Memory effect in time fractional Schr?dinger equation

A significant obstacle impeding the advancement of the time fractional Schr?dinger equation lies in the challenge of determining its precise mathematical formulation. In order to address this, we undertake an exploration of the time fractional Schr?dinger equation within the context of a non-Markovian environment. By leveraging a two-level atom as an illustrative case, we find that the choice to raise i to the order of the time derivative is inappropriate. In contrast to the conventional approac...     

Exponential rational function method for space–time fractional differential equations

In this paper, exponential rational function method is applied to obtain analytical solutions of the space–time fractional Fokas equation, the space–time fractional Zakharov Kuznetsov Benjamin Bona Mahony, and the space–time fractional coupled Burgers’ equations. As a result, some exact solutions for them are successfully established. These solutions are constructed in fractional complex transform to convert fractional differential equations into ordinary differential equations. The fractional derivatives are described in Jumarie’s modified Riemann–Liouville sense. The exact solutions obtained by the proposed method indicate that the approach is easy to implement and effective.     

Space–time fractional diffusion equations and asymptotic behaviors of a coupled continuous time random walk model

In this paper, we consider a type of continuous time random walk model where the jump length is correlated with the waiting time. The asymptotic behaviors of the coupled jump probability density function in the Fourier–Laplace domain are discussed. The corresponding fractional diffusion equations are derived from the given asymptotic behaviors. Corresponding to the asymptotic behaviors of the joint probability density function in the Fourier–Laplace space, the asymptotic behaviors of the waiting time probability density and the conditional probability density for jump length are also discussed.     

Nonperturbative analytical solution of the time fractional nonlinear Burger’s equation

A non-perturbative analytical solution is derived for the time fractional nonlinear Burger’s equation by using Adomian Decomposition Method (ADM). The present method performs extremely well in terms of accuracy, efficiency and simplicity.     

A reliable analytical algorithm for space–time fractional cubic isothermal autocatalytic chemical system

The nucleus-acoustic shock waves (NASWs) propagating in a white dwarf plasma system, which contain non-relativistically or ultrarelativistically degenerate electrons, non-relativistically degenerate, viscous fluid of light nuclei, and immobile nuclei of heavy elements, have been theoretically investigated. We have used the reductive perturbation method, which is valid for small but finite-amplitude NASWs to derive the Burgers equation. The NASWs are, in fact, associated with the nucleus-acoustic (NA) waves in which the inertia is provided by the light nuclei, and restoring force is provided by the degenerate pressure of electrons. On the other hand, the stationary heavy nuclei participate only in maintaining the background charge neutrality condition at equilibrium. It is found that the viscous force acting in the fluid of light nuclei is a source of dissipation, and is responsible for the formation of NASWs. It is also observed that the basic features (polarity, amplitude, width, etc.) of the NASWs are significantly modified by the presence of heavy nuclei, and that NASWs are formed with either positive or negative potential depending on the values of the charge density of the heavy nuclei. The basic properties are also found to be significantly modified by the effects of ultrarelativistically degenerate electrons. The implications of our results in white dwarfs are briefly discussed.     

Analytical treatments of the space–time fractional coupled nonlinear Schrödinger equations

Under investigation in this work is a (\(2+1\))-dimensional the space–time fractional coupled nonlinear Schrödinger equations, which describes the amplitudes of circularly-polarized waves in a nonlinear optical fiber. With the aid of conformable fractional derivative and the fractional wave transformation, we derive the analytical soliton solutions in the form of rational soliton, periodic soliton, hyperbolic soliton solutions by four integration method, namely, the extended trial equation method, the \(\exp (-\,\Omega (\eta ))\)-expansion method and the improved \(\tan (\phi (\eta )/2)\)-expansion method and semi-inverse variational principle method. Based on the the extended trial equation method, we derive the several types of solutions including singular, kink-singular, bright, solitary wave, compacton and elliptic function solutions. Under certain condition, the 1-soliton, bright, singular solutions are driven by semi-inverse variational principle method. Based on the analytical methods, we find that the solutions give birth to the dark solitons, the bright solitons, combine dark-singular, kink, kink-singular solutions with fractional order for nonlinear fractional partial differential equations arise in nonlinear optics.     

Lie symmetry analysis and explicit solutions for the time fractional generalized Burgers–Huxley equation

Optical and Quantum Electronics - Investigated in this paper is the modified Hirota equation with variable coefficients, which can describe the amplification or absorption of pulses propagating in...     

Dispersive traveling wave solutions to the space–time fractional equal-width dynamical equation and its applications

In this article, some new traveling wave solutions to the space–time fractional equal-width equation are constructed with the help of the extended Fan sub-equation method. A simple transformation is introduced to convert the fractional order partial differential equation into an ordinary differential equation. As a result, the bright, dark, singular and combined wave solitons are observed for different values of two parameters. Moreover, the graphical representations are also depicted.     

Analytic solutions of the space–time conformable fractional Klein–Gordon equation in general form

ABSTRACT

The Klein–Gordon equation plays an important role in mathematical physics. In this paper, a direct method which is very effective, simple, and convenient, is presented for solving the conformable fractional Klein–Gordon equation. Using this analytic method, the exact solutions of this equation are found in terms of the Jacobi elliptic functions. This method is applied to both time and space fractional equations. Some solutions are also illustrated by the graphics.     

Exact solutions to the space–time fractional Schrödinger–Hirota equation and the space–time modified KDV–Zakharov–Kuznetsov equation

In this paper, the first integral method and the functional variable method are used to establish exact traveling wave solutions of the space–time fractional Schrödinger–Hirota equation and the space–time fractional modified KDV–Zakharov–Kuznetsov equation in the sense of conformable fractional derivative. The results obtained confirm that proposed methods are efficient techniques for analytic treatment of a wide variety of the space–time fractional partial differential equations.     

Optical soliton perturbation for time fractional resonant nonlinear Schrödinger equation with competing weakly nonlocal and full nonlinearity

In this article, we retrieve optical soliton solutions of the perturbed time fractional resonant nonlinear Schrödinger equation having competing weakly nonlocal and full nonlinearity. We study the equation for two different forms of nonlinearity, namely Kerr law and anti-cubic law. The F-expansion method along with fractional complex transformation is used to obtain the optical solitons. Moreover, the existence of these solitons are guaranteed with the constraint relations between the model coefficients and the traveling wave frequency coefficient.     

Lie symmetry analysis,conservation laws and analytic solutions of the time fractional Kolmogorov–Petrovskii–Piskunov equation

In this paper, we consider the invariance properties of the multiple-term fractional Kolmogorov–Petrovskii–Piskunov (FKPP) equation. By employing the Lie symmetry analysis method, we explicitly investigate the vector fields and symmetry reductions of the FKPP equation. Moreover, an effective method is proposed to succinctly derive the exact power series solutions with their convergence analysis of the equation. Finally, by using the new conservation theorem, the conservation laws associated with Lie symmetries of the equation are well constructed with a detailed analysis.     

A study of fractional Schrödinger equation composed of Jumarie fractional derivative

In this paper we have derived the fractional-order Schrödinger equation composed of Jumarie fractional derivative. The solution of this fractional-order Schrödinger equation is obtained in terms of Mittag–Leffler function with complex arguments, and fractional trigonometric functions. A few important properties of the fractional Schrödinger equation are then described for the case of particles in one-dimensional infinite potential well. One of the motivations for using fractional calculus in physical systems is that the space and time variables, which we often deal with, exhibit coarse-grained phenomena. This means infinitesimal quantities cannot be arbitrarily taken to zero – rather they are non-zero with a minimum spread. This type of non-zero spread arises in the microscopic to mesoscopic levels of system dynamics, which means that, if we denote x as the point in space and t as the point in time, then limit of the differentials dx (and dt) cannot be taken as zero. To take the concept of coarse graining into account, use the infinitesimal quantities as (Δx) ^α (and (Δt) ^α ) with 0 < α < 1; called as ‘fractional differentials’. For arbitrarily small Δx and Δt (tending towards zero), these ‘fractional’ differentials are greater than Δx (and Δt), i.e. (Δx) ^α > Δx and (Δt) ^α > Δt. This way of defining the fractional differentials helps us to use fractional derivatives in the study of dynamic systems.     

New exact solutions to the space–time fractional nonlinear wave equation obtained by the ansatz and functional variable methods

In this paper, the ansatz method and the functional variable method are employed to find new analytic solutions for the space–time nonlinear fractional wave equation, the space–time fractional Kadomtsev–Petviashvili–Benjamin–Bona–Mahony equation and the space–time fractional modified Korteweg–de Vries–Zakharov–Kuznetsov equation. As a result, some exact solutions are obtained in terms of hyperbolic and periodic functions. It is shown that the proposed methods provide a more powerful mathematical tool for constructing exact solutions for many other nonlinear fractional differential equations occurring in nonlinear physical phenomena. We have also presented the numerical simulations for these equations by means of three dimensional plots.     

Crank–Nicolson method for the fractional diffusion equation with the Riesz fractional derivative

We examine a numerical method to approximate to a fractional diffusion equation with the Riesz fractional derivative in a finite domain, which has second order accuracy in time and space level. In order to approximate the Riesz fractional derivative, we use the “fractional centered derivative” approach. We determine the error of the Riesz fractional derivative to the fractional centered difference. We apply the Crank–Nicolson method to a fractional diffusion equation which has the Riesz fractional derivative, and obtain that the method is unconditionally stable and convergent. Numerical results are given to demonstrate the accuracy of the Crank–Nicolson method for the fractional diffusion equation with using fractional centered difference approach.     

Optical solitons and other solutions to the conformable space–time fractional complex Ginzburg–Landau equation under Kerr law nonlinearity

This study reveals the dark, bright, combined dark–bright, singular, combined singular optical solitons and singular periodic solutions to the conformable space–time fractional complex Ginzburg–Landau equation. We reach such solutions via the powerful extended sinh-Gordon equation expansion method (ShGEEM). Constraint conditions that guarantee the existence of valid solitary wave solutions are given. Under suitable choice of the parameter values, interesting three-dimensional graphs of some of the obtained solutions are plotted.     

Compact finite difference schemes for the backward fractional Feynman–Kac equation with fractional substantial derivative

The fractional Feynman–Kac equations describe the distributions of functionals of non-Brownian motion, or anomalous diffusion, including two types called the forward and backward fractional Feynman–Kac equations, where the nonlocal time–space coupled fractional substantial derivative is involved. This paper focuses on the more widely used backward version. Based on the newly proposed approximation operators for fractional substantial derivative, we establish compact finite difference schemes for the backward fractional Feynman–Kac equation. The proposed difference schemes have the q-th(q = 1, 2, 3, 4) order accuracy in temporal direction and fourth order accuracy in spatial direction, respectively. The numerical stability and convergence in the maximum norm are proved for the first order time discretization scheme by the discrete energy method, where an inner product in complex space is introduced. Finally, extensive numerical experiments are carried out to verify the availability and superiority of the algorithms. Also, simulations of the backward fractional Feynman–Kac equation with Dirac delta function as the initial condition are performed to further confirm the effectiveness of the proposed methods.     

On generalized fractional Cattaneo’s equations

This paper studies a family of generalized fractional Cattaneo’s equations for which passive (i.e., spontaneous) transport is possible. This is done by using fractional substitutions in integer-order rational transfer functions and showing conditions for positive realness.