Time-fractional Derivative Of Order Research Articles

Solutions of fractional gas dynamics equation by a new technique

In this paper, a novel technique is formed to obtain the solution of a fractional gas dynamics equation. Some reproducing kernel Hilbert spaces are defined. Reproducing kernel functions of these spaces have been found. Some numerical examples are shown to confirm the efficiency of the reproducing kernel Hilbert space method. The accurate pulchritude of the paper is arisen in its strong implementation of Caputo fractional order time derivative on the classical equations with the success of the highly accurate solutions by the series solutions. Reproducing kernel Hilbert space method is actually capable of reducing the size of the numerical work. Numerical results for different particular cases of the equations are given in the numerical section.

Modeling nonlinear fractional diffusion-wave equations with Burgers-type evolution equations

Fractional partial differential equations (FPDEs) with a time derivative of fractional order are used to describe wave motion in complex viscoelastic media with non-traditional equations of motion. Kappler et al. [Phys. Rev. Fluids 2, 114804 (2017)] derived a fractional diffusion-wave equation for a nonlinear Lucassen wave propagating along an elastic layer coupled to a viscous substrate. The fractional time derivative of order 3/2 in the linear form of this equation lies midway between order 1 for a diffusion process described by a parabolic equation, and order 2 for the traditional hyperbolic wave equation. The inclusion of nonlinear elasticity tends to inhibit purely progressive wave motion that is associated with classical nonlinear plane waves in fluids and solids, and which is described accurately by parabolic-like, Burgers-type evolution equations. In this work, a general FPDE is analyzed in a parameter space consisting of varying nonlinearity and time fractional orders. The focus is on conditions under which the FPDE can be modeled accurately with a Burgers-type evolution equation for progressive wave motion. The method of lines in combination with a general Runge-Kutta method for forward integration is used for numerical analysis. [B.E.S. is supported by the ARL:UT McKinney Fellowship in Acoustics.]Fractional partial differential equations (FPDEs) with a time derivative of fractional order are used to describe wave motion in complex viscoelastic media with non-traditional equations of motion. Kappler et al. [Phys. Rev. Fluids 2, 114804 (2017)] derived a fractional diffusion-wave equation for a nonlinear Lucassen wave propagating along an elastic layer coupled to a viscous substrate. The fractional time derivative of order 3/2 in the linear form of this equation lies midway between order 1 for a diffusion process described by a parabolic equation, and order 2 for the traditional hyperbolic wave equation. The inclusion of nonlinear elasticity tends to inhibit purely progressive wave motion that is associated with classical nonlinear plane waves in fluids and solids, and which is described accurately by parabolic-like, Burgers-type evolution equations. In this work, a general FPDE is analyzed in a parameter space consisting of varying nonlinearity and time fractional orders. The focus is on conditions ...

A Collocation Method for the Numerical Solution of Nonlinear Fractional Dynamical Systems

Fractional differential problems are widely used in applied sciences. For this reason, there is a great interest in the construction of efficient numerical methods to approximate their solution. The aim of this paper is to describe in detail a collocation method suitable to approximate the solution of dynamical systems with time derivative of fractional order. We will highlight all the steps necessary to implement the corresponding algorithm and we will use it to solve some test problems. Two Mathematica Notebooks that can be used to solve these test problems are provided.

On skewed grid point iterative method for solving 2D hyperbolic telegraph fractional differential equation

This paper presents the development of a new iterative method for solving the two-dimensional hyperbolic telegraph fractional differential equation (2D-HTFDE) which is central to the mathematical modeling of transmission line satisfying certain relationship between voltage and current waves in specific distance and time. This equation can be obtained from the classical two-dimensional hyperbolic telegraph partial differential equation by replacing the first and second order time derivatives by the Caputo time fractional derivatives of order 2α and α respectively, with 1/2< alpha < 1 . The iterative scheme, called the fractional skewed grid Crank–Nicolson FSkG(C-N), is derived from finite difference approximations discretized on a skewed grid rotated clockwise 450 from the standard grid. The skewed finite difference scheme combined with Crank–Nicolson discretization formula will be shown to be unconditionally stable and convergent by the Fourier analysis. The developed FSkG(C-N) scheme will be compared with the fractional Crank–Nicolson scheme on the standard grid to confirm the effectiveness of the proposed scheme in terms of computational complexities and computing efforts. It will be shown that the new proposed scheme demonstrates more superior capabilities in terms of the number of iterations and CPU timings compared to its counterpart on the standard grid but with the same order of accuracy.

Some results for the large time behavior of Hamilton-Jacobi equations with Caputo time derivative

We obtain some Holder regularity estimates for an Hamilton-Jacobi with fractional time derivative of order α ∈ (0, 1) cast by a Caputo derivative. The Holder seminorms are independent of time, which allows to investigate the large time behavior of the solutions. We focus on the Namah-Roquejoffre setting whose typical example is the Eikonal equation. Contrary to the classical time derivative case α = 1, the convergence of the solution on the so-called projected Aubry set, which is an important step to catch the large time behavior, is not straightforward. Indeed, a function with nonpositive Caputo derivative for all time does not necessarily converge; we provide such a counterexample. However, we establish partial results of convergence under some geometrical assumptions.

Numerical Analysis of Initial-Boundary Value Problem for a Sobolev-Type Equation with a Fractional-Order Time Derivative

The paper is concerned with initial-boundary value problems for a Sobolev-type equation with a Gerasimov–Caputo fractional derivative with memory effect. A priori estimates of the solutions are obtained in the differential and difference forms, which imply their uniqueness and stability with respect to the initial data and the right-hand side, as well as the convergence of the solution of the difference problem to the solution of the differential problem.

Error analysis of the L1 method on graded and uniform meshes for a fractional-derivative problem in two and three dimensions

An initial-boundary value problem with a Caputo time derivative of fractional order $\alpha\in(0,1)$ is considered, solutions of which typically exhibit a singular behaviour at an initial time. For this problem, we give a simple framework for the analysis of the error of L1-type discretizations on graded and uniform temporal meshes in the $L_\infty$ and $L_2$ norms. This framework is employed in the analysis of both finite difference and finite element spatial discretiztions. Our theoretical findings are illustrated by numerical experiments.

Anomalous Grain Boundary Diffusion: Fractional Calculus Approach

Grain boundary (GB) diffusion in engineering materials at elevated temperatures often determines the evolution of microstructure, phase transformations, and certain regimes of plastic deformation and fracture. Interpreting experimental data with the use of the classical Fisher model sometimes encounters contradictions that can be related to violation of Fick’s law. Here, we generalize the Fisher model to the case of non-Fickian (anomalous) diffusion ubiquitous in disordered materials. The process is formulated in terms of the subdiffusion equations with time-fractional derivatives of order α∈(0,1] and β∈(0,1] for grain volume and GB, respectively. It is shown that propagation along GB for the case of a localized instantaneous source and weak localization in GB (β&gt;α/2) is approximately described by distributed-order subdiffusion with exponents α/2 and β. The mean square displacement is calculated with the use of the alternating renewal process model. The tail of the impurity concentration profiles along the z axis is approximately described by the dependence ∝exp(-Az6/5) for all 0&lt;α≤1, as in the case of normal GB diffusion, so the 6/5-law itself can serve as an identifier of a more general phenomenon, namely, anomalous GB diffusion.

Application of Caputo-Fabrizio Fractional Order Derivative (NFDt) in Simulating the MHD Flow of the Third Grade non-Newtonian Fluid in the Porous Artery

In this paper, the third grade non-Newtonian MHD blood flow in the porous arteries subjected to the periodic pressure gradient was studied using the Caputo-Fabrizio (NFDt ) time fractional order derivative. The time fractional model was solved by taking the Laplace and the finite Hankel transforms. Results were compared with those reported in the previous studies and good agreement was found. The Mathematica software was used to simulate the velocity profile and the Bessel functions with zero order and first order of first kind. The correlations between the flow velocity and the third grade non-Newtonian fluid parameter, the magnetic field and the porosity were negative. Nevertheless, the flow velocity increased with respect to the Womersely number.

Mild solutions to the time fractional Navier-Stokes delay differential inclusions

In this paper, we study a Navier-Stokes delay differential inclusion with time fractional derivative of order \begin{document} $\alpha\in(0,1)$ \end{document} . We first prove the local and global existence, decay and regularity properties of mild solutions when the initial data belongs to \begin{document} $C([-h,0];D(A_r^\varepsilon))$ \end{document} . The fractional resolvent operator theory and some techniques of measure of noncompactness are successfully applied to obtain the results.

Regularity of Solutions to Space–Time Fractional Wave Equations: A PDE Approach

Abstract We consider an evolution equation involving the fractional powers, of order s ∈ (0, 1), of a symmetric and uniformly elliptic second order operator and Caputo fractional time derivative of order γ ∈ (1, 2]. Since it has been shown useful for the design of numerical techniques for related problems, we also consider a quasi–stationary elliptic problem that comes from the realization of the spatial fractional diffusion as the Dirichlet-to-Neumann map for a nonuniformly elliptic problem posed on a semi–infinite cylinder. We provide existence and uniqueness results together with energy estimates for both problems. In addition, we derive regularity estimates both in time and space; the time–regularity results show that the usual assumptions made in the numerical analysis literature are problematic.

Subordination in a Class of Generalized Time-Fractional Diffusion-Wave Equations

Abstract Motivated by recently proposed generalizations of the diffusion-wave equation with the Caputo time fractional derivative of order α ∈ (1, 2), in the present survey paper a class of generalized time-fractional diffusion-wave equations is introduced. Its definition is based on the subordination principle for Volterra integral equations and involves the notion of complete Bernstein function. Various members of this class are surveyed, including the distributed-order time-fractional diffusion-wave equation and equations governing wave propagation in viscoelastic media with completely monotone relaxation moduli.

A fractional spline collocation-Galerkin method for the time-fractional diffusion equation

AbstractThe aim of this paper is to numerically solve a diffusion differential problem having time derivative of fractional order. To this end we propose a collocation-Galerkin method that uses the fractional splines as approximating functions. The main advantage is in that the derivatives of integer and fractional order of the fractional splines can be expressed in a closed form that involves just the generalized finite difference operator. This allows us to construct an accurate and efficient numerical method. Several numerical tests showing the effectiveness of the proposed method are presented.

A class of fractional differential hemivariational inequalities with application to contact problem

In this paper, we study a class of generalized differential hemivariational inequalities of parabolic type involving the time fractional order derivative operator in Banach spaces. We use the Rothe method combined with surjectivity of multivalued pseudomonotone operators and properties of the Clarke generalized gradient to establish existence of solution to the abstract inequality. As an illustrative application, a frictional quasistatic contact problem for viscoelastic materials with adhesion is investigated, in which the friction and contact conditions are described by the Clarke generalized gradient of nonconvex and nonsmooth functionals, and the constitutive relation is modeled by the fractional Kelvin–Voigt law.

Mittag-Leffler function and fractional differential equations

Abstract We adopt a procedure of operational-umbral type to solve the (1 + 1)-dimensional fractional Fokker-Planck equation in which time fractional derivative of order α (0 &lt; α &lt; 1) is in the Riemann-Liouville sense. The technique we propose merges well documented operational methods to solve ordinary FP equation and a redefinition of the time by means of an umbral operator. We show that the proposed method allows significant progress including the handling of operator ordering.

A General fractional-order heat transfer model for Photovoltaic/Thermal hybrid systems and its observer design

This paper presents the results of heat transfer process modelling for Photovoltaic/Thermal PV/T hybrid system in state-space. The heat transfer process in solid materials can be described by an integer-order partial differential equation. However, in Photovoltaic (PV/T) hybrid systems, which is characterized by heterogeneous media due to the multilayers that make up the system, it can be described by a fractional-order partial differential equation. The major objectives of this study were to establish a new mathematical model of the Photovoltaic/Thermal (PV/T) hybrid systems and their fractional-order observers in time domain. This model has been constructed in the context of a new consideration of heat conduction with a time fractional-order derivative. On the other hand, after obtained the general model of the system in state-space, the existence conditions of the fractional-order observer of such systems are given. Then, a necessary and sufficient condition for the asymptotic stability of the estimation error is given in a Linear Matrix Inequality (LMI) formulation.

An efficient Galerkin spectral method for two-dimensional fractional nonlinear reaction–diffusion-wave equation

The aim of this paper is to develop an efficient numerical treatment for the two-dimensional fractional nonlinear reaction–diffusion-wave equation with the time-fractional derivative of order α (1<α<2). For this purpose, we employ the alternating direction implicit (ADI) method based on the Crank–Nicolson scheme for the time stepping, while we apply the Legendre–Galerkin spectral method for the space discretization. The stability and convergence analysis are rigorously set up. In addition, the proposed method is extended to solve the time-fractional Klein–Gordon and sine-Gordon models. Numerical experiments are included, which verifies the theoretical predictions.

An Unconditionally Stable and High-Order Convergent Difference Scheme for Stokes’ First Problem for a Heated Generalized Second Grade Fluid with Fractional Derivative

AbstractThis article is intended to fill in the blank of the numerical schemes with second-order convergence accuracy in time for nonlinear Stokes’ first problem for a heated generalized second grade fluid with fractional derivative. A linearized difference scheme is proposed. The time fractional-order derivative is discretized by second-order shifted and weighted Gr¨unwald-Letnikov difference operator. The convergence accuracy in space is improved by performing the average operator. The presented numerical method is unconditionally stable with the global convergence order of in maximum norm, where τ and h are the step sizes in time and space, respectively. Finally, numerical examples are carried out to verify the theoretical results, showing that our scheme is efficient indeed.

Numerical simulation of time fractional dual-phase-lag model of heat transfer within skin tissue during thermal therapy

In this paper, we investigated the thermal behavior in living biological tissues using time fractional dual-phase-lag bioheat transfer (DPLBHT) model subjected to Dirichelt boundary condition in presence of metabolic and electromagnetic heat sources during thermal therapy. We solved this bioheat transfer model using finite element Legendre wavelet Galerkin method (FELWGM) with help of block pulse function in sense of Caputo fractional order derivative. We compared the obtained results from FELWGM and exact method in a specific case, and found a high accuracy. Results are interpreted in the form of standard and anomalous cases for taking different order of time fractional DPLBHT model. The time to achieve hyperthermia position is discussed in both cases as standard and time fractional order derivative. The success of thermal therapy in the treatment of metastatic cancerous cell depends on time fractional order derivative to precise prediction and control of temperature. The effect of variability of parameters such as time fractional derivative, lagging times, blood perfusion coefficient, metabolic heat source and transmitted power on dimensionless temperature distribution in skin tissue is discussed in detail. The physiological parameters has been estimated, corresponding to the value of fractional order derivative for hyperthermia treatment therapy.

Determination of Two Unknown Thermal Coefficients Through an Inverse One-Phase Fractional Stefan Problem

Abstract We consider a semi-infinite one-dimensional phase-change material with two unknown constant thermal coefficients among the latent heat per unit mass, the specific heat, the mass density and the thermal conductivity. Aiming at the determination of them, we consider an inverse one-phase Stefan problem with an over-specified condition at the fixed boundary and a known evolution for the moving boundary. We assume that it is given by a sharp front and we consider a time fractional derivative of order α (0 &lt; α &lt; 1) in the Caputo sense to represent the temporal evolution of the temperature as well as the moving boundary. This might be interpreted as the consideration of latent-heat memory effects in the development of the phase-change process. According to the choice of the unknown thermal coefficients, six inverse fractional Stefan problems arise. For each of them, we determine necessary and sufficient conditions on data to obtain the existence and uniqueness of a solution of similarity type. Moreover, we present explicit expressions for the temperature and the unknown thermal coefficients. Finally, we show that the results for the classical statement of this problem, associated with α = 1, are obtained through the fractional model when α → 1—.