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.     

Turing pattern in the fractional Gierer–Meinhardt model

It is well-known that reaction–diffusion systems are used to describe the pattern formation models. In this paper,we will investigate the pattern formation generated by the fractional reaction–diffusion systems. We first explore the mathematical mechanism of the pattern by applying the linear stability analysis for the fractional Gierer–Meinhardt system.Then, an efficient high-precision numerical scheme is used in the numerical simulation. The proposed method is based on an exponential time differencing Runge–Kutta method in temporal direction and a Fourier spectral method in spatial direction. This method has the advantages of high precision, better stability, and less storage. Numerical simulations show that the system control parameters and fractional order exponent have decisive influence on the generation of patterns. Our numerical results verify our theoretical results.     

Optical solitons to the fractional perturbed Radhakrishnan–Kundu–Lakshmanan model

This study reaches the dark, bright, mixed dark-bright, singular, mixed singular optical solitons and singular periodic wave solutions to the time-fractional Radhakrishnan–Kundu–Lakshmanan equation. The parametric conditions that guarantee the existence of valid solitons and other solutions are stated. By choosing some suitable values of parameters, the 2- and 3-dimensional surfaces to some of the reported solutions are plotted. The reported solutions may be useful in expalining the physical meaning of the Radhakrishnan–Kundu–Lakshmanan equation and other related nonlinear models arising in nonlinear sciences.     

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 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.     

Numerical calculation of the neutral fermion gap at the ν=5/2 fractional quantum Hall state

We present the first numerical computation of the neutral fermion gap, Δ(F), in the ν=5/2 quantum Hall state, which is analogous to the energy gap for a Bogoliubov-de Gennes quasiparticle in a superconductor. We find Δ(F)≈0.027e(2)/ε?(0), comparable to the charge gap. We also deduce an effective Fermi velocity v(F) for neutral fermions from the low-energy spectra for odd numbers of electrons, and thereby obtain a correlation length ξ(F)=v(F)/Δ(F)≈1.3?(0). We comment on implications for experiments, topological quantum information processing, and electronic mechanisms of superconductivity.     

Soliton solutions of the conformable fractional Zakharov–Kuznetsov equation with dual-power law nonlinearity

In this article, the Riccati sub equation method is employed to solve fractional Zakharov–Kuznetsov equation with dual-power law nonlinearity in the sense of the conformable derivative. By using this method, new exact solutions involving parameters, expressed by generalized hyperbolic functions are obtained. This method presents a wider applicability for handling nonlinear fractional wave equations.     

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...     

On the generating function e xt+yϕ(t) and its fractional calculus

In this article, we derive the coefficient set {H _m (x,y)} _m=1 ^∞ using the generating function e ^xt+y?(t). When the complex function ?(t) is entire, using the inverse Mellin transform, and when ?(t) has singular points, using the inverse Laplace transform, the coefficient set is obtained. Also, bi-orthogonality of this set with its associated functions and its applications in the explicit solutions of partial fractional differential equations is discussed.     

Analysis of the seventh-order Caputo fractional KdV equation: applications to the Sawada–Kotera–Ito and Lax equations

In this study, we investigate the seventh-order nonlinear Caputo time-fractional KdV equation.The suggested model’s solutions, which have a series form, are obtained using the hybrid ZZtransform under the aforementioned fractional operator. The proposed approach combines the homotopy perturbation method(HPM) and the ZZ-transform. We consider two specific examples with suitable initial conditions and find the series solution to test their applicability. To demonstrate the utility of the presented...     

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.     

On the continuing relevance of Mandelbrot’s non-ergodic fractional renewal models of 1963 to 1967

The problem of “1∕f” noise has been with us for about a century. Because it is so often framed in Fourier spectral language, the most famous solutions have tended to be the stationary long range dependent (LRD) models such as Mandelbrot’s fractional Gaussian noise. In view of the increasing importance to physics of non-ergodic fractional renewal models, and their links to the CTRW, I present preliminary results of my research into the history of Mandelbrot’s very little known work in that area from 1963 to 1967. I speculate about how the lack of awareness of this work in the physics and statistics communities may have affected the development of complexity science, and I discuss the differences between the Hurst effect, “1∕f” noise and LRD, concepts which are often treated as equivalent.     

Measurement of the ν=1/3 fractional quantum hall energy gap in suspended graphene

We report on magnetotransport measurements of multiterminal suspended graphene devices. Fully developed integer quantum Hall states appear in magnetic fields as low as 2 T. At higher fields the formation of longitudinal resistance minima and transverse resistance plateaus are seen corresponding to fractional quantum Hall states, most strongly for ν=1/3. By measuring the temperature dependence of these resistance minima, the energy gap for the 1/3 fractional state in graphene is determined to be at ～20 K at 14 T.     

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.