Second-order Time Derivatives Research Articles

A GFDM with supplementary nodes for thin elastic plate bending analysis under dynamic loading

This work proposes a meshless generalized finite difference method (GFDM) with supplementary nodes for the solution of thin elastic plate bending under dynamic loading. The first- and second-order time derivatives in equilibrium equation are firstly discretized with the Houbolt method. The numerical solution of the resulting spatial problem at each time step is then calculated by the GFDM. The supplementary nodes on the boundary of the problem domain are introduced in the GFDM to yield the well-determined linear system of equations. Numerical example is provided to verify the accuracy and the stability of the developed approach, and these preliminary results reveal the potential of the method.

Time-dispersion correction for arbitrary even-order Lax-Wendroff methods and the application on full-waveform inversion

A second-order accurate finite-difference (FD) approximation is commonly used to approximate the second-order time derivative of a wave equation. The second-order accurate FD scheme may introduce time dispersion in wavefield extrapolation. Lax-Wendroff methods can suppress such dispersion by replacing the high-order time FD error terms with space FD error-correcting terms. However, the time dispersion cannot be completely eliminated and the computation cost dramatically increases with increasing order of (temporal) accuracy. To mitigate the problem, we have extended the existing time-dispersion correction scheme for the second- or fourth-order Lax-Wendroff method to a scheme for arbitrary even-order methods, which uses the forward and inverse time-dispersion transform (FTDT and ITDT) to add and remove the time dispersion from synthetic data. We test the correction scheme using a homogeneous model and the Sigsbee2A model. The modeling examples suggest that the use of derived FTDT and ITDT pairs on high-order Lax-Wendroff methods can effectively remove time-dispersion errors from high-frequency waves while using longer time steps than allowed in low-order Lax-Wendroff methods. We investigate the influence of the time dispersion on full-waveform inversion (FWI) and show an antidispersion workflow. We apply the FTDT to source terms and recorded traces before inversion, resulting in the source and adjoint wavefields containing equal time dispersion from source-side wave propagation and the modeled and observed traces accumulating equal time dispersion from source- and receiver-side wave propagation. The inversion results reveal that the antidispersion workflow is capable of increasing the accuracy of FWI for arbitrary even-order Lax-Wendroff methods. In addition, the high-order method can obtain better inversion results compared to the second-order method with the same antidispersion workflow.

New (3+1)-dimensional integrable fourth-order nonlinear equation: lumps and multiple soliton solutions

PurposeThis paper aims to introduce a new (3 + 1)-dimensional fourth-order integrable equation characterized by second-order derivative in time t. The new equation models both right- and left-going waves in a like manner to the Boussinesq equation.Design/methodology/approachThis formally uses the simplified Hirota’s method and lump schemes for determining multiple soliton solutions and lump solutions, which are rationally localized in all directions in space.FindingsThis paper confirms the complete integrability of the newly developed (3 + 1)-dimensional model in the Painevé sense.Research limitations/implicationsThis paper addresses the integrability features of this model via using the Painlevé analysis.Practical implicationsThis paper presents a variety of lump solutions via using a variety of numerical values of the included parameters.Social implicationsThis work formally furnishes useful algorithms for extending integrable equations and for the determination of lump solutions.Originality/valueTo the best of the author’s knowledge, this paper introduces an original work with newly developed integrable equation and shows useful findings of solitons and lump solutions.

Application of the Fractional Sturm–Liouville Theory to a Fractional Sturm–Liouville Telegraph Equation

In this paper, we consider a non-homogeneous time–space-fractional telegraph equation in n-dimensions, which is obtained from the standard telegraph equation by replacing the first- and second-order time derivatives by Caputo fractional derivatives of corresponding fractional orders, and the Laplacian operator by a fractional Sturm–Liouville operator defined in terms of right and left fractional Riemann–Liouville derivatives. Using the method of separation of variables, we derive series representations of the solution in terms of Wright functions, for the homogeneous and non-homogeneous cases. The convergence of the series solutions is studied by using well known properties of the Wright function. We show also that our series can be written using the bivariate Mittag-Leffler function. In the end of the paper some illustrative examples are presented.

Load Frequency Control for Multi-Area Power Plants with Integrated Wind Resources

To provide a more practical method of controlling the frequency and tie-line power flow of a multi-area interconnected power system (MAIPS), a state observer based on sliding mode control (SOboSMC) acting under a second-order time derivative is proposed. The proposed design is used to study load frequency control against load disturbance, matched and mismatched uncertainty and parameter measurement difficulties of power systems that exist in the modern power plant, such as multi-area systems integrated with wind plants. Firstly, the state observer is designed to optimally estimate system state variables. The estimated states are applied to construct the model of the MAIPS. Secondly, a SOboSMC is designed with an integral switching surface acting on the second-order time derivative to forcefully drive the dynamic errors to zero and eliminate chattering, which can occur in the first-order approach to sliding mode control. In addition, the stability of the total power system is demonstrated with the Lyapunov stability theory based on a new linear matrix inequality (LMI) technique. To extend the validation of the proposed design control for practical purposes, it was tested in a New England system with 39 bus power against random load disturbances. The simulation results confirm the superiority of the proposed SOboSMC over other recent controllers with respect to overshoot and settling time.

A space-time finite element method for solving linear Riesz space fractional partial differential equations

In this paper, numerical solutions for linear Riesz space fractional partial differential equations with a second-order time derivative are considered. A space-time finite element method is proposed to solve these equations numerically. In the time direction, the C0-continuous Galerkin method is used to approximate the second-order time derivative. In the space direction, the usual linear finite element method is developed to approximate the space fractional derivative. The matrix equivalent form of this numerical method is deduced. The stability of the discrete solution is established and the optimal error estimates are investigated. Some numerical tests are given to validate the theoretical results.

Vibration Reduction of SRM Through Controlling the Average of Applied Demagnetization Voltage

Although Switched Reluctnace Motors have many advantages, vibration is known as their most severe drawback acting as an impediment limiting their widespread use in industrial and domestic applications. Through electromechanical relationships, it is shown that the first-order time derivative of applied phase voltage and second-order time derivative of radial force directly affect the amplitude of generated vibration, especially at the beginning of demagnetization instance. Therefore, a set of criteria is stated in this paper meeting which ensures having low vibration intensity. In this paper, three types of demagnetization voltages are proposed which comprehensively meet the stated criteria. These voltages share the same characteristics contributing to the vibration reduction in Switched Reluctance Motor. Two vital parameters of Synchronous Counter and Coefficient are included in the proposed demagnetization voltages which must be meticulously chosen in order to be effective. Noticeable contribution of the proposed methods to the vibration alleviation is validated through simulation and experimental tests conducted under different working conditions.

Does frequency-dependent cell proliferation exhibit a Fano-type resonance?

We examined PC12 cell proliferation in environments with temporally varying epidermal growth factor concentrations by means of a microfluidic system. Our measurements revealed frequency-dependent cell behaviour over an observation period of three days. The cell population either increased, decreased or remained constant depending on the frequency of epidermal growth factor applied. A plot of the apparent proliferation rate as a function of growth-factor frequency was mathematically described by the Fano line-shape formula. In the context of linear response theory, these results imply that the PC12 cells compute zero, first and second-order time derivatives of the ligand concentration and utilise this information to decide to proliferate or die. We discuss a physical model based on periodic forcing of coupled oscillators that accounts for these observations. Our results and analysis suggest the possibility to influence cell fate by controlling the dynamics of the extracellular environment.

Numerical simulation of thermal wave propagation and collision in thin film using finite element solution

Numerical simulation of propagation and collision of thermal wave phenomena in the order of nanometer is carried out. The finite element procedure is developed for Cattaneo–Vernotte heat conduction equation. In this simulation, non-homogeneous boundary conditions are used. Newmark’s scheme and constant average acceleration method are used to solve second-order time derivatives. The non-dimensional results are compared with the analytical solutions of the same kind of problem solved by using superposition principle with solution structure theorems and experimental values. It is noted that this current procedure had good agreement with the results given by analytical and experimental solutions. In this study, the finite element modeling of hyperbolic equation is solved for thermal wave propagation and collision of thermal waves. The transition of hyperbolic to parabolic nature of thermal wave propagation for different layer lengths and the characteristics of nodal temperatures near to the boundary are discussed. Also this work examined the temperature variation at the center of the layer at all collisions, which shows the increase in temperature after the first collision when time is 0.5 and then propagates toward boundary with further increase in temperature. This is well above the applied boundary temperature and decrease in temperature at boundary for subsequent collisions. The transition of thermal waves from hyperbolic to parabolic nature is analyzed and is found that the transition happens for the film thickness greater than 5. This phenomenon is repeated until it reaches steady state when time is 5 for hyperbolic whereas 1.5 for parabolic mode.

Coercive space-time finite element methods for initial boundary value problems

We propose and analyse new space-time Galerkin-Bubnov-type finite element formulations of parabolic and hyperbolic second-order partial differential equations in finite time intervals. Using Hilbert-type transformations, this approach is based on elliptic reformulations of first- and second-order time derivatives, for which the Galerkin finite element discretisation results in positive definite and symmetric matrices. For the variational formulation of the heat and wave equations, we prove related stability conditions in appropriate norms, and we discuss the stability of related finite element discretisations. Numerical results are given which confirm the theoretical results.

The Linear Peridinamic Model in Elasticity Theory

The peridinamics theory, proposed by Silling in 2000, is a nonlocal theory of continuum mechanics based on an integro-differential equation without spatial derivatives and may cope with discontinuous displacement fields commonly occurring in fracture mechanics. Instead of spatial differential operators, integration over differences of the displacement field is used to describe the existing, possibly nonlinear, forces between particles of the solid body. Beside an overview of the peridynamics modelling and its application, results concerning the mathematical solution of the governing equation, which is a partial integro-differential equation with second-order time derivative, are presented. In this paper we consider well-posed Cauchy problem for the singular periodic integro-differential equation of peridinamics. In case of two-dimensional space the existence and uniqueness of solution are proved in Sobolev spaces. In multidimensional space the unique solvability of the problem in logarithmic scale of Hilbert spaces is proved.

A PDE approach to fractional diffusion: a space-fractional wave equation

We study solution techniques for an evolution equation involving second order derivative in time and the spectral fractional powers, of order $$s \in (0,1)$$ , of symmetric, coercive, linear, elliptic, second-order operators in bounded domains $$\varOmega $$ . We realize fractional diffusion as the Dirichlet-to-Neumann map for a nonuniformly elliptic problem posed on the semi-infinite cylinder $$\mathcal {C}= \varOmega \times (0,\infty )$$ . We thus rewrite our evolution problem as a quasi-stationary elliptic problem with a dynamic boundary condition and derive space, time, and space–time regularity estimates for its solution. The latter problem exhibits an exponential decay in the extended dimension and thus suggests a truncation that is suitable for numerical approximation. We propose and analyze two fully discrete schemes. The discretization in time is based on finite difference discretization techniques: the trapezoidal and leapfrog schemes. The discretization in space relies on the tensorization of a first-degree FEM in $$\varOmega $$ with a suitable hp-FEM in the extended variable. For both schemes we derive stability and error estimates. We consider a first-degree FEM in $$\varOmega $$ with mesh refinement near corners and the aforementioned hp-FEM in the extended variable and extend the a priori error analysis of the trapezoidal scheme for open, bounded, polytopal but not necessarily convex domains $$\varOmega \subset {\mathbb {R}}^2$$ . We discuss implementation details and report several numerical examples.

An efficient wavelet collocation method for nonlinear two-space dimensional Fisher–Kolmogorov–Petrovsky–Piscounov equation and two-space dimensional extended Fisher–Kolmogorov equation

In this paper, we consider two-space dimensional nonlinear Fisher–Kolmogorov–Petrovsky–Piscounov (Fisher–KPP) equation and two-space dimensional nonlinear fourth-order extended Fisher–Kolmogorov (EFK) equation which have a lot of applications in different branches of science, especially in mathematical biology. We present a wavelet collocation method based on Chebyshev wavelets combined with two different time discretization schemes. To generate more accurate results for Fisher–KPP equation, in time variable discretization, forward Euler timestepping scheme along with Taylor series expansion is used. Resultant time-discretized scheme has second-order accuracy and includes second-order time derivative which is evaluated from governing equation. On the other hand for EFK equation forward Euler timestepping scheme with first-order accuracy is used. Then space variables situated in the semi-discrete schemes are discretized with Chebyshev wavelet series expansion. In this way a full discrete scheme is obtained. By this approach obtainment of numerical solution of considered partial differential equations is turned into an operation of finding solution of an algebraic system of equations. Actually the solution of the algebraic system of equations is wavelet coefficients in wavelet series expansion. Putting these coefficients into Chebyshev wavelet series expansion the numerical solution of considered partial differential equations can be obtained successively. The main objective of this paper is to demonstrate that Chebyshev wavelet-based method is accurate, efficient, and reliable for nonlinear two-space dimensional high-order partial differential equations. Six test problems are considered and $$L_2$$, $$L_{\infty }$$ error norms are calculated for comparison of our numerical results with exact results whenever they are available. Also numerical results are plotted for comparison with reference solutions. The obtained results certify the applicability and efficiency of the suggested method for Fisher–KPP and EFK equations.

Magnetization nutation induced by surface effects in nanomagnets

We investigate the magnetization dynamics of ferromagnetic nanoparticles in the atomistic approach taking account of surface anisotropy and the spin misalignment it causes. We demonstrate that such inhomogeneous spin configurations induce nutation in the dynamics of the particle's magnetization. More precisely, in addition to the ordinary precessional motion with frequency $f_{p}\sim10\,{\rm GHz}$, we find that the dynamics of the net magnetic moment exhibits two more resonance peaks with frequencies $f_{c}$ and $f_{n}$ which are higher than the frequency $f_{p} : f_{c}=4\times f_{p}\sim40\,{\rm GHz}$ is related with the oscillations of the particle's magnetic moment between the minima of the effective potential induced by weak surface anisotropy. On the other hand, the much higher frequency $f_{n}\sim1\,{\rm THz}$ is attributed to the magnetization fluctuations at the atomic level driven by exchange interaction. We have compared our results on nutation induced by surface effects with those rendered by the macroscopic approach based on the Landau-Lifshitz-Gilbert equation augmented by an inertial term (proportional to the second-order time derivative of the macroscopic moment) with a phenomenological coefficient. The good agreement between the two models have allowed us to estimate the latter coefficient in terms of the atomistic parameters such as the surface anisotropy constant. We have thus proposed a new origin for the magnetization nutations as being induced by surface effects and have interpreted the corresponding resonance peaks and their frequencies.

Numerical Simulation of Thermal Processes in a Domain of Thin Metal Film Subjected to an Ultrashort Laser Pulse.

A thin metal film subjected to an ultrashort laser pulse is considered. With a sufficiently high laser intensity the process of the film heating may cause metal melting and even ablation. In this work, the numerical model of the melting and resolidification processes is presented. The mathematical model is based on the dual phase lag equation in which two positive constants appear, this means the relaxation and thermalization times. The considered equation contains a second-order time derivative and higher order mixed derivative in both time and space and should be supplemented by the appropriate boundary and initial conditions. The model of the melting and resolidification is presented in two versions. The first can be called ‘the introduction of the artificial mushy zone sub-domain’, while the second ‘the two forms of the basic energy equation’. At the stage of numerical computations, the implicit scheme of the finite difference method is used. The numerical algorithm is tested for the two proposed models which are applied to the computations concerning the thermal processes occurring in the cylindrical micro-domain (chromium, gold) subjected to an ultrashort laser pulse.

Large eddy simulation and investigation on the laminar-turbulent transition and turbulence-cavitation interaction in the cavitating flow around hydrofoil

The intrinsic 3D vortical structures of turbulent cavitating flow are difficult to be measured in experiment. In the present work, the compressible large eddy simulation (LES) was performed to investigate the cavitating flow around NACA66 hydrofoil, with a special emphasis put on the laminar-turbulent transition, the turbulence-cavitation interaction and the dynamics features. The mesh is constructed with sufficient resolution to capture the coherent structure and energy spectrum of turbulence, with the prerequisites of y+, Δx+ and Δz+ fulfilled to resolve at least 80% of the turbulence kinetic energy. The LES captured the subtle coherent structures of laminar-turbulent transition, including the evolution of pressure disturbance, 3D vortical wave and hairpin vortices. It manifests that, the cavity shedding produces super large-scaled vortices in the wake, and the Batchelor's pressure spectrum of conventional mono-phase turbulence is no longer met in turbulent cavitating flows. It is found that the conventional cavity shedding mechanism by re-entrant jet is not accurate if strong turbulence is present. The sweeping effect of the jet front expands the turbulent fluctuation along the cavity surface which eventually makes it broken. The various vorticity transport mechanisms are analyzed to clarify the role of cavitation effects in the evolution of the vortical structures. The counterintuitive drastic oscillation of lift and drag is studied, finding that the pressure is focused during the cavity collapses and a violent pressure rise is induced. By formulating approximate one-dimensional analytic formulas, the far-field pressure disturbance is found to be in strict accordance with the second-order time derivative of cavity volume.

Structure-preserving finite difference schemes for a semilinear thermoelastic system with second order time derivative

In this article we study a semilinear thermoelastic system which consists of heat equation and beam equation with second-order time derivative. Although in Yoshikawa (ZAMM 95(12):1393–1410, 2015; IMA J Numer Anal 37:477–504, 2017) the second author proposed the structure-preserving finite difference scheme for the system transformed to first order system with respect to time variable, here we propose structure-preserving finite difference schemes for the original system with second-order time derivative. The main purpose of this article is to give mathematical analysis for the scheme such as existence of solution and error estimate.

Two new integrable fourth-order nonlinear equations: multiple soliton solutions and multiple complex soliton solutions

In this paper, we develop two new fourth-order integrable equations represented by nonlinear PDEs of second-order derivative in time t. The new equations model both right- and left-going waves in a like manner to the Boussinesq equation. We will employ the Painleve analysis to formally show the complete integrability of each equation. The simplified Hirota’s method is used to derive multiple soliton solutions for this equation. We introduce a complex form of the simplified Hirota’s method to develop multiple complex soliton solutions. More exact traveling wave solutions for each equation will be derived as well.

Ghost-free theories with arbitrary higher-order time derivatives

We construct no-ghost theories of analytic mechanics involving arbitrary higher-order derivatives in Lagrangian. It has been known that for theories involving at most second-order time derivatives in the Lagrangian, eliminating linear dependence of canonical momenta in the Hamiltonian is necessary and sufficient condition to eliminate Ostrogradsky ghost. In the previous work we showed for the specific quadratic model involving third-order derivatives that the condition is necessary but not sufficient, and linear dependence of canonical coordinates corresponding to higher time-derivatives also need to be removed appropriately. In this paper, we generalize the previous analysis and establish how to eliminate all the ghost degrees of freedom for general theories involving arbitrary higher-order derivatives in the Lagrangian. We clarify a set of degeneracy conditions to eliminate all the ghost degrees of freedom, under which we also show that the Euler-Lagrange equations are reducible to a second-order system.

Caputo–Riesz-Feller fractional wave equation: analytic and approximate solutions and their continuation

In this article a time-space fractional wave equation is studied. In the new proposed fractional model, the second-order time derivative is replaced with a fractional derivative in Caputo sense, and the second-order space derivative is replaced with a Riesz-Feller derivative defined on infinite space domain. The fundamental solution of fractional wave equation is obtained in terms of Mittag-Leffler function in two parameters, that is by using the joint Laplace–Fourier transform method. We prove the continuation of the solution of the generalized Riesz wave equation as the skewness parameter tends to zero to the one of the corresponding fractional wave equation with classical Riesz derivative, this is accomplished by using the Lebesgue’s dominated convergence theorem. The optimal homotopy analysis method (OHAM) is employed to obtain semi-analytic solution of a newly proposed initial-value fractional wave problem, considering three numerical simulations. The continuation of the optimal solution and its dependence on the fractional derivative parameters are investigated. The study reveals that the OHAM is reliable and effective in case of fractional Riesz-Feller operator represents the fractional Laplacian operator.