Order Derivative Terms Research Articles

Improved numerical AC superposition method for electron energy distribution functions

An improved AC superposition method is proposed to obtain the electron energy distribution functions (EEPFs) by reducing the effect of the high order derivative terms. In the AC superposition method, a small AC voltage is applied to the current-voltage curve, and the EEPF is obtained by the second harmonic current which is proportional to the second derivative of the current. If the AC voltage is not small compared to the electron temperature, the EEPF can be distorted by high derivative terms that are not ignorable. Thus, the proposed method uses the DC component and the second harmonics of the current to reduce the influence of the high order derivative terms. The fourth derivative term is removed and the coefficients of the higher order derivative terms become ignorable. The EEPF by this method shows that the resolution of the low energy part in the EEPFs is greatly improved.

Stability analysis and error estimates of local discontinuous Galerkin methods with implicit-explicit time-marching for the time-dependent fourth order PDEs

The main purpose of this paper is to give stability analysis and error estimates of the local discontinuous Galerkin (LDG) methods coupled with three specific implicit-explicit (IMEX) Runge–Kutta time discretization methods up to third order accuracy, for solving one-dimensional time-dependent linear fourth order partial differential equations. In the time discretization, all the lower order derivative terms are treated explicitly and the fourth order derivative term is treated implicitly. By the aid of energy analysis, we show that the IMEX-LDG schemes are unconditionally energy stable, in the sense that the time step τ is only required to be upper-bounded by a constant which is independent of the mesh size h. The optimal error estimate is also derived by the aid of the elliptic projection and the adjoint argument. Numerical experiments are given to verify that the corresponding IMEX-LDG schemes can achieve optimal error accuracy.

On Saulyev's Methods

In 1957 V. K. Saulyev proposed two so-called asymmetric methods for solving parabolic equations. We study these methods w.r.t. their stability and consistency, how to include first order derivative terms, how to apply boundary conditions with a derivative, and how to extend the methods to two space dimensions. We also prove that the various modifications proposed by Saulyev, Barakat and Clark, and Larkin also (as was to be expected) require k = o(h) in order to be consistent. As a curiosity we show that the two original Saulyev methods in fact solve two different differential equations.

Vibrations of asymptotically and variationally based Uflyand–Mindlin plate models

In this paper, we provide alternative Uflyand–Mindlin's plate equations taking into account rotary inertia and shear deformation, based on both asymptotic expansion and variational arguments. The aim is to derive truncated versions of Uflyand–Mindlin's equations, specifically without the fourth order derivative term with respect to time. The truncated version of Uflyand–Mindlin's plate model may be derived starting from three-dimensional elasticity equations, by using asymptotic arguments based on expansion of displacements with respect to a small geometrical parameter. This expansion method also leads to a proper identification of the shear correction factor. It is shown that suitably modified variational derivation leads to an additional term which is shown to be negligible for determination of the fundamental natural frequency of the all-round simply supported plates, but may contribute significantly in estimation of higher natural frequencies. It is argued that the proposed version of Uflyand–Mindlin's plate equations is simpler and more consistent than the original Uflyand–Mindlin equations. Likewise, it is advantageous over the equation that stems from neglecting the fourth order time derivative in original Uflyand–Mindlin equations. The two alternative truncated models serve as intermediate theories between the classical plate theory and the original Uflyand–Mindlin theory their usefulness depending on the problem at hand.

ACCURATE SOLUTION AND GRADIENT COMPUTATION FOR ELLIPTIC INTERFACE PROBLEMS WITH VARIABLE COEFFICIENTS.

A new augmented method is proposed for elliptic interface problems with a piecewise variable coefficient that has a finite jump across a smooth interface. The main motivation is not only to get a second order accurate solution but also a second order accurate gradient from each side of the interface. The key of the new method is to introduce the jump in the normal derivative of the solution as an augmented variable and re-write the interface problem as a new PDE that consists of a leading Laplacian operator plus lower order derivative terms near the interface. In this way, the leading second order derivatives jump relations are independent of the jump in the coefficient that appears only in the lower order terms after the scaling. An upwind type discretization is used for the finite difference discretization at the irregular grid points near or on the interface so that the resulting coefficient matrix is an M-matrix. A multi-grid solver is used to solve the linear system of equations and the GMRES iterative method is used to solve the augmented variable. Second order convergence for the solution and the gradient from each side of the interface has also been proved in this paper. Numerical examples for general elliptic interface problems have confirmed the theoretical analysis and efficiency of the new method.

Canonical invariance of spatially covariant scalar-tensor theory

We investigate invariant canonical transformations of a spatially covariant scalar-tensor theory of gravity, called the XG theory, by which the action or the Hamiltonian and the primary constraints keep their forms invariant. We derive the Hamiltonian in a non perturbative manner and complete the Hamiltonian analysis for all regions of the theory. We confirm that the theory has at most 3 degrees of freedom in all regions of the theory as long as the theory has the symmetry under the spatial diffeormorphism. Then, we derive the invariant canonical transformation by using the infinitesimal transformation. The invariant metric transformation of the XG theory contains a vector product as well as the disformal transformation. The vector product and the disformal factor can depend on the higher order derivative terms of the scalar field and the metric. In addition, we discover the invariant canonical transformation which transforms the momentum of the metric. Using the invariant transformation, we study the relation between the Horndeski theory and the GLPV theory, and find that we can not obtain the arbitrary GLPV theory from the Horndeski theory through the invariant canonical transformation we have found.

Exact versus Taylor-expanded energy density in the study of the neutron star crust–core transition

The importance of the fourth and higher order terms in the Taylor series expansion of energy of isospin asymmetric nuclear matter in studies of the neutron star crust–core phase transition is investigated using the finite-range simple effective interaction. Analytic expressions for the evaluation of the second and fourth order derivative terms in the Taylor series expansion for any general finite-range interaction of Yukawa, exponential or Gaussian form have been obtained. The effect of the nuclear matter incompressibility, symmetry energy and slope parameters on the predictions for the crust–core transition density is examined. The crustal moment of inertia is calculated and the prediction for the radius of the Vela pulsar is analyzed using different equations of state.

Perona–Malik Model with a New Diffusion Coefficient for Image Denoising

This paper presents a new diffusion coefficient which is based on second order derivative and local entropy information for image denoising. In the proposed model, a second order derivative term is introduced, which reduces the staircasing effect and preserves edge in a processed image. The local entropy information can preserve texture. The Perona–Malik model with a new diffusion coefficient improves the denoised effects, and prevents edges from being over-smoothed. Comparative experiments show that the proposed model obtains more satisfied results than the other two existing models.

Quantum massive conformal gravity

We first find the linear approximation of the second plus fourth order derivative massive conformal gravity action. Then we reduce the linearized action to separated second order derivative terms, which allows us to quantize the theory by using the standard first order canonical quantization method. It is shown that quantum massive conformal gravity is renormalizable but has ghost states. A possible decoupling of these ghost states at high energies is discussed.

An extended form of Boussinesq-type equations for nonlinear water waves

An extended form of Boussinesq-type equations with improved nonlinearity is presented for the water wave propagation and transformation in coastal areas. To improve the nonlinearity of lower order Boussinesq-type equations, without including higher order derivative terms, an extended form of Boussinesq-type equations is derived by a generalized method. The Stokes-type analysis of the extended equations shows that the accuracy of the second order and third order nonlinear characteristics is improved greatly. The numerical test is also carried out to investigate the practical performance of the new equations under different wave conditions. Better agreement with experimental data is found in the regions of strong nonlinear wave-wave interaction and harmonic generation.

Four interpretations of temporal memory operators in the wave equation

Attenuation and dispersion in ultrasound and elastography may be modeled with convolution memory operators in the wave equation. When the attenuation follows an arbitrary frequency power law the memory is a time domain power law. Since that is also a fractional time derivative, a large body of literature on fractional derivatives then becomes available for analysis and simulation. It can also be shown that, e.g., an elementary grain shearing process in an unconsolidated saturated sediment results in fractional wave equations for both compressional and shear waves. Second, some of these wave equations can be derived from constitutive equations with memory operators, ensuring satisfaction of causality and the Kramers-Kronig relations. The fractional Kelvin-Voigt and Zener models and their resulting attenuation and dispersion will in particular be discussed. The fractional operators can also be interpreted as the result of an infinite number of basic processes. Therefore, third, a fractional wave equation can be shown to be the result of an infinite number of elementary relaxation processes. Fourth, the fractional constitutive equation can also be expressed as an infinite sum of integer order derivatives of higher order which is also equivalent to a wave equation with higher order derivative terms.

Computation of fixed boundary tokamak equilibria using a method based on approximate particular solutions

In this work a meshless method based on the approximate particular solutions is applied to the computation of fixed boundary tokamak equilibria using Grad–Shafranov (GS) equation. The GS equation is solved for different choices of the right hand side of the equation: (i) when it is not a function of magnetic flux (i.e., Solov’ev solutions), (ii) when it is a linear function of magnetic flux, and (iii) when it is a nonlinear function of magnetic flux. For all these cases the first order derivative term in the GS equation is transferred to the right hand side such that the left hand side consists only the Laplace operator. This enables us to use the Radial Basis Functions (RBFs) in the calculation of approximate particular solutions. A linear combination of these particular solutions is taken as the solution of the GS equation and the resulting system of algebraic equations is solved iteratively because of the presence of the magnetic flux on the right hand side in all three choices. Furthermore, we use least squares approach in solving the overdetermined system of algebraic equations which alleviates the problem of ill-conditioning to a certain extent. The numerical results obtained using this method are in good agreement with the analytical solutions (where available). We find that the method is convergent, accurate and easily applicable to the irregular geometries due to its meshless character.

Steady-state bifurcation analysis of a strong nonlinear atmospheric vorticity equation

The atmospheric vorticity equation studied in the present paper is a simplified form of the atmospheric circulation model introduced by Charney and DeVore (1979) [4] on the existence of multiple steady states to the understanding of the persistence of atmospheric blocking. The fluid motion defined by the equation is driven by a zonal thermal forcing and an Ekman friction forcing measured by κ. It is proved that the steady-state solution is globally unique for large κ values while multiple steady-state solutions branch off the basic steady-state solution for κ<κcrit where the critical value κcrit is less than one. Without involvement of viscosity, the equation has strong non-linear property as its non-linear part contains the highest order derivative term. Steady-state bifurcation analysis is essentially based on the compactness, which can be simply obtained for semilinear equations such as the Navier–Stokes equations but is not available for the strong nonlinear vorticity equation in the Euler formulation. Therefore the Lagrangian formulation of the equation is employed to gain the required compactness.

A novel optimal PID plus second order derivative controller for AVR system

This paper proposes a novel controller for automatic voltage regulator (AVR) system. The controller is a four term control type consisting of proportional, integral, derivative, and second order derivative terms (PIDD2). The four parameters of the proposed controller are optimized using particle swarm optimization (PSO) algorithm. The performance of the proposed PIDD2 is compared with various PID controllers tuned by modern heuristic optimization algorithms. In addition, a comparison with the fractional order PID (FOPID) controller tuned by Chaotic Ant Swarm (CAS) algorithm is also performed. Furthermore, a frequency response, zero-pole map, and robustness analysis of the AVR system with PIDD2 is performed. Practical implementation issues of the proposed controller are also addressed. Simulation results showed a superior response performance of the PIDD2 controller in comparison to PID and FOPID controllers. Moreover, the proposed PIDD2 can highly improve the system robustness with respect to model uncertainties.

Subharmonic Response of Linear Vibroimpact System with Fractional Derivative Damping to a Randomly Disrobed Periodic Excitation

The subharmonic response of single-degree-of-freedom vibroimpact oscillator with fractional derivative damping and one-sided barrier under narrow-band random excitation is investigated. With the help of a special Zhuravlev transformation, the system is reduced to one without impacts, thereby permitting the applications of asymptotic averaging over the period for slowly varying random process. The analytical expression of the response amplitude is obtained in the case without random disorder, while only the approximate analytical expressions for the steady-state moments of the response amplitude are obtained in the case with random disorder. The effects of the fractional order derivative term, damping term, random disorder, and the coefficient of restitution and other system parameters on the system response are discussed. Theoretical analyses and numerical simulations show that fractional derivative makes both the system damping and stiffness coefficients increase, such that it changes the system parameters region at which the response amplitude reaches the maximum. The system energy loss in collision is equivalent to increasing the damping coefficient of the system. System response amplitude will increase when the excitation frequency is close to the resonant frequency and will decay rapidly when the excitation frequency gradually deviates from the resonance frequency.

Linearized fluid/gravity correspondence: from shear viscosity to all order hydrodynamics

In ref. \cite{1406.7222}, we reported a construction of all order linearized fluid dynamics with strongly coupled $\mathcal{N}=4$ super-Yang-Mills theory as underlying microscopic description. The linearized fluid/gravity correspondence makes it possible to resum all order derivative terms in the fluid stress tensor. Dissipative effects are fully encoded by the shear term and a new one, emerging starting from third order in hydrodynamic derivative expansion. In this work, we provide all computational details omitted in \cite{1406.7222} and present additional results. We derive closed-form linear holographic RG flow-type equations for momenta-dependent transport coefficient functions. Generalized Navier-Stokes equations are shown to emerge from the constraint components of the bulk Einstein equations. We perturbatively solve the RG equations for the viscosity functions, up to third order in derivative expansion, and up to this order compute spectrum of small fluctuations. Finally, we solve the RG equations numerically, thus accounting for all order derivative terms in the boundary stress tensor.

High-order spline filter: Design and application to surface metrology

The differences between the transmission characteristics of the standard cubic spline filter and the Gaussian filter lead to different evaluation results even for the same profile. There is indeed an adverse impact on the comparison of measurement results and the applications of the related international standards. A novel high-order spline filter is proposed to resolve this practical problem of approximating the Gaussian filtering characteristic. The design of the new filter is based on an improved variational approach by adding the high order derivative terms to the bending energy part whose structural parameters are determined by the aid of the universal Taylor series, so as to realize the convergence to the function of the Gaussian filter. In addition, a cascade algorithm in terms of the low-order filters is also developed in order to ensure stable performance of the high-order filter. Its effectiveness and application were verified by the experiments.

Optimization analysis of Duffing oscillator with fractional derivatives

The improved version of the constrained optimization harmonic balance method is presented to solve the Duffing oscillator with two kinds of fractional order derivative terms. The analytical gradients of the objective function and nonlinear quality constraints with respect to optimization variables are formulated and the sensitivity information of the Fourier coefficients can also obtained. A new stability analysis method based on the analytical formulation of the nonlinear equality constraints is presented for the nonlinear system with fractional order derivatives. Furthermore, the robust stability boundary of periodic solution can be determined by the interval eigenvalue problem. In addition, the sensitivity information mixed with the interval analysis method is used to quantify the response bounds of periodic solution. Numerical examples show that the proposed approach is valid and effective for analyzing fractional derivative nonlinear system in the presence of uncertainties. It is illustrated that the bifurcation solution in the fractional nonlinear systems may not be sensitive to the variation of the influence parameters.

Vacuum condition and the relation between response parameter and anomaly coefficient in (1 + 3) dimensions

The role of Israel-Hartle-Hawking vacuum is discussed for anomalous fluid in presence of both the gauge and gravitational anomalies in ($1+3$) dimensions. I show that imposition of this vacuum condition leads to the relation $\tilde{c}_{4d}=-8\pi^2c_m$ between the response parameter ($\tilde{c}_{4d}$) and the anomaly coefficient ($c_m$). This establishes a connection between the coefficients appearing in a first order and a third order derivative terms in the constitutive relation.

Chimera states in a two–population network of coupled pendulum–like elements

More than a decade ago, a surprising coexistence of synchronous and asynchronous behavior called the chimera state was discovered in networks of nonlocally coupled identical phase oscillators. In later years, chimeras were found to occur in a variety of theoretical and experimental studies of chemical and optical systems, as well as models of neuron dynamics. In this work, we study two coupled populations of pendulum-like elements represented by phase oscillators with a second derivative term multiplied by a mass parameter $m$ and treat the first order derivative terms as dissipation with parameter $\epsilon>0$. We first present numerical evidence showing that chimeras do exist in this system for small mass values $0<m<<1$. We then proceed to explain these states by reducing the coherent population to a single damped pendulum equation driven parametrically by oscillating averaged quantities related to the incoherent population.