is studied. Three criteria of these equations are obtained for oscillation. And examples are given to show the meanings of the theorems.

Oscillations in difference equations with continuous time caused by several deviating arguments

Sufficient conditions for the oscillation of all solutions of difference equations with continuous time caused by several deviating arguments are presented. Examples are provided to illustrate the results and compare them to relevant results in the literature.