Time-fractional Derivative Of Order Research Articles

The Cauchy problem for fractional Navier–Stokes equations in Sobolev spaces

In this paper we use the tools from harmonic analysis to study the Cauchy problem for Navier–Stokes equations with time-fractional derivative of order α ∈ (0, 1), which can be used to simulate the anomalous diffusion in fractal media. Two main results concerning the local existence of solutions for the given problem in Sobolev space are addressed.

Time fractional Cattaneo-Christov anomalous diffusion in comb frame with finite length of fingers

Fractional Cattaneo-Christov flux model is used in analyzing the anomalous diffusion in comb frame subject to finite length of fingers. Formulated governing equation contains Dirac delta function and mixed partial derivatives. Solutions are obtained by numerical discretization method where the time fractional derivative of order α in (0, 1] is approximated by L1-scheme. The correctness of numerical method is verified by introducing a source item to construct an exact solution. Results show that, when there exists relaxation, the particles distribution displays a parabolic feature with the spatial evolution at small times while a hyperbolic feature appears with a larger relaxation parameter at large times, the particles distribution and the total number of particles on x axis are oscillating with the temporal evolution. Moreover, the effects of involved parameters on dynamic characteristics are graphically analyzed and discussed in detail.

Unsteady translational motion of a slip sphere in a viscous fluid using the fractional Navier-Stokes equation

In this paper, we investigate the translational motion of a slip sphere with time-dependent velocity in an incompressible viscous fluid. The modified Navier-Stokes equation with fractional order time derivative is used. The linear slip boundary condition is applied on the spherical boundary. The integral Laplace transform technique is employed to solve the problem. The solution in the physical domain is obtained analytically by inverting the Laplace transform using the complex inversion formula together with contour integration. An exact formula for the drag force exerted by the fluid on the spherical object is deduced. This formula is applied to some flows, namely damping oscillation, sine oscillation and sudden motion. The numerical results showed that the order of the fractional derivative contributes considerably to the drag force. The increase in this parameter resulted in an increase in the drag force. In addition, the values of the drag force increased with the increase in the slip parameter.

The Time Fractional Schr\xf6dinger Equation on Hilbert Space

We study the linear fractional Schrödinger equation on a Hilbert space, with a fractional time derivative of order 0<alpha <1, and a self-adjoint generator A. Using the spectral theorem we prove existence and uniqueness of strong solutions, and we show that the solutions are governed by an operator solution family { U_{alpha }(t)}_{tge 0}. Moreover, we prove that the solution family U_{alpha }(t) converges strongly to the family of unitary operators e^{-itA}, as alpha approaches to 1.

Electromagnetic Interpretation of Fractional-Order Elements

Fractional circuits have attracted extensive attention of scholars and researchers for their superior performance and potential applications. Recently, the fundamentals of the conventional circuit theory were extended to include the new generalized elements and fractional-order elements. As is known to all, circuit theory is a limiting special case of electromagnetic field theory and the characterization of classical circuit elements can be given an elegant electromagnetic interpretation. In this paper, considering fractional-order time derivatives, an electromagnetic field interpretation of fractional-order elements: fractional-order inductor, fractional-order capacitor and fractional-order mutual inductor is presented, in terms of a quasi-static expansion of the fractional Maxwell’s equations. It shows that fractional-order elements can also be interpreted as a fractional electromagnetic system. As the element order equals to 1, the interpretation of fractional-order elements matches that of the classical circuit elements: L, C, and mutual inductor, respectively.

Fractional nonlinear dynamics of DNA breathing

The classical Lagrangian formulation for the nonlinear dynamics of a homogeneous Peyrard–Bishop DNA molecular chain is reviewed and extended to include coordinates with time derivative of fractional order γ (0 < 2γ < 2), which can be viewed as memory effect. We obtain the equations of motion depending on γ. The analytical procedure for obtaining nonlinear waves solutions is performed through the application of a powerful fractional perturbation technique. The results show that both the amplitude and the velocity of waves increase when γ decreases. Accordingly, for low values of γ, the system exhibits highly localized waves with high amplitude and velocity. The numerical results agree with the theoretical ones and show that the system can support fractional breather-like modes.

Fractional formalism to DNA chain and impact of the fractional order on breather dynamics

We have investigated the impact of the fractional order derivative on the dynamics of modulated waves of a homogeneous DNA chain that is based on site-dependent finite stacking and pairing enthalpies. We have reformulated the classical Lagrangian of the system by including the coordinates depending on the Riemann-Liouville time derivative of fractional order γ. From the Lagrange equation, we derived the fractional nonlinear equation of motion. We obtained the fractional breather as solutions by means of a fractional perturbation technique. The impact of the fractional order is investigated and we showed that depending on the values of γ, there are three types of waves that propagate in DNA. We have static breathers, breathers of small amplitude and high velocity, and breathers of high amplitude and small velocity.

Heat conduction with fractional Cattaneo–Christov upper-convective derivative flux model

Fractional order derivatives are global operators for which the time fractional order derivative possesses memory character while the space ones reflects non-local behavior. In this paper, a new time and space fractional Cattaneo–Christov upper-convective derivative flux heat conduction model is suggested where the space fractional derivative is characterized by the weight coefficient of forward versus backward transition probability. Governing equation is formulated and solved by L1-approximation and shifted Grünwald formula. Results show that the fractional parameters, time and location parameters, relaxation parameter, weight coefficient and convection velocity have remarkable impacts on heat transfer characteristics. Temperature distribution profiles are monotonically decreasing in a concave form versus time fractional parameter with existing of relaxation parameter, while in a convex form with space fractional parameter evolution under three special conditions, i.e., the right region, the larger weight coefficient (γ ≥ 0.5) and smaller convection parameter u.

Solution for fractional distributed optimal control problem by hybrid meshless method

We use a hybrid local meshless method to solve the distributed optimal control problem of a system governed by parabolic partial differential equations with Caputo fractional time derivatives of order α ∈ (0, 1]. The presented meshless method is based on the linear combination of moving least squares and radial basis functions in the same compact support, this method will change between interpolation and approximation. The aim of this paper is to solve the system of coupled fractional partial differential equations, with necessary and sufficient conditions, for fractional distributed optimal control problems using a combination of moving least squares and radial basis functions. To keep matters simple, the problem has been considered in the one-dimensional case, however the techniques can be employed for both the two- and three-dimensional cases. Several test problems are employed and results of numerical experiments are presented. The obtained results confirm the acceptable accuracy of the proposed method.

On the regional gradient observability of time fractional diffusion processes

This paper for the first time addresses the concepts of regional gradient observability for the Riemann–Liouville time fractional order diffusion system in an interested subregion of the whole domain without the knowledge of the initial vector and its gradient. The Riemann–Liouville time fractional order diffusion system which replaces the first order time derivative of normal diffusion system by a Riemann–Liouville time fractional order derivative of order α∈(0,1] is used to well characterize those anomalous sub-diffusion processes. The characterizations of the strategic sensors when the system under consideration is regional gradient observability are explored. We then describe an approach leading to the reconstruction of the initial gradient in the considered subregion with zero residual gradient vector. At last, to illustrate the effectiveness of our results, we present several application examples where the sensors are zone, pointwise or filament ones.

High-order local discontinuous Galerkin method combined with WSGD-approximation for a fractional subdiffusion equation

In this paper, a high-order local discontinuous Galerkin (LDG) method combined with weighted and shifted Grünwald difference (WSGD) approximation is developed and discussed for a Caputo time-fractional subdiffusion equation. The time fractional derivative of order α, 0<α<1, is approximated by a third-order method based on the idea of WSGD operator, while the spatial operator is approximated by the LDG method. Some useful lemmas are first introduced and proved, then the analysis of stability and optimal error estimate O(Δt3+hk+1) are obtained for the LDG method. Extensive numerical results using Pk,k=0,1,2,3, elements are presented to validate our theoretical analysis.

Weak solutions of the time-fractional Navier–Stokes equations and optimal control

In this paper, we deal with the Navier–Stokes equations with the time-fractional derivative of order α∈(0,1), which can be used to simulate anomalous diffusion in fractal media. We firstly give the concept of the weak solutions and establish the existence criterion of weak solutions by means of Galerkin approximations in the case that the dimension n≤4. Moreover, a complete proof of the uniqueness is given when n=2. At last we give a sufficient condition of optimal control pairs.

Boundary feedback stabilisation for the time fractional‐order anomalous diffusion system

In this study, the authors attempt to explore the boundary feedback stabilisation for an unstable heat process described by fractional-order partial differential equation (PDE), where the first-order time derivative of normal reaction–diffusion equation is replaced by a Caputo time fractional derivative of order α∈(0, 1]. By designing an invertible coordinate transformation, the system under consideration is converted into a Mittag–Leffler stability linear system and the boundary stabilisation problem is transformed into a problem of solving a linear hyperbolic PDE. It is worth mentioning that with the help of this invertible coordinate transformation, they can explicitly obtain the closed-loop solutions of the original problem. The output feedback problem with both anti-collocated and collocated actuator/sensor pairs in one-dimensional domain is also presented. A numerical example is given to test the effectiveness of the authors' results.

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.

On the time-fractional Navier–Stokes equations

This paper is concerned with the Navier–Stokes equations with time-fractional derivative of order α∈(0,1). This type of equations can be used to simulate anomalous diffusion in fractal media. We establish the existence and uniqueness of local and global mild solutions in Hβ,q. Meanwhile, we also give local mild solutions in Jq. Moreover, we prove the existence and regularity of classical solutions for such equations in Jq.

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.

Fractional Cauchy problems on compact manifolds

ABSTRACTWe investigate anomalous diffusion on compact Riemannian manifolds, modeled by time-changed Brownian motions. These stochastic processes are governed by equations involving the Laplace–Beltrami operator and a time-fractional derivative of order β ∈ (0, 1). We also consider time dependent random fields that can be viewed as random fields on randomly varying manifolds.

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.

Hopf lemma for the fractional diffusion operator and its application to a fractional free-boundary problem

We consider a one-dimensional moving-boundary problem for the time-fractional diffusion equation, where the time-fractional derivative of order α∈(0,1) is taken in the Caputo sense. A generalization of the Hopf lemma is proved and then used to prove a monotonicity property for the free-boundary when a fractional free-boundary Stefan problem is investigated.

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.