Order Time Derivatives Research Articles

Local and global existence of mild solutions of time-fractional Navier–Stokes system posed on the Heisenberg group

This paper is a development of the results and techniques of the two papers (Carvalho-Neto and Planas in J Differ Equ 259:2948–2980, 2015; Oka in J Math Anal Appl 473:382–407, 2019) for the aim of addressing the existence and uniqueness of local and global mild solutions, on the Heisenberg group $$\mathbb {H}^d$$ , of the time-fractional Navier–Stokes system with time derivative of order $$\alpha \in (0, 1)$$ . The proof relies on Schaefer’s fixed point theorem.

Data-driven identification of 2D Partial Differential Equations using extracted physical features

Many scientific phenomena are modeled by Partial Differential Equations (PDEs). The development of data gathering tools along with the advances in machine learning (ML) techniques have raised opportunities for data-driven identification of governing equations from experimentally observed data. We propose an ML method to discover the terms involved in the equation from two-dimensional spatiotemporal data. Robust and useful physical features are extracted from data samples to represent the specific behaviors imposed by each mathematical term in the equation. Compared to the previous models, this idea provides us with the ability to discover 2D equations with time derivatives of different orders, and also to identify new underlying physics on which the model has not been trained. Moreover, the model can work with small sets of low-resolution data while avoiding instability caused by numerical differentiations. The results indicate robustness of the features extracted based on prior knowledge in comparison to automatically detected features by a Three-dimensional Convolutional Neural Network (3D CNN) given the same amounts of data. Although particular PDEs are studied in this work, the idea of the proposed approach could be extended for reliable identification of various PDEs.

Effects of fractional order time derivative on the solitary wave dynamics of the generalized ZK–Burgers equation

The generalized ZK-Burgers equation has nonlinear characteristics of the governing equation and helps to understand nonlinear waves in many fluid problems of plasma mediums. This work aims to study the effects of fractional order of derivative on the wave solutions of this interesting equation via exact and analytical approximate solution techniques. The analytical approximate wave solutions are retrieved using homotopy analysis method taking the fractional derivative in Caputo’s sense, while exact solutions are reaped with the help of first integral method taking the fractional derivative in Jumarie’s modified Riemann–Liouville sense. The obtained solutions will be helpful to understand the dynamical framework of many nonlinear physical phenomena.

A rigorous derivation and energetics of a wave equation with fractional damping

We consider a linear system that consists of a linear wave equation on a horizontal hypersurface and a parabolic equation in the half space below. The model describes longitudinal elastic waves in organic monolayers at the water–air interface, which is an experimental setup that is relevant for understanding wave propagation in biological membranes. We study the scaling regime where the relevant horizontal length scale is much larger than the vertical length scale and provide a rigorous limit leading to a fractionally damped wave equation for the membrane. We provide the associated existence results via linear semigroup theory and show convergence of the solutions in the scaling limit. Moreover, based on the energy–dissipation structure for the full model, we derive a natural energy and a natural dissipation function for the fractionally damped wave equation with a time derivative of order 3/2.

Axisymmetric Slow Motion of a Porous Spherical Particle in a Viscous Fluid Using Time Fractional Navier–Stokes Equation

In this paper, we present the unsteady translational motion of a porous spherical particle in an incompressible viscous fluid. In this case, the modified Navier–Stokes equation with fractional order time derivative is used for conservation of momentum external to the particle whereas modified Brinkman equation with fractional order time derivative is used internal to the particle to govern the fluid flow. Stress jump condition for the tangential stress along with continuity of normal stress and continuity of velocity vectors is used at the porous–liquid interface. The integral Laplace transform technique is employed to solve the governing equations in fluid and porous regions. Numerical inversion code in MATLAB is used to obtain the solution of the problem in the physical domain. Drag force experienced by the particle is obtained. The numerical results have been discussed with the aid of graphs for some specific flows, namely damping oscillation, sine oscillation and sudden motion. Our result shows a significant contribution of the jump coefficient and the fractional order parameter to the drag force.

Modelling charge carrier transport with anomalous diffusion and heat conduction in amorphous semiconductors using fractional calculus

An ordered semiconductor has a crystalline lattice in which charge carriers move around by the Gaussian process of normal diffusion. The mean square displacement (MSD) of these charge carriers is proportional to time. On the contrary, the movement of carriers in a material with a non-crystalline structure such as amorphous semiconductors is considered to be non-Gaussian in nature. In this case, MSD is proportional to some power of time. Diffusion in this type of transport mechanism is classified as anomalous diffusion. The usual drift-diffusion equation (DDE) cannot adequately describe this process because it has non-Gaussian and dispersive transport mechanisms. Fractional calculus has been used to generalize the standard DDE to a time fractional equation in order to include the hereditary effects of the carrier transport. For power devices, the distribution and conduction of heat is the primary criteria considered when making a device. Therefore, an equation for heat conduction is added to the model for inclusion of variable temperature. The coupled system is solved using a Numerical scheme wherein Finite Difference method has been employed to discretize the Riemann - Liouville time derivative of order α and the space variable. The effects of different physical factors such as light intensity, heat and applied electric field are discussed with the help of graphical illustrations.

A new blow-up criterion of the strong solution to the quantum hydrodynamic model

In this paper, we establish a new blow-up criterion of the strong solution to the quantum hydrodynamic model. This blow-up criterion is different from the one established in our former paper (Wang and Guo, 2020). This blow-up criterion only depends on the gradient of the velocity, but also the first order time derivative of the square root of the density.

The time fractional Schrödinger equation with a nonlinearity of Hartree type

The aim of this work is to show existence, uniqueness and regularity properties of local in time mild solutions for the nonlinear space-time fractional Schrodinger equation (1.1), with a fractional time derivative of order $$\alpha \in (0,1),$$ and with a nonlinear term of Hartree type.

On the existence, uniqueness and stability results for time-fractional parabolic integrodifferential equations

We study the time-fractional parabolic integrodifferential equations with the Caputo fractional time derivative of order α∈(0,1) in a bounded convex polygonal domain in ℝd. We prove the existence and uniqueness of the solution using the eigenfunction expansion and establish a priori bounds for the solution under various regularity assumptions on the initial data and the source function. Our study includes the initial data in the spaces Ḣ2(Ω), H01(Ω) and L2(Ω) while the source function belongs to the class of Hölder continuous and bounded functions. It is shown that the solution of the corresponding homogeneous problem is infinitely differentiable with respect to time t when the initial function is an element of L2(Ω). Finally, we derive a general stability results for the solution of the homogeneous problem.

An Efficient Method for Solving Coupled Time Fractional Nonlinear Evolution Equations with Conformable Fractional Derivatives

In this article, an efficient reliable method, which is the residual power series method (RPSM), is used in order to investigate the approximate solutions of conformable time fractional nonlinear evolution equations with conformable derivatives under initial conditions. In particular, two types of equations are considered, which are time coupled diffusion-reaction equations (CD-REs) and MKdv equations coupled with conformable fractional time derivative of order α. The attitude of RPSM and the influence of different values of α are shown graphically.

On simulating the short and long memory of ergodic Markov and Non-Markov genetic diffusion processes on the long run

Abstract The theory of classical ergodic Markov genetic diffusion process, Feller genetic diffusion, and the genetic drift diffusion with and without selection and mutation for a single unlinked locus with two alleles, Kimura models, is studied. These genetic problems are numerically investigated on the long run. The approximates solutions of these models are described as the conditional probabilities to find the specific gene with frequencies between zero and one at generation T . As a surprise, we notice that the summation of these approximate solutions lose their unity rapidly as the number of generations increase than five, i.e. T > > 5 . The earlier biologists did due the reason to the migration, immigration and death of individuals. Till now, there is no mathematical proof to this phenomenon. In this paper, we solve this problem by extending the first order time derivative to the time fractional derivative, to study the effect of the memory on these models. This extension ensures that the summations of the approximate solutions, i.e. of the Non-Markov cases, are one at any number of generations. The time evolution of the approximate solutions are simulated and compared with other published papers, for different values of generations and for the Markov and Non-Markov cases with different values of the fractional order β . The convergence of the approximate descrete solutions and the reversibility property of these stochastic processes for both the Markov cases and Non-Markov cases are also numerically simulated and discussed.

An Adaptive Analytic Continuation Method for Computing the Perturbed Two-Body Problem State Transition Matrix

In this work, the Taylor series based technique, Analytic Continuation is implemented to develop a method for the computation of the gravity and drag perturbed State Transition Matrix (STM) incorporating adaptive time steps and expansion order. Analytic Continuation has been developed for the two-body problem based on two scalar variables f and gp and their higher order time derivatives using Leibniz rule. The method has been proven to be very precise and efficient in trajectory propagation. The method is expanded to include the computation of the STM for the perturbed two-body problem. Leibniz product rule is used to compute the partials for the recursive formulas and an arbitrary order Taylor series is used to compute the STM. Four types of orbits, LEO, MEO, GTO and HEO, are presented and the simulations are run for 10 orbit periods. The accuracy of the STM is evaluated via RMS error for the unperturbed cases, symplectic check for the gravity perturbed cases and error propagation for the gravity and drag perturbed orbits. The results are compared against analytical and high order numerical solvers (ODE45, ODE113 and ODE87) in terms of accuracy. The results show that the method maintains double-precision accuracy for all test cases and 1-2 orders of magnitude improvement in linear prediction results compared to ODE87. The present approach is simple, adaptive and can readily be expanded to compute the full spherical harmonics gravity perturbations as well as the higher order state transition tensors.

Time-Fractional Allen–Cahn Equations: Analysis and Numerical Methods

In this work, we consider a time-fractional Allen–Cahn equation, where the conventional first order time derivative is replaced by a Caputo fractional derivative with order $$\alpha \in (0,1)$$ . First, the well-posedness and (limited) smoothing property are studied, by using the maximal $$L^p$$ regularity of fractional evolution equations and the fractional Grönwall’s inequality. We also show the maximum principle like their conventional local-in-time counterpart, that is, the time-fractional equation preserves the property that the solution only takes value between the wells of the double-well potential when the initial data does the same. Second, after discretizing the fractional derivative by backward Euler convolution quadrature, we develop several unconditionally solvable and stable time stepping schemes, such as a convex splitting scheme, a weighted convex splitting scheme and a linear weighted stabilized scheme. Meanwhile, we study the discrete energy dissipation property (in a weighted average sense), which is important for gradient flow type models, for the two weighted schemes. In addition, we prove the fractional energy dissipation law for the gradient flow associated with a convex free energy. Finally, using a discrete version of fractional Grönwall’s inequality and maximal $$\ell ^p$$ regularity, we prove that the convergence rates of those time-stepping schemes are $$O(\tau ^\alpha )$$ without any extra regularity assumption on the solution. We also present extensive numerical results to support our theoretical findings and to offer new insight on the time-fractional Allen–Cahn dynamics.

Thermodynamic constraints on matter creation models

Entropy is a fundamental concept from Thermodynamics and it can be used to study models on context of Creation Cold Dark Matter (CCDM). From conditions on the first (dot{S}ge 0) (throughout the present work we will use dots to indicate time derivatives and dashes to indicate derivatives with respect to scale factor) and second order (ddot{S}<0) time derivatives of total entropy in the initial expansion of Sitter through the radiation and matter eras until the end of Sitter expansion, it is possible to estimate the intervals of parameters. The total entropy (S_{t}) is calculated as sum of the entropy at all eras (S_{gamma } and S_{m}) plus the entropy of the event horizon (S_h). This term derives from the Holographic Principle where it suggests that all information is contained on the observable horizon. The main feature of this method for these models are that thermodynamic equilibrium is reached in a final de Sitter era. Total entropy of the universe is calculated with three terms: apparent horizon (S_{h}), entropy of matter (S_{m}) and entropy of radiation (S_{gamma }). This analysis allows to estimate intervals of parameters of CCDM models.

Limitations on nonlinear fractional calculus models used in biomedical acoustics

Fractional calculus models used for biomedical ultrasound are associated with attenuation proportional to ω y, where y is typically in the range 1 < y < 2. To determine whether the attenuation and accompanying dispersion are sufficient to stabilize shock formation, the models are formulated as a Burgers equation with the traditional loss term replaced by a fractional derivative of order y. For y < 1 the resulting equation predicts unphysical solutions beyond the shock-formation distance. The second example pertains to nonlinear Lucassen interface waves, a model equation for which has been proposed to describe mechanical perturbations that accompany the transmission of nerve impulses. Linear Lucassen waves are defined by a second-order space derivative and a fractional time derivative of order 3/2, which falls between order 2 in the wave equation and order 1 in the diffusion equation. The resulting attenuation is proportional to ω3/4, and the corresponding nonlinear “fractional diffusive waves,” while strongly attenuated on the scale of a wavelength, may be lacking essential physics beyond the predicted shock-formation distance. Calculations are presented that determine wave amplitudes and propagation distances for which these two fractional calculus models may be of questionable physical significance due to nonlinearity. [B.E.S. is supported by the ARL:UT McKinney Fellowship in Acoustics.]

Chaotic and spatiotemporal oscillations in fractional reaction-diffusion system

• Fractional derivative approximation in 2D. • Linear stability analysis for temporal system. • Alternating direction implicit finite difference method. • Numerical simulations. • Chaotic and spatiotemporal phenomenon. This paper focuses on the design and analysis of an efficient numerical method based on the novel implicit finite difference scheme for the solution of the dynamics of reaction-diffusion models. The work replaces the integer first order derivative in time with the Caputo fractional derivative operator. The dynamics of activator-inhibitor as encountered in chemistry, physics and engineering processes, and predator-prey models are two cases addresses in this study. In order to provide a good guidelines on the correct choice of parameters for the numerical simulation of full fractional reaction-diffusion system, its linear stability analysis is also examined. The resulting scheme is applied to solve cross-diffusion problem in two-dimensions. In the experimental results, a number of spatiotemporal and chaotic patterns that are related to Turing pattern are observed. It was discovered in the simulation experiments that the species predator-prey model distribute in almost same fashion, while that of the activator-inhibitor dynamics behaved differently regardless of the value of fractional order chosen.

Time Fractional Fisher-KPP and Fitzhugh-Nagumo Equations.

A standard reaction–diffusion equation consists of two additive terms, a diffusion term and a reaction rate term. The latter term is obtained directly from a reaction rate equation which is itself derived from known reaction kinetics, together with modelling assumptions such as the law of mass action for well-mixed systems. In formulating a reaction–subdiffusion equation, it is not sufficient to know the reaction rate equation. It is also necessary to know details of the reaction kinetics, even in well-mixed systems where reactions are not diffusion limited. This is because, at a fundamental level, birth and death processes need to be dealt with differently in subdiffusive environments. While there has been some discussion of this in the published literature, few examples have been provided, and there are still very many papers being published with Caputo fractional time derivatives simply replacing first order time derivatives in reaction–diffusion equations. In this paper, we formulate clear examples of reaction–subdiffusion systems, based on; equal birth and death rate dynamics, Fisher–Kolmogorov, Petrovsky and Piskunov (Fisher–KPP) equation dynamics, and Fitzhugh–Nagumo equation dynamics. These examples illustrate how to incorporate considerations of reaction kinetics into fractional reaction–diffusion equations. We also show how the dynamics of a system with birth rates and death rates cancelling, in an otherwise subdiffusive environment, are governed by a mass-conserving tempered time fractional diffusion equation that is subdiffusive for short times but standard diffusion for long times.

Modelling of Chaotic Processes with Caputo Fractional Order Derivative.

Chaotic dynamical systems are studied in this paper. In the models, integer order time derivatives are replaced with the Caputo fractional order counterparts. A Chebyshev spectral method is presented for the numerical approximation. In each of the systems considered, linear stability analysis is established. A range of chaotic behaviours are obtained at the instances of fractional power which show the evolution of the species in time and space.

A study of fractional order dual-phase-lag bioheat transfer model.

In this study, we have established a space-time fractional DPL bioheat transfer model in the presence of temperature-dependent metabolic and space-time dependent electromagnetic heat sources. Applying the Legendre wavelet collocation method, the fractional order partial differential equation is reduced into the system of algebraic equations, which has been solved using the Newton iteration method. The error bound as well as stability analysis and numerical scheme validation are provided. The time to achieve for the position of hyperthermia is discussed in three cases: the DPL model, the time-fractional DPL model, and the space-time-fractional DPL model. The effect of variability of time and space fractional derivative orders (α and β), transmitted power (P) and lagging times on the temperature profile in biological tissue at a different time are discussed in detail. We conclude that a suitable value of α, β, τT, τq, and P provides a desirable temperature at a particular time in thermal therapies. Such knowledge will be very useful in the clinical therapeutic application.

Asymptotics of Fundamental Solutions for Time Fractional Equations with Convolution Kernels

Abstract The paper deals with the large time asymptotic of the fundamental solution for a time fractional evolution equation with a convolution type operator. In this equation we use a Caputo time derivative of order α ∈ (0, 1), and assume that the convolution kernel of the spatial operator is symmetric, integrable and shows a super-exponential decay at infinity. Under these assumptions we describe the point-wise asymptotic behavior of the fundamental solution in all space-time regions.