Space-time Fractional Derivatives Research Articles

Some nonexistence results for space–time fractional Schrödinger equations without gauge invariance

In this paper, we consider the Cauchy problem in \(\mathbb {R}^N\), \(N\ge 1\), for semi-linear Schrödinger equations with space–time fractional derivatives. We discuss the nonexistence of global \(L^1\) or \(L^2\) weak solutions in the subcritical and critical cases under some conditions on the initial data and the nonlinear term. Furthermore, the nonexistence of local \(L^1\) or \(L^2\) weak solutions in the supercritical case are studied.

Classification of All Single Traveling Wave Solutions of ( 3 + 1 )-Dimensional Jimbo-Miwa Equation with Space-Time Fractional Derivative

In this paper, the complete discrimination system method is used to construct the single traveling wave solutions for the ( 3 + 1 )-dimensional Jimbo-Miwa equations with space-time fractional derivative. As a result, we get the exact traveling wave solutions of the ( 3 + 1 )-dimensional Jimbo-Miwa equation with space-time fractional derivative, which include rational function solutions, Jacobian elliptic function solutions, hyperbolic function solutions, and trigonometric function solutions. Some graphical representations of the solutions are also provided. Finally, the obtained solution is compared with the existing literature.

Oscillatory states and patterns formation in a two-cell cubic autocatalytic reaction-diffusion model subjected to the Dirichlet conditions

<p style='text-indent:20px;'>The approximate solutions of a two-cell reaction-diffusion model equation subjected to the Dirichlet conditions are obtained. The reaction is assumed to occur in the presence of cubic autocatalyst which decays to an inert compound in the first cell. Coupling with the reactant is assumed to be cubic in the concentrations. A linear exchange in the concentration of the reactant is taken between the two cells. The formal exact solution is found analytically. Here, in this investigation, use is made of the Picard iterative scheme which is constructed and applied after the exact one. The results obtained are compared with those found by means of a numerical method. It is observed that the solution obtained here is symmetric with respect to the mid-point of the container.The travelling wave is expected due to the parity of the space operator and the symmetric boundary conditions. Symmetric patterns, including among them a parabolic one, are observed for a large time.</p><p style='text-indent:20px;'>When the initial conditions are periodic, the most dominant modes travel at a constant speed for a large time. This phenomenon is highly affected by the rate of decay of the autocatalyst to an inert compound. The present work is of remarkably significant interest in chemical engineering as well as in other physical sciences. For example, in chemical industry, the objective is to achieve a great yield of a given product, which is carried by controlling the initial concentration of the reactant. Furthermore, in the last section on conclusions, we have cited many potentially useful recent works related to the subject-matter of this investigation in order to provide incentive and motivation for making further advances by using space-time fractional derivatives along the lines of the problem of finding approximate analytical solutions of the reaction-diffusion model equations which we have discussed in this article.</p>

A finite-difference procedure to solve weakly singular integro partial differential equation with space-time fractional derivatives

The main aim of the current paper is to propose an efficient numerical technique for solving space-time fractional partial weakly singular integro-differential equation. The temporal variable is based on the Riemann–Liouville fractional derivative and the spatial direction is based on the Riesz fractional derivative. Thus, to achieve a numerical technique, the time variable is discretized using a finite difference scheme with convergence order $${{\mathcal {O}}}(\tau ^{\frac{3}{2}})$$. Also, the space variable is discretized using a finite difference scheme with second-order accuracy. Furthermore, for the time-discrete and the full-discrete schemes error estimate has been presented to show the unconditional stability and convergence of the developed numerical method. Finally, two test problems have been illustrated to verify the efficiency, applicability and simplicity of the proposed technique.

On the soliton solutions to the space-time fractional simplified MCH equation

In this paper, with the aid of the Maple software, the new extended direct algebraic method is used as a powerful method to constructs some new solutions to the well-known nonlinear models, namely, the simplified modified Camassa-Holm equation. The space-time fractional derivatives are defined in the sense of the new conformable fractional derivative. It is shown that the method is straightforward and effective mathematical tool for solving nonlinear evolution equations in mathematical physics and engineering.

Non-Fickian Transport in Porous Media: Always Temporally Anomalous?

The methanol transport through the porous media, e.g., pelletized zeolite-based catalyst, was investigated. Standard Fickian diffusion equation was found to be inapplicable to a description of the experimental mass transfer kinetics. The diffusion equation with the fractional-order temporal and spatial derivatives was used in order to fit the experimental data. It is demonstrated that only the diffusion equation, which contains the time-fractional derivative or both, space–time-fractional derivatives, may thoroughly describe the experimental data. The measured time-fractional order is 0.85, which reveals the non-Fickian sub-diffusive transport through the pellet. The obtained fractional order allows the application of the continuous-time random walk model as a physical ground for the explanation of the measured non-Fickian transport. Contrary, using the space-fractional diffusion equation for the experimental data description leads to the inconsistent results similarly to the standard diffusion equation. The obtained findings reveal that the diffusate flux and the flux conservation law cannot be purely space fractional.

Thiele modulus having regard to the anomalous diffusion in a catalyst pellet

In the present paper, Thiele modulus (TM) for a catalytic reaction with the anomalous diffusion of a reagent in a catalyst pellet is introduced. Different cases of the TM are considered related to the anomalous diffusion process governed by a diffusion equation with the space-fractional, time-fractional, and space-time fractional derivatives. In addition, each fractional derivative is used according to the Caputo and the Riemann–Liouville definitions. Closed-form expressions of the TM for each definition of the fractional derivative are provided. For the time-fractional derivative, the TM is obtained under the assumption of the reaction dynamics nonlinearity. We demonstrate and critically discuss the applicability of the TM obtained for the reaction-diffusion equation with non-integer order derivatives to the evaluation of the parameters of the heterogeneous catalytic process.

Analytical study of exact solutions of the nonlinear Korteweg–de Vries equation with space–time fractional derivatives

This paper gives an analytical study of dynamic behavior of the exact solutions of nonlinear Korteweg–de Vries equation with space–time local fractional derivatives. By using the improved [Formula: see text]-expansion method, the explicit traveling wave solutions including periodic solutions, dark soliton solutions, soliton solutions and soliton-like solutions, are obtained for the first time. They can better help us further understand the physical phenomena and provide a strong basis. Meanwhile, some solutions are presented through 3D-graphs.

Fractional Dynamics for Coupled CTRW Optimal Random Search Algorithm

In this manuscript, the coupled Continuous Time Random Walk (CTRW) optimal search strategy is designed to find the target site. In which, the search lengths distribution satisfies a power-law function. The Levy flight optimal search strategy which has been adopted in a variety of fields can been seen as a special case of the proposed algorithm. Also, the search efficiency function is derived to evaluate the algorithm’s performance. Based on fractional calculus technique, its governing equation containing the coupled space-time fractional derivative is further given. Several numerous simulations are obtained to verify the veracity of the derived results.

New structures for the space-time fractional simplified MCH and SRLW equations

In this paper, we constructed new solitary structures for the space-time fractional simplified modified Camassa-Holm (MCH) equation and space-time fractional symmetric regularized long wave (SRLW) equation using the modified extended tanh method. The space-time fractional derivatives are defined in the sense of the new conformable fractional derivative. Further, with the help of Mathematica software, the set of over-determined algebraic equations obtained after reducing the equations to ordinary differentials equations are treated. We finally provide graphical illustrations for some structures.

On some applications of the space-time fractional derivative

In this work, we study and extend the one-dimensional fractional derivative to the multidimensional space-time fractional derivative, determine the exact solution via the Laplace transform, and develop mathematical foundations of the respective operators. The multidimensional space-time fractional derivative is developed to augment the equations. The results show that this method is effective, convenient, and easily executable. The main advantage of the proposed approach is that it is an accurate analytical method (the Laplace transform method) that can be implemented for both space and time discretizations of the fractional derivatives and allows us to present new solutions to problems by certain applications for solving space-time fractional derivatives.

Positivity and boundedness preserving schemes for space–time fractional predator–prey reaction–diffusion model

The semi-implicit schemes for the nonlinear predator–prey reaction–diffusion model with the space–time fractional derivatives are discussed, where the space fractional derivative is discretized by the fractional centered difference and WSGD scheme. The stability and convergence of the semi-implicit schemes are analyzed in the L∞ norm. We theoretically prove that the numerical schemes are stable and convergent without the restriction on the ratio of space and time stepsizes and numerically further confirm that the schemes have first order convergence in time and second order convergence in space. Then we discuss the positivity and boundedness properties of the analytical solutions of the discussed model, and show that the numerical solutions preserve the positivity and boundedness. The presented numerical examples confirm the theoretical results and convergence orders.

ANALYTICAL APPROXIMATE SOLUTION FOR NONLINEAR SPACE-TIME FRACTIONAL CAHN-HILLIARD EQUATION

The fractional derivatives in the sense of Caputo and the homotopy analysis method are used to construct an approximate solution for the nonlinear space-time fractional derivatives Cahn-Hilliard equation. The numerical results show that the approaches are easy to implement and accurate when applied to the non- linear space-time fractional derivatives Cahn-Hilliard equation. This method intro- duces a promising tool for solving many space-time fractional partial differential equations. This method is efficient and powerful in solving wide classes of nonlinear evolution fractional order equat. The HAM contains a certain auxiliary parameter h which provides us with a simple way to adjust and control the convergence region and rate of convergence of the series solution.

Analytical Solutions of a Space-Time Fractional Derivative of Groundwater Flow Equation

The classical Darcy law is generalized by regarding the water flow as a function of a noninteger order derivative of the piezometric head. This generalized law and the law of conservation of mass are then used to derive a new equation for groundwater flow. Two methods including Frobenius and Adomian decomposition method are used to obtain an asymptotic analytical solution to the generalized groundwater flow equation. The solution obtained via Frobenius method is valid in the vicinity of the borehole. This solution is in perfect agreement with the data observed from the pumping test performed by the institute for groundwater study on one of their boreholes settled on the test site of the University of the Free State. The test consisted of the pumping of the borehole at the constant discharge rateQand monitoring the piezometric head for 350 minutes. Numerical solutions obtained via Adomian method are compared with the Barker generalized radial flow model for which a fractal dimension for the flow is assumed. Proposition for uncertainties in groundwater studies was given.

An approximate solution to a moving boundary problem with space–time fractional derivative in fluvio-deltaic sedimentation process

A mathematical model of the movement of the shoreline in a sedimentary ocean basin is discussed. The model includes space–time fractional derivative in Caputo sense and variable latent heat term. An approximate solution of the problem is obtained by Adomian decomposition method and the results thus obtained are compared graphically with an exact solution of integer order (β=1, α=1). Three particular cases, the standard diffusion, the time-fractional and the space-fractional diffusions are also discussed. The model and solution are generalization of previous works.

Analytical approximate solution for nonlinear space—time fractional Klein—Gordon equation

The fractional derivatives in the sense of Caputo and the homotopy analysis method are used to construct an approximate solution for the nonlinear space—time fractional derivatives Klein—Gordon equation. The numerical results show that the approaches are easy to implement and accurate when applied to the nonlinear space—time fractional derivatives Klein—Gordon equation. This method introduces a promising tool for solving many space—time fractional partial differential equations. This method is efficient and powerful in solving wide classes of nonlinear evolution fractional order equations.

Analytical Solutions of the Space-Time Fractional Derivative of Advection Dispersion Equation

Fractional advection-dispersion equations are used in groundwater hydrology to model the transport of passive tracers carried by fluid flow in porous medium. A space-time fractional advection-dispersion equation (FADE) is a generalization of the classical ADE in which the first-order space derivative is replaced with Caputo or Riemann-Liouville derivative of order , and the second-order space derivative is replaced with the Caputo or the Riemann-Liouville fractional derivative of order . We derive the solution of the new equation in terms of Mittag-Leffler functions using Laplace transfrom. Some examples are given. The results from comparison let no doubt that the FADE is better in prediction than ADE.

Fractional governing equations for coupled random walks

In a continuous time random walk (CTRW), a random waiting time precedes each random jump. The CTRW is coupled if the waiting time and the subsequent jump are dependent random variables. The CTRW is used in physics to model diffusing particles. Its scaling limit is governed by an anomalous diffusion equation. Some applications require an overshoot continuous time random walk (OCTRW), where the waiting time is coupled to the previous jump. This paper develops stochastic limit theory and governing equations for CTRW and OCTRW. The governing equations involve coupled space–time fractional derivatives. In the case of infinite mean waiting times, the solutions to the CTRW and OCTRW governing equations can be quite different.

Analytical solutions to the moving boundary problems with space–time-fractional derivatives in drug release devices

A mathematical model of a solute release from a planar polymer matrix is presented and the analytical solution in terms of the Fox H function is given. The equation of space–time-fractional diffusion and a generalized Fick's law are used in the paper. Three particular cases, the standard diffusion, the time-fractional and the space-fractional diffusions are discussed in detail. The model and the solution are the generalization of the previous works and include them as special cases.

Generalized fractional Schrödinger equation with space-time fractional derivatives

In this paper the generalized fractional Schrödinger equation with space and time fractional derivatives is constructed. The equation is solved for free particle and for a square potential well by the method of integral transforms, Fourier transform and Laplace transform, and the solution can be expressed in terms of Mittag-Leffler function. The Green function for free particle is also presented in this paper. Finally, we discuss the relationship between the cases of the generalized fractional Schrödinger equation and the ones in standard quantum.