Time-dependent Ordinary Differential Equations Research Articles

SMS: Spiking marching scheme for efficient long time integration of differential equations

We propose a Spiking Neural Network (SNN)-based explicit numerical scheme for long time integration of time-dependent Ordinary and Partial Differential Equations (ODEs, PDEs). The core element of the method is a SNN, trained to use spike-encoded information about the solution at previous timesteps to predict spike-encoded information at the next timestep. After the network has been trained, it operates as an explicit numerical scheme that can be used to compute the solution at future timesteps, given a spike-encoded initial condition. A decoder is used to transform the evolved spiking-encoded solution back to function values. We present results from numerical experiments of using the proposed method for ODEs and PDEs of varying complexity.

Weakly non-linear stability analysis of g-jitter induced Rayleigh–Bénard convection under inclined surfaces

This study investigated the effect of g-jitter on the Rayleigh–Bénard convection between inclined surfaces whose lower layer is concentrated with a solute C. The problem is formulated using the Ginzburg–Landau (GL) equation, which is time-dependent non-autonomous ordinary differential equation and derived from a weakly non-linear theory and Fredholm solvability condition. The graphical solution of the GL equation is obtained with the help of MATHEMATICA by using the Runge–Kutta fourth-order method. The impact of various flow parameters on the convective rate of heat and mass transport has been discussed. The heat and mass transport is found to be advanced by the non-dimensional parameters viz. modulations’ amplitude (δ), Lewis number (Le), and solute Rayleigh number (RC), but it is suppressed by the angle of inclination ϕ.

Effect of porosity on the nonlinear free vibrational behavior of two-directional functionally graded porous cone-shaped shells resting on elastic substrates

The present article describes an analytical approach to investigate the nonlinear free vibrational behavior of a two-directional functionally graded porous (TDFGP) cone-shaped shell resting on elastic substrates. A mixture model with the power-law distribution functions is utilized to obtain the TDFGP shell’s mechanical properties. In addition, it is assumed that the internal porosities in the matrix materials can be dispersed into two independent patterns, either even or uneven porosity distribution. The field displacement in the TDFGP shell is approximated using first-order shear deformation theory. Hamilton’s principle and von Karman’s large deformation assumptions give a dynamic model for the TDFGP shell. The Galerkin decomposition method is employed to discretize the governing partial differential equations into time-dependent ordinary differential equations. The harmonic balance method is used to analytically solve the final time-dependent equations and acquire the nonlinear frequency response of the TDFGP shell. Research has been conducted on porosity, gradient indexes, elastic substrates, boundary conditions, and geometric ratios through parametric studies to investigate the outcomes and their applicability to real-world issues.

Galerkin-Bernstein Approximations of the System of Time Dependent Nonlinear Parabolic PDEs

The purpose of the research is to find the numerical solutions to the system of time dependent nonlinear parabolic partial differential equations (PDEs) utilizing the Modified Galerkin Weighted Residual Method (MGWRM) with the help of modified Bernstein polynomials. An approximate solution of the system has been assumed in accordance with the modified Bernstein polynomials. Thereafter, the modified Galerkin method has been applied to the system of nonlinear parabolic PDEs and has transformed the model into a time dependent ordinary differential equations system. Then the system has been converted into the recurrence equations by employing backward difference approximation. However, the iterative calculation is performed by using the Picard Iterative method. A few renowned problems are then solved to test the applicability and efficiency of our proposed scheme. The numerical solutions at different time levels are then displayed numerically in tabular form and graphically by figures. The comparative study is presented along with L2 norm, and L∞ norm.

Dynamic Analysis of Elastically Supported Functionally Graded Sandwich Beams Resting on Elastic Foundations Under Moving Loads

Vibration behaviors of elastically supported functionally graded (FG) sandwich beams resting on elastic foundations under moving loads are investigated. The transformed-section method is first applied to establish the bending vibration equations of FG sandwich beams, then the Chebyshev collocation method is used to study free and forced vibrations. Two types of sandwich beams with FG faces-isotropic core and isotropic faces-FG core are considered. The material properties of FG materials are assumed to vary across the beam thickness according to a simple power function. Regarding the free vibration analysis, bending vibration frequencies are calculated numerically by forming a matrix eigenvalue equation. As for the forced vibration analysis, the backward differentiation formula method is employed to solve the time-dependent ordinary differential equations to obtain the dynamic deformations of the beam. To ensure the accuracy of the proposed model, some calculated results are compared with those in the published literature. Parametric studies are then performed to demonstrate the effects of material gradient indexes, stack types, layer thickness ratios, slenderness ratios, excitation frequency and speed of moving loads, and foundation and support stiffness parameters on the dynamic characteristics of FG sandwich beams.

Nonlinear free vibrations analysis of truncated conical shells made of bidirectional functionally graded materials

For the first time, present paper addresses the nonlinear vibrations analysis of a truncated conical shell made of bidirectional functionally graded (BDFG) materials whose mechanical properties varied continuously through the thickness and the length directions of the conical shell based on the power law distribution. Accordingly, the set of nonlinear equation of motions of the conical shell are derived based on first-order shear deformation theory (FSDT) and von Karman strain-displacement relations using Hamilton’s principle. System of partial differential equations of the BDFG conical shell are transformed into time-dependent ordinary differential equations using Galerkin approach. Afterward, the nonlinear equations are analytically solved by means of modified Poincaré–Lindstedt method, and the nonlinear frequency of the BDFG conical shell is obtained. For verification purpose, the present outcomes are compared with those available in previous researches. Eventually, the effect of longitudinal and transverse gradient indexes, vibration amplitude, and geometrical parameters on the nonlinear frequency of the BDFG conical shell is investigated. The present research may give benchmark solutions and some guidelines for designing BDFG truncated conical shells.

The RBF-FD and RBF-FDTD Methods for Solving Time-Domain Electrical Transient Problems in Power Systems

In this paper, the development and application of the radial basis function-finite difference (RBF-FD) method and the RBF-finite difference time domain (RBF-FDTD) method for solving electrical transient problems in power systems that are defined by the time-dependent ordinary differential equations (ODEs) and the time-dependent partial differential equations (PDEs), respectively, are presented. RBFs such as Gaussian (GA), Multiquadric (MQ), Inverse Quadric (IQ), and Inverse Multiquadric (IMQ) are used in these numerical methods to formulate the central finite difference approximations of the first- and second-order derivatives of a function. The algorithm of selecting “optimal” shape parameters for our basis functions is also applied, specifically to increase the accuracy of the suggested methods with regard to high accuracy needs. Finally, the accuracy, effectiveness, and applicability of our new approaches are evaluated through simulations of the switching transient voltages on a typical electrical circuit and a 220 kV single-phase transmission line, lightning-induced voltages on a 110 kV single-phase overhead distribution line, and transient voltages along two horizontal grounding electrodes excited by lightning impulse sources. The obtained numerical results demonstrate that our proposed RBF-based numerical approaches compare favorably to the traditional numerical methods.

Using a neural network approach to accelerate disequilibrium chemistry calculations in exoplanet atmospheres

ABSTRACT In this era of exoplanet characterization with JWST, the need for a fast implementation of classical forward models to understand the chemical and physical processes in exoplanet atmospheres is more important than ever. Notably, the time-dependent ordinary differential equations to be solved by chemical kinetics codes are very time-consuming to compute. In this study, we focus on the implementation of neural networks to replace mathematical frameworks in one-dimensional chemical kinetics codes. Using the gravity gradient, temperature-pressure profiles, initial mixing ratios, and stellar flux of a sample of hot-Jupiter’s atmospheres as free parameters, the neural network is built to predict the mixing ratio outputs in steady state. The architecture of the network is composed of individual autoencoders for each input variable to reduce the input dimensionality, which is then used as the input training data for an LSTM-like neural network. Results show that the autoencoders for the mixing ratios, stellar spectra, and pressure gradients are exceedingly successful in encoding and decoding the data. Our results show that in 90 per cent of the cases, the fully trained model is able to predict the evolved mixing ratios of the species in the hot-Jupiter atmosphere simulations. The fully trained model is ∼103 times faster than the simulations done with the forward, chemical kinetics model while making accurate predictions.

High-fidelity simulations for Turing pattern formation in multi-dimensional Gray–Scott reaction-diffusion system

In this study, the authors present high-fidelity numerical simulations for capturing the Turing pattern in a multi-dimensional Gray–Scott reaction-diffusion system. For this purpose, an explicit mixed modal discontinuous Galerkin (DG) scheme based on multi-dimensional structured meshes is employed. This numerical scheme deals with high-order derivatives in diffusion, and highly nonlinear functions in reaction terms. Spatial discretization is accomplished using hierarchical basis functions based on scaled Legendre polynomials. A new reaction term treatment is also detailed, showing an intrinsic property of the DG scheme and preventing incorrect solutions due to excessively nonlinear reaction terms. The system is reduced to a set of time-dependent ordinary differential equations, which are solved with an explicit third-order Total Variation Diminishing (TVD) Runge–Kutta method. The Saddle-Node, Hopf, and Turing bifurcations’ boundary conditions are also examined for the system, and subsequently, Turing space is identified. The developed numerical scheme is applied to various multidimensional Gray–Scott systems to assess its ability to capture Turing pattern formation. Several test problems, such as stationary and non-stationary waves, pulse-splitting waves, self-replicating waves, and different types of spots, are chosen from the literature to demonstrate the different Turing patterns.

Modeling, simulation, and optimization of hydrogen production from microalgae in compact photobioreactors

This study investigates the possibility of hydrogen production from microalgae depending on temperature, and cultivation medium microalgae, oxygen and carbon dioxide concentrations in compact photobioreactors (PBR). A mathematical model generated a system of time dependent ordinary differential equations (ODE) to predict hydrogen generation in the indirect biophotolysis process originating from microalgae cultivation in tubular compact PBR. The indirect biophotolysis consists of two stages: i) aerobic (microalgae growth with air supply) and ii) anaerobic (consumption of microalgae biomass with hydrogen generation without air supply). A Michaelis-Menten type expression was used to model the rate of H2 generation considering that oxygen and sulfur could inhibit the process. The model was adjusted and experimentally validated against measured H2 production from green microalgae. The maximum rate of hydrogen production from local wild microalgae Tetradesmus obliquus was 6.8×10−7kgH2kgmed−1d−1. The thermodynamic optimization of the system determined that the optimum time of the anaerobic stage was 11 days and 13 h. The herein obtained results demonstrated that it is reasonable to state that a fundamental anaerobic stage optimum time for maximum large-scale H2 production should be expected in any indirect H2 biophotolysis process, no matter the complexity of the actual system.

Uniform stability and chaotic dynamics in nonhomogeneous linear dissipative scalar ordinary differential equations

The paper analyzes the structure and the inner long-term dynamics of the invariant compact sets for the skewproduct flow induced by a family of time-dependent ordinary differential equations of nonhomogeneous linear dissipative type. The main assumptions are made on the dissipative term and on the homogeneous linear term of the equations. The rich casuistic includes the uniform stability of the invariant compact sets, as well as the presence of Li-Yorke chaos and Auslander-Yorke chaos inside the attractor.

On the thermo-vibrational response of multi-layer viscoelastic skin tissue to laser irradiation

This paper investigates temperature and vibration responses of human multi-layer skin tissue to laser irradiation. Due to the viscoelastic property of skin, a standard linear solid model known as the Zener model with two relaxation times with dual phase-lag heat conduction model has been used to describe the temperature and displacement in a one-dimensional form. The obtained differential equations have been discretized in the weak form to balance thermal and vibrational energies. The extracted time-dependent ordinary differential equations have been solved numerically by integration over time. The investigations are based on the effect of four relaxation times of temperature, heat, strain, and stress with laser effect in the form of a single step or repeated pulses on the behavior of the skin. By examining their results, the effect of phase lags has also been discussed, showing that the heat phase lag has a complex impact on thermal response and depends on the type of laser radiation. The phase lag of temperature reduces the temperature change rate, and the strain phase lag spreads the displacement at later times, gives inertia to the system, and dampens the system's vibration.

Analysis of Applied Mathematics

Mathematics applied to applications involves using mathematics for issues that arise in various fields, e.g., science, engineering, engineering, or other areas, and developing new or better techniques to address the demands of the unique challenges. We consider it applied math to apply maths to problems in the real world with the double purpose of describing observed phenomena and forecasting new yet unknown phenomena. Thus, the focus is on math, e.g., creating new techniques to tackle the issues of the unique challenges and the actual world. The issues arise from a variety of applications, including biological and physical sciences as well as engineering and social sciences. They require knowledge of different branches of mathematics including the analysis of differential equations and stochastics. They are based on mathematical and numerical techniques. Most of our faculty and students work directly with the experimentalists to watch their research findings come to life. This research team investigates mathematical issues arising out of geophysical, chemical, physical, and biophysical sciences. The majority of these problems are explained by time-dependent partial integral or ordinary differential equations. They are also accompanied by complex boundary conditions, interface conditions, and external forces. Nonlinear dynamical systems provide an underlying geometrical and topological model for understanding, identifying, and quantifying the complex phenomena in these equations. The theory of partial differential equations lets us correctly formulate well-posed problems and study the behavior of solutions, which sets the stage for effective numerical simulations. Nonlocal equations result from the macroscopically modeling stochastic dynamical systems characterized by Levy noise and the modeling of long-range interactions. They also provide a better understanding of anomalous diffusions.

Recovering the carrier number conservation in SPICE simulation of PIN diodes and IGBT devices

In SPICE simulations of PIN diodes and IGBT devices using finite difference method, one discretizes an undepleted N− region into several equally spaced nodes with a time-dependent distance of Δx(t), and then transforms the ambipolar diffusion equation, a time–space partial differential equation, into a set of time-dependent ordinary differential equations. However, the time-dependent property of Δx(t) destroys the carrier number conservation. In this paper, we propose an approach to correct the effect of the Δx(t) by introducing an auxiliary system. It has the same total current and the total carrier number in the undepleted N− region as the real system, but has different electron and hole current components. The difference is caused by adding compensation current terms with the equal amplitude and opposite sign to the electron and hole current terms in the auxiliary system. These compensation current terms are proportional to the boundary speed of the undepleted N− region and do not change the total current. The auxiliary system can be easily solved using SPICE behavior models and its carrier density is a good approximation to the real one. Our simulations show that the compensation current correction is important for fast switching PIN diodes, but may not be very important in IGBT devices due to their large gate-related capacitance.

Nonlinear forced vibration of rectangular plates by modified multiple scale method

In the present study, the nonlinear forced vibration of a rectangular plate is investigated analytically using modified multiple scales method for the first time. The plate is subjected to transversal harmonic excitation, and the boundary condition is assumed to be simply supported. The von Karman nonlinear strain displacement relations are used. The extended Hamilton principle and classical plate theory are applied to derive the partial differential equations of motions. This research focuses on resonance case with 3:1 internal resonance. By applying Galerkin method, the nonlinear partial differential equations are transformed into time dependent nonlinear ordinary differential equations, which are then solved analytically by modified multiple scales method. This proposed approach is very simple and straightforward. The obtained results are then compared with both the traditional multiple scales method and previous studies, and excellent compatibility is noticed. The effect of some of the main parameters of the system is also examined.

An analytical solution for nonlinear vibration of floating plate on the fluid by modified multiple scales method

The aim of the present paper is to analytically study the nonlinear forced vibration of a rectangular plate floating on the fluid by Modified Multiple Time Scales method for the first time. It is assumed that the fluid is stationary, incompressible, non-viscous, and non-rotational, and the plate is subjected to transversal excitation. The boundary condition is considered to be simply supported. Using von Karman nonlinear strain displacement relationships, the extended Hamilton principle, and FSTD plate theory, the partial differential equations of motion are derived. The fluid is mathematically modeled by Bernoulli equation and the velocity potential function. Galerkin method is then applied for converting the nonlinear partial differential equations into time-dependent nonlinear ordinary differential equations. The resulted equations are solved analytically by the Modified Multiple Scales Method, thereafter. Despite the large number of derivatives and calculations of the conventional multiple scale method, this approach is very simple and straightforward. The results reveal an excellent agreement with the traditional Multiple Scales method results and existing studies, and are more accurate than other available results. The effect of the presence of fluid near the plate on natural frequency and amplitude of vibration of plate are studied. The effects of some key parameters of the system are also examined.

Sixth-Order Combined Compact Finite Difference Scheme for the Numerical Solution of One-Dimensional Advection-Diffusion Equation with Variable Parameters

A high-accuracy numerical method based on a sixth-order combined compact difference scheme and the method of lines approach is proposed for the advection–diffusion transport equation with variable parameters. In this approach, the partial differential equation representing the advection-diffusion equation is converted into many ordinary differential equations. These time-dependent ordinary differential equations are then solved using an explicit fourth order Runge–Kutta method. Three test problems are studied to demonstrate the accuracy of the present methods. Numerical solutions obtained by the proposed method are compared with the analytical solutions and the available numerical solutions given in the literature. In addition to requiring less CPU time, the proposed method produces more accurate and more stable results than the numerical methods given in the literature.

Physics-guided deep learning framework for predictive modeling of bridge vortex-induced vibrations from field monitoring

Vortex-induced vibrations (VIVs) with large amplitudes have been observed on long-span bridges worldwide. Classic semi-empirical VIV models that depend on wind tunnel tests are challenged when required to predict the VIV response of real bridges due to the complexity of real winds, high Reynolds number effects, and uncertainty of bridge structures. The prediction accuracy by these laboratory-based models may, thus, be reduced for real large-scale bridges. Emerging field monitoring systems on prototype bridges allow one to reconsider modeling of bridge VIVs with considerations of real natural winds and full-scale structures by massive monitoring data. In this research, first, we derive a general form of time-dependent ordinary differential equation based on Scanlan's semi-empirical model and field observed bridge VIVs to describe VIV dynamics. Second, guided by the formulation and field observation, we propose a deep learning framework to identify the VIV dynamics, leading to a data-driven model. We demonstrate the proposed framework on a real long-span bridge by performing long-time prediction of the VIV response under real natural winds.

Nonlinear free vibration analysis of piezoelectric laminated plate with random actuation electric potential difference and thermal loading

Solutions to nonlinear free vibration analysis of an arbitrarily laminated thin rectangular plate subjected to thermo-piezoelectric load is investigated. The governing equation of motion is based on von Kármán’s deflection of thin plates to capture the effects of moderate geometric nonlinearity. Induced strain actuation assumption is confined to a linear model for applied low electric fields and varying temperature distribution along the thickness direction. The solution for lateral displacement field for simply supported and clamped condition edges are determined by displacement-stress; mixed formulation. The applied Galerkin’s projection reduces the governing nonlinear partial differential equation to a time dependent ordinary differential equation. Solutions are obtained using a predictor-corrector numerical integration scheme. The effects of randomly applied actuation potential and thermal loading on nonlinear frequency ratio are examined. The marginal values provided using various probabilistic methods are found to be in good agreement with each other.

Vibration Analysis of Axially Functionally Graded Timoshenko Beams with Non-uniform Cross-section

The present paper investigates the transverse vibration of a non-uniform axially functionally graded Timoshenko beam with cross-sectional and material properties varying in the beam length direction. The Chebyshev collocation method is used to spatially discretize the governing partial differential equations of motion of the beam into time-dependent ordinary differential equations in terms of Chebyshev differentiation matrices. An algebraic eigenvalue equation in matrix form is then formed to study the free vibration behavior of non-uniform axially functionally graded Timoshenko beams. Several results of natural frequencies of the beams are evaluated and compared with those in the published literature to assure the accuracy of the proposed model. The effects of taper ratio, material graded index, slenderness ratio, material compositions and restraint types on the natural frequencies of tapered axially functionally graded Timoshenko beams are examined.