Order Time Derivatives Research Articles

Heat transfer analysis in a second grade fluid over and oscillating vertical plate using fractional Caputo–Fabrizio derivatives

This paper presents a Caputo–Fabrizio fractional derivatives approach to the thermal analysis of a second grade fluid over an infinite oscillating vertical flat plate. Together with an oscillating boundary motion, the heat transfer is caused by the buoyancy force induced by temperature differences between the plate and the fluid. Closed form solutions of the fluid velocity and temperature are obtained by means of the Laplace transform. The solutions of ordinary second grade and Newtonian fluids corresponding to time derivatives of integer and fractional orders are obtained as particular cases of the present solutions. Numerical computations and graphical illustrations are used in order to study the effects of the Caputo–Fabrizio time-fractional parameter \(\upalpha \), the material parameter \(\alpha _2 \), and the Prandtl and Grashof numbers on the velocity field. A comparison for time derivative of integer order versus fractional order is shown graphically for both Newtonian and second grade fluids. It is found that fractional fluids (second grade and Newtonian) have highest velocities. This shows that the fractional parameter enhances the fluid flow.

Investigation of initial‐boundary value problems of gradient elasticity in the realm of implicit second order evolution equations

The initial‐boundary value problems encountered in the regime of the dynamic theory of gradient elasticity are characterized by several crucial and intrinsic issues. One of the most important characteristics is that the involving differential equation is of fourth order and mainly that the second order time derivative – appearing in it – is not given explicitly but in the contrary is incorporated implicitly in mixed‐type differential terms (via spatial‐time derivatives). The aim of the present work is the investigation of the emerged initial‐boundary value problems of gradient elasticity in one spatial dimension and the establishment of the suitable functional setting assuring existence and uniqueness of weak (or strong) solutions. Furthermore, the spectral analysis of the gradient elastic operator is investigated and compared with the well‐known results of classical elasticity. Copyright © 2016 John Wiley & Sons, Ltd.

Stability and convergence analysis of a variable order replicator–mutator process in a moving medium

A more generalized approach, the concept of variable order derivative, is used to study the well-known replicator–mutator dynamics taking place in a moving medium. The biological relevance of the variable order context is explored via the language learning in social groups and stability of fixed points for the generalized model is recalled and discussed. Related graphs are plotted for different values of the derivative order γ. It happens that the threshold condition for learning accuracy symbolized by a function of payoff is a monotonically increasing function irrespective of the value of the time derivative order. Also, the limit cycles and their amplitudes are shown to vary with the value of the derivative order γ. These amplitudes become bigger as γ grows but the stability of the system is not affected. The generalized model, namely the variable order replicator–mutator dynamics in a moving medium is numerically solved via Crank–Nicholson scheme whose stability and convergence results are provided in details. An application to a variable order replicator–mutator dynamics of a population with three strategies is presented and numerical simulations are performed for some fixed values of the position variable r and the grid points. They display limit cycles appearing and disappearing in function of the values of the position r. The amplitudes of limit cycles are also proved to proportionally depend on r and the stability of the system remains unaffected. This shows the impressive effect of the transport process on the bifurcation dynamics of the model.

A Schrödinger approach to Newton-Cartan and Hořava-Lifshitz gravities

We define a `non-relativistic conformal method', based on a Schr\"odinger algebra with critical exponent z = 2, as the non-relativistic version of the relativistic conformal method. An important ingredient of this method is the occurrence of a complex compensating scalar field that transforms under both scale and central charge transformations. We apply this non-relativistic method to derive the curved space Newton-Cartan gravity equations of motion with twistless torsion. Moreover, we reproduce z = 2 Ho\v{r}ava-Lifshitz gravity by classifying all possible Schr\"odinger invariant scalar field theories of a complex scalar up to second order in time derivatives.

Regional gradient controllability of sub-diffusion processes

By sub-diffusion systems we mean to replace the first order time derivative of normal diffusion system with a Riemann–Liouville time fractional derivative of order α∈(0,1). In this paper, we first introduce a new definition of regional gradient controllability for the sub-diffusion system with the control inputs appearing in the differential equations as distributed inputs, which recovers the usual definition of regional gradient controllability as α→1. The sufficient and necessary conditions on actuators to achieve the gradient controllability in a given subregion of the whole domain are presented. An approach to guarantee the regional gradient controllability of the problems studied within the considered subregion using minimum energy control effort is presented. Several examples are presented in the end to illustrate the effectiveness of our results, where zone actuators, pointwise actuators or filament actuators are respectively considered.

Effects of the fractional order and magnetic field on the blood flow in cylindrical domains

In this paper, based on the magnetohydrodynamics approach, the blood flow along with magnetic particles through a circular cylinder is studied. The fluid is acted by an oscillating pressure gradient and an external magnetic field. The study is based on a mathematical model with Caputo fractional derivatives. The model of ordinary fluid, corresponding to time-derivatives of integer order, is obtained as a particular case. Closed forms of the fluid velocity and magnetic particles velocity are obtained by means of the Laplace and finite Hankel transforms. Effects of the order of Caputo's time-fractional derivatives and of the external magnetic field on flow parameters of both blood and magnetic particles are studied. Numerical simulations and graphical illustrations are used in order to study the influence of the fractional parameter α, Reynolds number and Hartmann number on the fluid and particles velocity. The results highlights that, models with fractional derivatives bring significant differences compared to the ordinary model. This fact can be an important advantage for some practical problems. It also results that the blood velocity, as well as that of magnetic particles, is reduced under influence of the exterior magnetic field.

Hyperbolic and pseudo-hyperbolic equations in the theory of vibration

Longitudinal vibration of bars is usually considered in mathematical physics in terms of a classical model described by the wave equation under the assumption that the bar is thin and relatively long. More general theories have been formulated taking into consideration the effect of the lateral motion of a relatively thick bar (beam). The mathematical formulation of these models includes higher-order derivatives in the equation of motion. Rayleigh derived the simplest generalization of the classical model in 1894, by including the effects of lateral motion and neglecting the shear stress. Bishop obtained the next generalization of the theory in 1952. The Rayleigh–Bishop model is described by a fourth-order partial differential equation not containing the fourth-order time derivative. He took into account the effects of shear stress. Both Rayleigh’s and Bishop’s theories consider lateral displacement being proportional to the longitudinal strain. The Bishop model was generalized by Mindlin and Herrmann. They considered the lateral displacement proportional to an independent function of time and longitudinal coordinate. This result is formulated as a system of two differential equations of second order, which could be replaced by a single equation of fourth order resolved with respect to the highest order time derivative. To obtain a more general class of equations, the longitudinal and lateral displacements are expressed in the form of a power series expansion in the lateral coordinate. In this paper, all of the above-mentioned equations are considered in the framework of a general theory of hyperbolic equations, with the aim of classifying the equations into general groups. The solvability of the corresponding problems is also discussed.

Recursive higher-order constraints for linkages with lower kinematic pairs

The geometric constraints imposed on the bodies of a linkage determine its function and mobility. They are the basis for any computational analysis of the kinematics and dynamics of a linkage, which ultimately requires higher-order time derivatives of the constraints. A tailored formulation for linkages comprising lower kinematic pairs is the product of exponentials (POE) formula, where rigid body motions are represented as curves in the Lie group SE(3). The corresponding velocity constraints involve the instantaneous joint screws, and their derivatives involve the derivatives of these screws. It is known that partial derivatives of arbitrary order of the instantaneous joint screws are given explicitly and algebraically in terms of Lie brackets (i.e. screw products). This, however, leads to complex expressions that are very difficult to use in actual computations. In this paper a recursive formulation for the time derivatives of arbitrary order of the velocity constraints for lower-pair linkages is presented. This formulation applies to multi-loop linkages. To this end, the linkage topology is represented by a topological graph, and loop closure constraints are formulated for the (topologically independent) fundamental cycles. It is briefly discussed that this provides the basis for the higher-order kinematic analysis of linkages.

Degenerate higher derivative theories beyond Horndeski: evading the Ostrogradski instability

Theories with higher order time derivatives generically suffer from ghost-like instabilities, known as Ostrogradski instabilities. This fate can be avoided by considering ``degenerate'' Lagrangians, whose kinetic matrix cannot be inverted, thus leading to constraints between canonical variables and a reduced number of physical degrees of freedom. In this work, we derive in a systematic way the degeneracy conditions for scalar-tensor theories that depend quadratically on second order derivatives of a scalar field. We thus obtain a classification of all degenerate theories within this class of scalar-tensor theories. The quartic Horndeski Lagrangian and its extension beyond Horndeski belong to these degenerate cases. We also identify new families of scalar-tensor theories with the property that they are degenerate despite the nondegeneracy of the purely scalar part of their Lagrangian.

Unique continuation property for anomalous slow diffusion equation

ABSTRACTA Carleman estimate and the unique continuation of solutions for an anomalous diffusion equation with fractional time derivative of order 0 < α <1 are given. The estimate is derived through some subelliptic estimates for an operator associated to the anomalous diffusion equation using calculus of pseudo-differential operators.

Difference methods for two-dimensional space-time fractional dispersion equation

The two-dimensional space-time fractional dispersion equation is obtained from the standard two-dimensional dispersion equation by replacing the first order time derivative by the Caputo fractional derivative,and the two second order space derivatives by the Riemann-Liouville fractional derivatives,respectively.Base on the shifted Grunwald finite difference approximation for the two space fractional derivatives,an implicit difference method and a practical alternate direction implicit difference method were proposed to approximate the fractional dispersion equation. The consistency,stability,and convergence of the two implicit difference methods were analyzed. By using mathematical induction method,it was proven that the two implicit difference methods were all unconditionally stable and convergent and the order of convergence were obtained. The convergence speed and computational complexity of the two implicit difference methods were compared. A numerical simulation for a space-time fractional dispersion equation with known exact solution was also presented,and correctness of the theoretical analysis was verified by the numerical results.

Effect of loading history on visco-elastic potato starch gel

Abstract In this work gelatinised potato starch is shown to retain the memory of past loading history. It exhibits a visco-elastic response which does not depend solely on instantaneous conditions. A simple squeeze flow experiment is performed, where loading is done in two steps with a time lag τ∼ seconds between the steps. The effect on the strain, of varying τ is reproduced by a three element visco-elastic solid model. Complexity is introduced through a generalised calculus approach by incorporating a non-integer order time derivative in the viscosity equation. A strain hardening proportional to the time lag between the two loading steps is also incorporated. This model reproduces the three salient features observed in the experiment, namely – the memory effect, slight initial oscillations in the strain as well as the long-time solid-like response. Dynamic visco-elasticity of the sample is also reported.

A Rotated Crank-Nicolson Iterative Method For The Solution of Two-Dimensional Time-Fractional Diffusion Equation

This paper aims to examine the effectiveness of a rotated iterative method to solve the two-dimensional time fractional diffusion equations, which are used when describing transport processes with long memory where the rate of diffusion is inconsistent with the classical Brownian motion model. This type of equation is obtained by replacing the first order time derivative in the standard di ff usion equation with a fractional derivative of order α, where 0 < α < 1, in accordance with Riemann-Liouville or Caputo. The developed method is derived from the standard Crank-Nicolson (S(C-N)) scheme by rotating clockwise 45 o with respect to the standard mesh. The study demonstrates the enhanced efficiency and superiority of the rotated Crank-Nicolson (R(C-N)) method, which overall reduces the CPU consumption time.

Finite element method for space-time fractional diffusion equation

In this paper, we consider two types of space-time fractional diffusion equations(STFDE) on a finite domain. The equation can be obtained from the standard diffusion equation by replacing the second order space derivative by a Riemann-Liouville fractional derivative of order s (1 <s ≤ 2), and the first order time derivative by a Caputo fractional derivative of order ? (0 <? ≤ 1). For the 0 <? < 1 case, we present two schemes to approximate the time derivative and finite element methods for the space derivative, the optimal convergence rate can be reached O(?2?? + h2) and O(?2 + h2), respectively, in which ? is the time step size and h is the space step size. And for the case ? = 1, we use the Crank-Nicolson scheme to approximate the time derivative and obtain the optimal convergence rate O(?2 + h2) as well. Some numerical examples are given and the numerical results are in good agreement with the theoretical analysis.

ON ADOMIAN'S DECOMPOSITION METHOD FOR SOLVING A FRACTIONAL ADVECTION-DISPERSION EQUATION

In this paper, the Adomian's decomposition method (ADM) is considered to solve a fractional advection-dispersion model. This model can be represented if the first order derivative in time is replaced by the Caputo fractional derivative of order � (0 < � ≤ 1). In addition, the space derivative orders are replaced by the alternative orders 0 < � ≤ 1 and 1 < ≤ 2. The obtained solutions are formulated in a convergent infinite series in terms of Mittage-Leffler functions. Finally, two illustrative examples are introduced to ensure the effectiveness of the used method. AMS Subject Classification: 34A08, 34K37

An effective phase shift diffusion equation method for analysis of PFG normal and fractional diffusions

Pulsed field gradient (PFG) diffusion measurement has a lot of applications in NMR and MRI. Its analysis relies on the ability to obtain the signal attenuation expressions, which can be obtained by averaging over the accumulating phase shift distribution (APSD). However, current theoretical models are not robust or require approximations to get the APSD. Here, a new formalism, an effective phase shift diffusion (EPSD) equation method is presented to calculate the APSD directly. This is based on the idea that the gradient pulse effect on the change of the APSD can be viewed as a diffusion process in the virtual phase space (VPS). The EPSD has a diffusion coefficient, Kβ(t)D radβ/sα, where α is time derivative order and β is a space derivative order, respectively. The EPSD equations of VPS are built based on the diffusion equations of real space by replacing the diffusion coefficients and the coordinate system (from real space coordinate to virtual phase coordinate). Two different models, the fractal derivative model and the fractional derivative model from the literature were used to build the EPSD fractional diffusion equations. The APSD obtained from solving these EPSD equations were used to calculate the PFG signal attenuation. From the fractal derivative model the attenuation is exp(−γβgβδβDf1tα), a stretched exponential function (SEF) attenuation, while from the fractional derivative model the attenuation is Eα,1(−γβgβδβDf2tα), a Mittag–Leffler function (MLF) attenuation. The MLF attenuation can be reduced to SEF attenuation when α=1, and can be approximated as a SEF attenuation when the attenuation is small. Additionally, the effect of finite gradient pulse widths (FGPW) is calculated. From the fractal derivative model, the signal attenuation including FGPW effect is exp-Df1∫0τKβ(t)dtα. The results obtained in this study are in good agreement with the results in literature. Several expressions that describe signal attenuation have not been reported and that can be of great importance for the PFG experiments. This EPSD equation method provides a new, simple path to calculate signal attenuation of PFG NMR experiments.

Fractional Pennes’ Bioheat Equation: Theoretical and Numerical Studies

Abstract In this work we provide a new mathematical model for the Pennes’ bioheat equation, assuming a fractional time derivative of single order. Alternative versions of the bioheat equation are studied and discussed, to take into account the temperature-dependent variability in the tissue perfusion, and both finite and infinite speed of heat propagation. The proposed bioheat model is solved numerically using an implicit finite difference scheme that we prove to be convergent and stable. The numerical method proposed can be applied to general reaction diffusion equations, with a variable diffusion coefficient. The results obtained with the single order fractional model, are compared with the original models that use classical derivatives.

COSMIC RAY TRANSPORT WITH MAGNETIC FOCUSING AND THE “TELEGRAPH” MODEL

Cosmic rays (CR), constrained by scattering on magnetic irregularities, are believed to propagate diffusively. But a well-known defect of diffusive approximation whereby some of the particles propagate unrealistically fast has directed interest towards an alternative CR transport model based on the "telegraph" equation. However, its derivations often lack rigor and transparency leading to inconsistent results. We apply the classic Chapman-Enskog method to the CR transport problem. We show that no "telegraph" (second order time derivative) term emerges in any order of a proper asymptotic expansion with systematically eliminated short time scales. Nevertheless, this term may formally be\emph{ converted} from the \emph{fourth} order hyper-diffusive term of the expansion. But, both the telegraph and hyperdiffusive terms may only be important for a short relaxation period associated with either strong pitch-angle anisotropy or spatial inhomogeneity of the initial CR distribution. Beyond this period the system evolves diffusively in both cases. The term conversion, that makes the telegraph and Chapman-Enskog approaches reasonably equivalent, is possible only after this relaxation period. During this period, the telegraph solution is argued to be unphysical. Unlike the hyperdiffusion correction, it is not uniformly valid and introduces implausible singular components to the solution. These dominate the solution during the relaxation period. As they are shown not to be inherent in the underlying scattering problem, we argue that the telegraph term is involuntarily acquired in an asymptotic reduction of the problem.

High accuracy solution of bi-directional wave propagation in continuum mechanics

Solution of partial differential equations by numerical method is strongly affected due to numerical errors, which are caused mainly by deviation of numerical dispersion relation from the physical dispersion relation. To quantify and control such errors and obtain high accuracy solutions, we consider a class of problems which involve second derivative of unknowns with respect to time. Here, we analyse numerical metrics such as the numerical group velocity, numerical phase speed and the numerical amplification factor for different methods in solving the model bi-directional wave equation (BDWE). Such equations can be solved directly, for example, by Runge–Kutta–Nyström (RKN) method. Alternatively, the governing equation can be converted to a set of first order in time equations and then using four-stage fourth order Runge–Kutta (RK4) method for time integration. Spatial discretisation considered are the classical second and fourth order central difference schemes, along with Lele's central compact scheme for evaluating second derivatives. In another version, we have used Lele's scheme for evaluating first derivatives twice to obtain the second derivative. As BDWE represents non-dissipative, non-dispersive dynamics, we also consider the canonical problem of linearised rotating shallow water equation (LRSWE) in a new formulation involving second order derivative in time, which represents dispersive waves along with a stationary mode. The computations of LRSWE with RK4 and RKN methods for temporal discretisation and Lele's compact schemes for spatial discretisation are compared with computations performed with RK4 method for time discretisation and staggered compact scheme (SCS) for spatial discretisation by treating it as a set of three equations as reported in Rajpoot et al. (2012) [1].

Cosmology of a Lorentz violating Galileon theory

We modify the scalar Einstein-aether theory by breaking the Lorentz invariance of a gravitational theory coupled to a Galileon type scalar field. This is done by introducing a Lagrange multiplier term into the action, thus ensuring that the gradient of the scalar field is time-like, with unit norm. The theory can also be considered as an extension to the mimetic dark matter theory, by adding some derivative self interactions to the action, which keeps the equation of motion at most second order in time derivatives. The cosmological implications of the model are discussed in detail. In particular, for pressure-less baryonic matter, we show that the universe experiences a late time acceleration. The cosmological implications of a special coupling between the scalar field and the trace of the energy-momentum tensor are also explored.