Derivative Terms Research Articles

Numerical Analysis and Computation of the Finite Volume Element Method for the Nonlinear Coupled Time-Fractional Schrödinger Equations

In this article, our aim is to consider an efficient finite volume element method combined with the L2−1σ formula for solving the coupled Schrödinger equations with nonlinear terms and time-fractional derivative terms. We design the fully discrete scheme, where the space direction is approximated using the finite volume element method and the time direction is discretized making use of the L2−1σ formula. We then prove the stability for the fully discrete scheme, and derive the optimal convergence result, from which one can see that our scheme has second-order accuracy in both the temporal and spatial directions. We carry out numerical experiments with different examples to verify the optimal convergence result.

Combined effect of track-bridge interaction on the dynamic response of a simply supported railway bridge

In this work, the objective is to understand the effect of the nonlinear behaviour of the ballasted track, or the mechanism of vertical and horizontal track-bridge interaction behaviour, on the dynamic response of a simply supported railway bridge, carrying a single ballasted track characterized by a fractional order viscoelastic nonlinear Winkler foundation type, and acted by a moving train load with constant speed. The main feature of the model is that the horizontal and vertical interactions at the interface between the bridge deck and the ballasted track, composed by rails, sleepers and ballast, is introduced through a nonlinear cubic force-displacement relation, schematizing the nonlinear stiffness, plus fractional derivative term for simulating the rail bed’s damping behaviour and a system of horizontally orientated spring-damper elements. By applying the two numerical methods, fourth-order Runge-Kutta method and Finite Difference Method, the validation of the dynamic behaviour of the rail-bridge system is performed, and numerical simulations demonstrate the accuracy of the proposed theoretical model for predicting the real behaviour of the studied bridge taking into account the track-bridge interaction effect. The influence of the fractional-order derivative and the distance between sleepers on the critical velocities, associated with resonance effects, is also addressed.

Neural network-based distributed consensus tracking control for uncertain Euler–Lagrange systems over directed topologies

This paper proposes a neural network-based robust control approach for driving disturbed Euler–Lagrange systems (e.g., robot manipulator) in a directed network to perform consensus tracking of a heterogeneous leader’s output signal. The parameter matrices and external disturbances in each follower’s Euler–Lagrange dynamics are unmeasurable, thus we unify them into one lumped uncertainty term. Neural networks are then employed to online estimate these uncertainties, with adaptive update laws ensuring the boundedness of the estimation errors. Subsequently, a set of distributed observers are developed to adaptively estimate the leader’s states as well as the unidentified parameter vector and output matrix in the leader’s model. With their help, the robust cooperative controller is designed. Its efficacy on directed networks is theoretically justified by designing a Lyapunov function with a special structure. This function produces symmetric positive definite matrices when its first derivative terms interact with the specified parameters in the observer and Laplacian matrices for directed graphs. Finally, numerical simulations based on two-link robot manipulators are carried out to verify that the tracking errors converge to zero when applying the neural network-based robust controller.

An Improved CH4 Profile Retrieving Method for Ground-Based Differential Absorption Lidar

Range-resolved CH4 concentration measurement is important prior data for atmospheric physical and chemical models. Ground-based differential absorption lidar (DIAL) can measure the vertical distribution of CH4 concentration in the atmosphere. The traditional method uses lidar observational data and the lidar equation to calculate profiles, but the inversion accuracy is greatly affected by noise. Although some denoising methods can improve accuracy at low altitudes, the low signal-to-noise ratio caused by the effect of aerosol Mie scattering and lower aerosol concentrations at high altitudes cannot be solved. Here, an improved cubic smoothing spline fitting CH4 concentration profile inversion method is proposed to address this challenge. By adding a penalty term of the second derivative of the conventional cubic spline function to the objective function, this penalty term acts to smooth the fitting, allowing the fitting function to avoid necessarily passing through those noisy sampling points. This avoids the large fluctuations caused by noisy sampling points, effectively suppresses noise, captures signals with lower noise levels, and thereby enhances the inversion accuracy of the profiles. Simulations and case studies demonstrated the superiority of the proposed method. Compared with the traditional method, cubic smoothing spline fitting can reduce the mean error of the whole CH4 profile by 85.54%. The standard deviation of CH4 concentration retrieved is 3.59 ppb–90.29 ppb and 0.01 ppb–6.75 ppb smaller than the traditional method and Chebyshev fitting, respectively. Three real cases also indicate that the CH4 concentration retrieved by cubic smoothing spline fitting is more consistent with in-situ measurements. In addition, long-term DIAL observations have also revealed notable diurnal and seasonal trends in CH4 concentration at observation sites.

A rotational velocity-correction projection method for the Micropolar Navier-Stokes equations

In this paper, we introduce a velocity correction projection method for the Micropolar Navier-Stokes Equations. The velocity correction method are adopted to approximate the time derivative term, stability analysis and error estimation of the first-order semi-discrete scheme are proved. At the same time, the optimal error estimate using the technique of dual norm are obtained. In this way, the divergence free of the velocity u can be conserved. Finally, the numerical results show the method has an optimal convergence order. The numerical results are consistent with our theoretical analysis, and our method is effective.

Numerical study on phase change propulsion mechanism of thermal underwater glider based on spectral method

The thermal underwater glider’s propulsion power is provided by the phase change energy storage tube. To improve the propulsion performance of the glider, it is essential to study the phase change heat transfer characteristics. This work proposes a collocation spectral method to handle the spatial derivative term and introduces a fourth-order Runge-Kutta method to handle the time derivative term. The temperature distribution and liquid volume fraction of the phase change material in the energy storage tube were numerically simulated under specific boundary conditions. The simulation results were then compared to the experimental data, and a good agreement was observed.Based on these findings, the impact of incoming free stream velocity and temperature difference intensity on phase change within the vessel was examined. The results underscored the dominance of convection in solid–liquid phase change and identified key factors influencing phase change in scenarios with Stefan numbers below 0.1342. Additionally, the study highlighted the limitations of fixed-value approximations and the nonlinear nature of phase change over time. These insights provide valuable guidance for developing phase change devices that harness volumetric potential energy aboard underwater thermal gliders.

Development of an Adaptive Fuzzy Integral-Derivative Line-of-Sight Method for Bathymetric LiDAR Onboard Unmanned Surface Vessel

Previous control methods developed by our research team cannot satisfy the high accuracy requirements of unmanned surface vessel (USV) path-tracking during bathymetric mapping because of the excessive overshoot and slow convergence speed. For this reason, this study developed an adaptive fuzzy integral-derivative line-of-sight (AFIDLOS) method for USV path-tracking control. Integral and derivative terms were added to counteract the effect of the sideslip angle with which the USV could be quickly guided to converge to the planned path for bathymetric mapping. To obtain high accuracy of the look-ahead distance, a fuzzy control method was proposed. The proposed method was verified using simulations and outdoor experiments. The results demonstrate that the AFIDLOS method can reduce the overshoot by 79.85%, shorten the settling time by 55.32% in simulation experiments, reduce the average cross-track error by 10.91% and can ensure a 30% overlap of neighboring strips of bathymetric LiDAR outdoor mapping when compared with the traditional guidance law.

Phase-field-based lattice Boltzmann method for two-phase flows with interfacial mass or heat transfer.

In this work, we develop a phase-field-based lattice Boltzmann (LB) method for a two-scalar model of the two-phase flows with interfacial mass or heat transfer. Through the Chapman-Enskog analysis, we show that the present LB method can correctly recover the governing equationsfor phase field, flow field, and concentration or temperature field. In particular, to derive the two-scalar equationsfor the mass or heat transfer, we propose a new LB model with an auxiliary source distribution function to describe the extra flux terms, and the discretizations of some derivative terms can be avoided. The accuracy and efficiency of the present LB method are also tested through several benchmark problems, and the influence of mass or heat transfer on the fluid viscosity is further considered by introducing an exponential relation. The numerical results show that the present LB method is suitable for the two-phase flows with interfacial mass or heat transfer.

L2$L^{2}$‐growth property for the wave equation with a higher derivative term

AbstractWe consider the Cauchy problem in for the wave equation with a higher derivative term. We derive sharp growth estimates of the ‐norm of the solution itself for the case of and . By imposing the weighted ‐initial velocity, we can get the lower and upper bound estimates of the solution itself. For the case of , we observe that the ‐growth behavior of the solution never occurs in the ‐framework of the initial data.

A sparse spectral method for fractional differential equations in one-spatial dimension

We develop a sparse spectral method for a class of fractional differential equations, posed on R\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathbb {R}$$\\end{document}, in one dimension. These equations may include sqrt-Laplacian, Hilbert, derivative, and identity terms. The numerical method utilizes a basis consisting of weighted Chebyshev polynomials of the second kind in conjunction with their Hilbert transforms. The former functions are supported on [-1,1]\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$[-1,1]$$\\end{document} whereas the latter have global support. The global approximation space may contain different affine transformations of the basis, mapping [-1,1]\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$[-1,1]$$\\end{document} to other intervals. Remarkably, not only are the induced linear systems sparse, but the operator decouples across the different affine transformations. Hence, the solve reduces to solving K independent sparse linear systems of size O(n)×O(n)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathcal {O}(n)\ imes \\mathcal {O}(n)$$\\end{document}, with O(n)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathcal {O}(n)$$\\end{document} nonzero entries, where K is the number of different intervals and n is the highest polynomial degree contained in the sum space. This results in an O(n)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathcal {O}(n)$$\\end{document} complexity solve. Applications to fractional heat and wave equations are considered.

Optimal thermometry theory of three-channel wide spectrum based on three-directional difference.

The constant spectral emissivity decoupling method within current wide-spectrum thermometry theories stands as a primary factor contributing to accuracy degradation. This creates a deadlock in the current radiation thermometry framework, where the system's two-dimensional analytical capabilities and resolution accuracy cannot be concurrently achieved, becoming a major theoretical obstacle in the development of this technique. Consequently, based on the Taylor series de-integration method under the wide spectral framework, and taking the first and second derivative terms of spectral emissivity as the starting point, a wide spectral optimization temperature solution theory based on three-directional difference method is proposed. It ensures compatibility and stable solving conditions for imaging systems, while fundamentally removing the dependency on the constantization of spectral emissivity treatment, and realizing the decoupling and inversion of three-channel spectral emissivity. The handling effects of different cutoff precision differential methods on spectral emissivity derivatives are discussed, and the temperature and spectral emissivity solving capabilities of the method are theoretically validated under various spectral emissivity models. Furthermore, this method is used to monitor the continuous temperature rise processes of two different samples. Maximum average relative temperature calculation errors below 6% and 5% are achieved, and the target spectral emissivity variation rate and trend are well reproduced, yielding conclusions consistent with simulations.

Investigation of Magneto-convection in Viscoelastic Fluid Saturated Anisotropic Porous Layer Under Local Thermal Non-equilibrium Condition

The problem of magneto-convection in viscoelastic fluid saturated anisotropic porous layer under local thermal non-equilibrium (LTNE) effect is investigated. Extended Darcy model with time derivative term for viscoelastic fluid of the Oldroyd type with an externally imposed vertical magnetic field is used to model the momentum equation. The entire investigation has been split into two parts: (i) linear stability analysis (ii) weakly non-linear stability analysis. We perform normal mode technique to examine linear stability analysis while truncated representation of Fourier series method is used for weakly non-linear stability analysis. The onset of convection is set in through oscillatory rather than stationary mode due to competition between the processes of thermal, magnetic effect and viscoelasticity. A comparative study between anisotropic and isotropic porous medium is made as a function of Q (Chandrasekhar number), 𝛤 (non dimensional inter phase heat transfer coefficient), 𝜆1 (Relaxation time) and λ2 (Retardation time). Apart from this, Q, 𝛤 and λ2 stabilize the system in oscillatory case while 𝜆1 destabilize the system. Furthermore 𝜉 (mechanical anisotropic parameter), 𝜂s (thermal anisotropic parameter for solid phase), destabilizes the system and 𝜂f (thermal anisotropic parameter for fluid phase) stabilizes the system. The effect of Q, 𝜆1, λ2, 𝛤, 𝜉, 𝜂f and 𝜂s on heat transfer is also examined.

Improved lattice Boltzmann model for immiscible multicomponent systems with high viscosity gradients at the interface.

We propose alternative discretization schemes for improving the lattice Boltzmann pseudopotential model for incompressible multicomponent systems, with the purpose of modeling the flow of immiscible fluids with a large viscosity ratio. Compared to the original model of Shan-Chen [Phys. Rev. E 47, 1815 (1993)1063-651X10.1103/PhysRevE.47.1815], the present discretization schemes consider: (i) an explicit force term, (ii) a second-order discretization of the stream term, (iii) a moments-based model for the kinetic nonequilibrium distributions, and (iv) a high-order discretization of the spatial derivative terms. To verify the accuracy of the proposed model, the effects of varying the viscosity ratio as well as both fluid's viscosities on spurious currents and capillary number are investigated for the problems dealing with a static bubble, two-component Poiseuille flow, and immiscible fluid-fluid displacement. The resulting algorithm maintains the simplicity of the pseudopotential model while allowing an easy implementation for multicomponent systems. The results of the model herein proposed show improved control of the interface region and interfacial tension, relatively smaller magnitudes of spurious current values with increasing viscosity ratio, and also a significantly wider stability range with respect to the previously best results in the literature.

A mixed finite element spatial discretization scheme and a higher‐order accurate temporal discretization scheme for a strongly discontinuous electromagnetic interface condition in ideal sliding electrical contact problems

AbstractSliding electrical contact involves multiple conductors sliding in contact at different speeds, with current flowing through the contact surfaces. The Lagrangian method is commonly used to describe the electromagnetic field in order to overcome the trouble of convective dominance, especially in high‐speed sliding electrical contact problems. However, to maintain correct field continuity, magnetic vector potential and scalar potential taken as variables cannot be continuous simultaneously at the ideal sliding electrical contact interface. This involves a strongly discontinuous condition for variables. Further, commonly used spatial–temporal discretization algorithms are invalid, for example, the classic finite element (CFEM) framework does not allow discontinuous variables, and the backward Euler method in time domain introduces a significant interface error source associated with the velocity of relative motion. To accurately handle strongly discontinuous conditions in numerical calculations, a mixed nodal finite element scheme and a higher‐order accurate temporal discretization scheme are introduced. In this scheme, classical finite element method is performed in each subdomain, and the derivative terms at the boundary are added as new variables. The effectiveness and accuracy of the above methods are verified by comparing them with a standard solution in a two‐dimensional railgun model and analyzing the current density distribution in a three‐dimensional railgun model.

Computation of three-dimensional incompressible flows using high-order weighted essentially non-oscillatory finite-difference lattice Boltzmann method

In the present work, an accurate and robust solution methodology based on the high-order weighted essentially non-oscillatory (WENO) finite-difference lattice Boltzmann method (LBM) in the three-dimensional generalized curvilinear coordinates is presented and applied for simulating the three-dimensional incompressible flows over complicated configurations with curved boundaries. Here, the incompressible form of the lattice Boltzmann equation in three dimensions is considered and the discretization of the spatial derivative terms is performed with the fifth-order WENO finite-difference method and the temporal derivative term is discretized with the fourth-order Runge–Kutta scheme to ensure the accuracy and stability of the solution method for both the steady and unsteady problems. The three-dimensional lattice Boltzmann equation applied here is based on a nineteen discrete velocity model for transforming the microscopic properties to the macroscopic ones. To assess the accuracy and robustness of the present three-dimensional high-order finite-difference LBM solver, different incompressible flow benchmarks and practical test cases are studied that are the cavity flow, the Beltrami flow, the flow in the curved ducts of rectangular cross sections, and the flow over a sphere for different flow conditions. The decay of the homogeneous isotropic turbulence is also computed to examine the suitability of the present solution method to be applied as the direct numerical simulation of turbulent flows. It is demonstrated that the solution methodology presented based on the high-order WENO finite-difference LBM in the three-dimensional generalized curvilinear coordinate can be used for accurately and effectively computing the three-dimensional practical incompressible flow problems.

Numerical approximation of a generalized time fractional partial integro-differential equation of Volterra type based on a meshless method

The moving least squares (MLS) approximation is used in this study to solve time-fractional partial integro-differential equations (TFPIDE). In our approach, we approximate the time Caputo fractional derivative term and the integral term by using a finite difference scheme and trapezoidal method, respectively, which yields an order of accuracy of O(τ+τ2−α), where 0<α<1. We thoroughly investigate the applicability and validity of the proposed method and provide an error estimate for its performance. Additionally, we solve various numerical problems to demonstrate the accuracy and computational efficiency of our approach, confirming the theoretical findings.

Self-adaptive and time divide-and-conquer physics-informed neural networks for two-phase flow simulations using interface tracking methods

Physics-informed neural networks (PINNs) are emerging as a promising artificial intelligence approach for solving complex two-phase flow simulations. A critical challenge in these simulations is an accurate representation of the gas–liquid interface using interface tracking methods. While numerous studies in conventional computational fluid dynamics (CFD) have addressed this issue, there remains a notable absence of research within the context of PINNs-based two-phase flow simulations. Therefore, this study aims to develop a robust and generic PINNs for two-phase flow by incorporating the governing equations with three advanced interface tracking methods—specifically, the Volume of Fluid, Level Set, and Phase-Field method—into an improved PINN framework that has been previously proposed and validated. To further enhance the performance of the PINNs in simulating two-phase flow, the phase field constraints, residual connection and the time divide-and-conquer strategies are employed for restricting neural network training within the scope of physical laws. This self-adaptive and time divide-and-conquer (AT) PINNs is then optimized by minimizing both the residual and loss terms of partial differential equation. By incorporating the three different interface tracking methods, it efficiently handles high-order derivative terms and captures the phase interface. The case of single rising bubble in two-phase flow is simulated to validate the robustness and accuracy of the AT PINNs. The simulation's accuracy is evaluated by comparing its performance in terms of velocity, pressure, phase field, center of mass, and rising velocity with that of conventional PINNs and CFD benchmarks. The results indicate that the AT PINNs coupled with these interface tracking methods offers a satisfactory performance in simulating rising bubble phenomenon.

New derivation of the amplitude of asymptotic oscillatory tails of weakly delocalized solitons

The computation of the amplitude, α, of asymptotic standing wave tails of weakly delocalized, stationary solutions in a fifth-order Korteweg–de Vries equation is revisited. Assuming the coefficient of the fifth order derivative term, ε2≪1, a new derivation of the “beyond all orders in ε” amplitude, α, is presented. It is shown by asymptotic matching techniques, extended to higher orders in ε, that the value of α can be obtained from the asymmetry at the center of the unique solution exponentially decaying in one direction. This observation, complemented by some fundamental results of Hammersley and Mazzarino [], not only sheds new light on the computation of α but also greatly facilitates its numerical determination to a remarkable precision for so small values of ε, which are beyond the capabilities of standard numerical methods. Published by the American Physical Society 2024

Establishing breather and N-soliton solutions for conformable Klein–Gordon equation

Abstract This article develops and investigates the behavior of soliton solutions for the spatiotemporal conformable Klein–Gordon equation (CKGE), a well-known mathematical physics model that accounts for spinless pion and de-Broglie waves. To accomplish this task, we deploy an effective analytical method, namely, the modified extended direct algebraic method (mEDAM). This method first develops a nonlinear ordinary differential equation (NODE) through the use of a wave transformation. With the help of generalized Riccati NODE and balancing nonlinearity with the highest derivative term, it then assumes a finite series-form solution for the resulting NODE, from which four clusters of soliton solutions – generalized rational, trigonometric, exponential, and hyperbolic functions – are derived. Using contour and three-dimensional visuals, the behaviors of the soliton solutions – which are prominently described as dark kink, bright kink, breather, and other N N -soliton waves – are examined and analyzed. These results have applications in solid-state physics, nonlinear optics, quantum field theory, and a more thorough knowledge of the dynamics of the CKGE.

A discrete spectral method for time fractional fourth-order 2D diffusion-wave equation involving [formula omitted]-Caputo fractional derivative

In this work, the ψ-Caputo fractional derivative, as a generalization of the classical Caputo fractional derivative in which the fractional derivative of a sufficiently differentiable function is defined with respect to another strictly increasing function, ψ, is utilized to define the time fractional fourth-order 2D diffusion-wave equation. A Chebyshev–Gauss–Lobatto scheme is developed to solve this equation. In this way, a new operational matrix for the ψ-Riemann–Liouville fractional integration of the Chebyshev polynomials is derived. The solution of the equation is obtained by determining the solution of the algebraic system extracted from approximation the ψ-Caputo fractional derivative term via a finite discrete Chebyshev series and employing the expressed operational matrix. The validity of the established approach is examined by solving two examples.