Order Time Derivatives Research Articles

Weyl curvature evolution system for GR

Starting from the chiral first-order pure connection formulation of General Relativity, we put the field equations of GR in a strikingly simple evolution system form. The two dynamical fields are a complex symmetric tracefree matrix , which encodes the self-dual part of the Weyl curvature tensor, as well as a spatial connection . The right-hand sides of the evolution equations also contain the triad for the spatial metric, and this is constructed non-linearly from the field and the curvature of the spatial connection . The evolution equations for this pair are first order in both time and spatial derivatives, and so simple that they could have been guessed without a computation. They are the most natural spin two generalisations of Maxwell’s spin one equations. We also determine the modifications of the evolution system needed to enforce the ‘constraint sweeping’, so that any possible numerical violation of the constraints present becomes propagating and gets removed from the computational grid.

Numerical simulation of fluid flow with a fast Boundary‐Domain Integral Method

AbstractFor the simulation of fluid flow the velocity‐vorticity formulation of the Navier‐Stokes equations can be used. Approximating the first order time derivative in these equations by finite differences, essentially, the Yukawa equation for the velocity has to be solved with an inhomogeneity governed by the vorticity. Applying the Boundary Element Method to such problems is not straightforward. However, the Boundary‐Domain Integral Method (BDIM) can be used, which requires to solve beside the boundary integrals as well a domain integral due to the inhomogeneity. Classical BEM formulations have a quadratic complexity with respect to the unknowns on the boundary and, for the BDIM additionally, for the unknowns in the domain. This prevents the BDIM from being applied to real‐world problems.To reduce the quadratic complexity in BEM, approximation methods like the fast multipole method, ℋ‐matrices in combination with the adaptive cross approximation or the ℋ2 ‐matrix technique can be used. All these methods reduce the complexity to an almost linear order. Here, the ℋ2 ‐matrix technique is applied to reduce storage and computing time for the right hand side, i.e., for the matrix of the domain integrals. Based on interpolatory kernel expansions and an heuristic criterium for too small matrix entries an almost linear complexity can be achieved. The numerical studies focus on a lid‐driven cavity. The complexity reduction is demonstrated.

Hidden jerk in universal creep and aftershocks.

Most materials exhibit creep memory under the action of a constant load. The memory behavior is governed by Andrade's creep law, which also has an inherent connection with the Omori-Utsu law of earthquake aftershocks. Both empirical laws lack a deterministic interpretation. Coincidentally, the Andrade law is similar to the time-varying part of the creep compliance of the fractional dashpot in anomalous viscoelastic modeling. Consequently, fractional derivatives are invoked, but since they lack a physical interpretation, the physical parameters of the two laws extracted from curve fit lack confidence. In this Letter, we establish an analogous linear physical mechanism that underlies both laws and relates its parameters with the material's macroscopic properties. Surprisingly, the explanation does not require the property of viscosity. Instead, it necessitates the existence of a rheological property that relates strain with the first order time derivative of stress, which involves jerk. Further, we justify the constant quality factor model of acoustic attenuation in complex media. The obtained results are validated in light of the established observations.

Study on reflected waves through visco-elastic solid rotating with fixed angular frequency

The work is about the characteristics study of reflected waves through viscoelastic solid. The solid used has a nanostructure and is semiconductor in nature, with uniform angular frequency. The concept of nonlocal thermoelasticity is considered to study the disturbances propagating through the medium. The refined theory of the multiphase lag heat conduction model is used to encounter the heat disturbance that travels through the medium. We have also considered the concept of fractional order time derivative while analyzing the heat conduction model. Expressions for dispersion relation for the reflected medium are obtained. It is found that four coupled waves reflected into the medium and each wave is neither purely transverse nor dilatational. The analytical results obtained are computed graphically for a particular medium. Characteristics like velocity, attenuation coefficient, and specific heat loss are computed and represented graphically for the medium. In addition to semiconductor nanostructure devices, the findings might be useful in visco-elastic semiconductor nanostructures involved in geology and seismology studies.

Characteristic analysis of waves through piezoelectric semiconductor medium with fractional order time derivative

The article aims to study the characteristics of the waves generated in a medium by reflection from the surface of the solid medium. The theory of non-local elasticity is adopted, in which stress acting at any point is a function of the strain of all points in the neighborhood. The selected medium is an n-type semiconductor and is piezoelectric in nature. Elastic deformation results in the propagation of thermal waves and the conduction process are encountered by the three-phase-lag heat conduction model. A secular equation is obtained for reflected waves generated when an injected wave hits the surface of the media. The analysis indicates four reflected waves with positive projection and one with vibrational mode having zero wave number. All the waves produced are coupled waves and quasi in nature. The obtained theoretical results are later represented graphically for a material with Piezoelectric material. The effect of the order of fractional derivative, non-local parameters, and carrier density on the amplitude ratio of waves is discussed.

Analysis of algebraic flux correction schemes for semi-discrete advection problems

Based on recent developments regarding the analysis of algebraic flux correction schemes, we consider a locally bound-preserving discretization of the time-dependent advection equation. Specifically, we analyze a monolithic convex limiting scheme based on piecewise (multi-)linear continuous finite elements in the semi-discrete formulation. To stabilize the discretization, we use low order time derivatives in the definition of raw antidiffusive fluxes. Our analytical investigation reveals that their limited counterparts should satisfy a certain compatibility condition. The conducted numerical experiments suggest that this prerequisite is satisfied unless the size of mesh elements is vastly different. We prove global-in-time existence of semi-discrete approximations and derive an a priori error estimate for finite time intervals with a worst-case convergence rate of frac{1}{2} w. r. t. the {textrm{L}}^2 error. This rate is optimal in the setting under consideration because we allow all correction factors of the flux-corrected scheme to become zero. In this case, the algorithm reduces to the bound-preserving discrete upwinding method but the limited counterpart of this scheme converges much faster, in practice. Additional numerical experiments are performed to verify the provable convergence rate for a few variants of the scheme.

NON-POLYNOMIAL CUBIC SPLINE APPROACH FOR NUMERICAL APPROXIMATION OF SECOND ORDER LINEAR KLEIN-GORDON EQUATION

Most of the fundamental theories and mathematical models of engineering and physical sciences are expressed in terms of partial differential equations (PDEs). Several studies were carried out for the numerical approximation of the second order linear Klein-Gordon equation. This study constructed a new numerical technique for the numerical approximation of second order linear Klein-Gordon equation. The new constructed scheme was based on employing non-polynomial cubic spline method (NPCSM). The second order time derivatives involved in the linear Klein-Gordon equation were decomposed into the first order derivatives. The decomposition generated a linear system of PDEs, where the first order time derivatives were approximated by the central finite differences of . Three test problems were considered for the numerical illustration of the developed scheme. For different values of spatial displacement , step size , and time step , the developed numerical technique produced encouraging results which were very much close to the analytical solution. For , , and , the best observed accuracy was close to the machine precision.

Higher derivative gravity’s anti-Newtonian limit and the Caldirola–Kanai oscillator

Fourth order gravity in dimensions is investigated in its anti-Newtonian limit where lightcones shrink to lines. The limiting theory is fourth order in time derivatives, still fully diffeomorphism invariant, and retains the original number of physical degrees of freedom. In an unimodular Hamiltonian formulation the dynamics can, however, be reduced to a second order one by regarding the unimodular metric as a composite field. In a gauge where the logarithm of the volume element is ‘time’ a Hamiltonian proper arises, which (for each spatial point) is a matrix generalization of a Caldirola–Kanai oscillator with inverted quartic potential. In an alternative ‘Wick-flipped’ Hamiltonian formulation an analogous reduced system arises with non-inverted quartic potential.

Limits of solutions to the semilinear plate equation with small parameter

We study the existence of the limits of solutions to the semilinear plate equation with boundary Dirichlet condition with a small parameter coefficient of the second order derivative in time. We establish the convergence of solutions to the perturbed problem and their derivatives in spacial variables to the corresponding solutions to the unperturbed problem as the small parameter tends to zero.

Effect of Fractional Order Time Derivative on the Out-Diffusion Profiles During Degassing a Thin Plate in Vacuum

Effect of Fractional Order Time Derivative on the Out-Diffusion Profiles During Degassing a Thin Plate in Vacuum

An An Interacting Dark Energy Model with Nonminimal Derivative Coupling in the Parameterized Post-Friedmannian Framework

We investigation the parameterization of the cosmological model with the nonminimal derivative coupling of a scalar field where gravity is coupled nonminimally with the derivatives of dark energy components in the form of a scalar field. We follow the parameterized post-Friedmannian approach for the interacting dark energy theories. We show how the big number of free functions can be reduced by limiting certain assumptions to a few non-zero coefficients. We only consider the case that the dark sector contains at most second order in time derivatives of the metric and scalar fields. In this paper, we demonstrate their use through an example of the dark sector interactions model and classify them according to the current literature.

Three Approximations of Numerical Solution's by Finite Element Method for Resolving Space-Time Partial Differential Equations Involving Fractional Derivative's Order

In this paper, we apply to a class of partial differential equation the finite element method when the problem is involving the Riemann-Liouville fractional derivative for time and space variables on a bounded domain with bounded conditions. The studied equation is obtained from the standard time diffusion equation by replacing the first order time derivative by  for 0<<1 and for the second standard order space derivative by  for 1<<2 respectively. The existence of the unique solution is proved by the Lax-Milgram Lemma. We present here three schemes to approximate numerically the time derivative and use the finite element method for the space derivative using the Hadamard finite part integral and the Diethlem's first degree compound quadrature formula, the second approach is based on the link between Riemann-Liouville and Caputo fractional derivative, when the third method was based on the approximation of the Riemann-Liouville by the Grunwald-Letnikov fractional derivative. For the approximation of the space fractional derivative, the finite element method is introduced for all the three approaches. Finally, to check the effectiveness of the three methods, a numerical example was given.

An accurate, robust, and efficient weak Galerkin finite element scheme with graded meshes for the time-fractional quasi-linear diffusion equation

A time-fractional quasi-linear diffusion equation with a Caputo time derivative of order 0 < α < 1 is considered. The weak Galerkin finite element method is used for the space discretization, the Caputo time derivative is discretized by L 1 method and Newton linear method is used for the quasi-linear term with graded meshes in the temporal direction. The stability and convergence for fully discrete weak Galerkin finite element scheme in L 2 -norm are proved. Numerical experiments are given to confirm the theoretical results.

Causality in strain gradient elasticity: An internal variables approach

The absence of higher order time derivatives in equations of motion causes the infinite velocity of wave propagation in strain gradient elasticity models. This issue needs to be avoided from both a theoretical and practical viewpoint. It is shown that dual internal variables approach solves the causality problem. Furthermore, the primary internal variable can be interpreted as the Laplacian of the strain gradient resulting in an Aifantis-type strain gradient model.

A nonlinear epidemic model for tuberculosis with Caputo operator and fixed point theory

Tuberculosis (TB) is one of the most dangerous infectious diseases which spreads from one person to another through the air and often attacks the lungs. A mathematical model of TB with control measures is considered in this paper. The integer order time derivative is modeled via the Caputo fractional order operator. A reliable numerical technique was applied to discretize the fractional derivative. The existence and uniqueness of solutions are examined. The effect of the control parameters was investigated through some numerical experiments. The effect of variation of the fractional-order parameters on the susceptible, early latent, infected, persistent latent, and recovered population with respect to time are given via figures and discussed. It was observed that a model with a fractional order parameter could serve as a good control measure to the spread of TB.

Novel soliton solutions for the fractional three-wave resonant interaction equations

Abstract In this article, we obtained new infinite sets of exact soliton solutions for the nonlinear evolution system of three-wave resonant interaction equations. The solved system contains the non-zero second-order dispersion coefficients, the non-zero phase velocity mismatch, and the conformable fractional time derivative of order between zero and one. The solution method is a constructed ansatz that consists of linear combinations of the tan and cotan hyperbolic functions with complex coefficients. We stated clear systematic steps toward writing an exact soliton solution for the studied system. To show the efficiency of this method, we introduced some numerical examples on each obtained set of solutions. The computations showed that similar solutions can be obtained if one replaces the tan and cotan hyperbolic functions with the tan and cotan trigonometric functions. The new obtained fractional solutions could be useful in studying the broad applications of triad resonances in plasma physics and in nonlinear optics.

Magneto-Thermoelastic Response in an Infinite Medium with a Spherical Hole in the Context of High Order Time-Derivatives and Triple-Phase-Lag Model.

The article presents the interactions of magneto-thermoelastic effects in an isotropic material with a spherical cavity. The spherical cavity is expected to be tractionless and subjected to both heat and magnetic fields. The motion equation contains the Lorentz force. Laplace’s transformation methodology is used with a refined multi-time-derivative triple-phase-lag thermoelasticity theory to develop the generalized magneto-thermoelastic coupled solution. Many results were obtained to serve as benchmarks for future comparisons. The effects of time, magnetic field, and electric permittivity under the thermal environment were investigated.

Time-dependent and Caputo derivative order-dependent quasi-uniform synchronization on fuzzy neural networks with proportional and distributed delays

This paper focuses on studying the quasi-uniform (Q-U) synchronization on fractional-order fuzzy neural networks (FOFNNs) with the proportional and distributed delays. The impacts of the Caputo derivative order, coefficients of network system and control gain constants on the synchronization performance are taken into account. Two novel time-dependent and Caputo derivative order-dependent algebraic criteria on the Q-U synchronization of FOFNNs are established by using Cauchy–Schwartz inequality, Minkowski inequality, Cp inequality and Hölder inequality, which reveal the internal influences of the time and fractional derivative order on Q-U synchronization with the derivative order interval (0,2). The simulation examples further confirm the effectiveness and practicability of the algebraic criteria in terms of the MATLAB toolbox.

Time-domain model identification of structural dynamics from spatially dense 3D vision-based measurements

A novel approach is presented for the time-domain system identification of structural dynamic components exploiting the high spatial density of vision-based measurements. By using spatially dense measurements, the number of spatial measurement points is much larger than the dimensionality of the underlying dominant dynamics (i.e., the number of measurement points needed to meet observability of the targeted dynamics). This opens up the potential to develop new experimental identification methods that use this spatial overdetermination which were out of reach with conventional discrete sensors. The new approach presented in this paper directly extracts a relatively noise-free, low-order set of dynamic states by projecting the spatially dense time-domain measurements on a low-order dominant deformation motion basis. The basis is constructed by applying an over-complete singular value decomposition on the measurements. These dynamic states together with their numerically calculated first and second order time-derivatives are then used in a two-step identification of the structural dynamic parameters. Here, the structure of the model to identify is based on a-priori physical knowledge of the underlying set of partial differential equations, unlike a purely data-driven method. This approach has the benefits of providing a mathematical formulation that is straightforward to understand and implement and is time-efficient to solve. The presented approach is experimentally validated on a clamped plate as well as on a flexibly suspended plate which requires a correction of the rigid body motion. The identified models have shown to provide an accuracy up to 1 × 10−5 m with respect to the dominant measured motion components in both validation cases.

Higher-Order Relative Kinematics of Rigid Body and Multibody Systems. A Novel Approach with Real and Dual Lie Algebras

In this paper, a computing method for the higher-order acceleration vector field properties in general rigid body motion and multibody system is proposed using algebraic properties of dual number algebras. The higher-order accelerations field of a rigid body in a general motion is uniquely determined of higher- order time derivative of a spatial twist. For the relative kinematics in multibody systems, equations that allow the determination of the higher-order order acceleration vector field are given, using an exponential Brockett-like formula in these algebras. This is obtained with no need for further differentiation of the body pose with respect to time variable. The properties for the fields of velocity, acceleration, jerk, and jounce are presented, as well as cases of the general theory. This approach uses the isomorphism between the Lie algebra of the rigid displacements ▪, of the Special Euclidean group, SE3, and the Lie algebra of dual vectors. The results are coordinate-free and in a closed-form.