The time fractional heat conduction equation in the general orthogonal curvilinear coordinate and the cylindrical coordinate systems

In this paper a time fractional Fourier law is obtained from fractional calculus. According to the fractional Fourier law, a fractional heat conduction equation with a time fractional derivative in the general orthogonal curvilinear coordinate system is built. The fractional heat conduction equations in other orthogonal coordinate systems are readily obtainable as special cases. In addition, we obtain the solution of the fractional heat conduction equation in the cylindrical coordinate system in terms of the generalized H-function using integral transformation methods. The fractional heat conduction equation in the case 0<α≤1 interpolates the standard heat conduction equation (α=1) and the Localized heat conduction equation (α→0). Finally, numerical results are presented graphically for various values of order of fractional derivative.     

Modified Kudryashov method via new exact solutions for some conformable fractional differential equations arising in mathematical biology

In this study, the modified Kudryashov method is used to construct new exact solutions for some conformable fractional differential equations. By implementing the conformable fractional derivative and compatible fractional complex transforms, the fractional generalized reaction duffing (RD) model equation, the fractional biological population model and the fractional diffusion reaction (DR) equation with quadratic and cubic nonlinearity are discussed. As an outcome, some new exact solutions are formally established. All solutions have been verified back into its corresponding equation with the aid of maple package program. We assure that the employed method is simple and robust for the estimation of the new exact solutions, and practically capable for reducing the size of computational work for solving a various class of fractional differential equations arising in applied mathematics, mathematical physics and biology.     

Radial anomalous diffusion in an annulus

In this study, time fractional radial diffusion has been modeled in cylindrical coordinates in order to analyze the anomalous diffusion in an annulus. By using an integral transform technique, the analytical solution of the concentration distribution formula is obtained. The establishing of the concentration distribution is found to be controlled by the fractional derivative α, and the influences of α on the concentration field, the total amount diffused and the quantity of mass passing through the inner wall are presented graphically and studied in detail. Asymptotic expressions for the exact solutions are developed in order to explain the numerical results at small and large time, respectively, and the physical mechanism explanation for the paradoxical behavior shown in the numerical results is given.     

A family of evolution equations with nonlinear diffusion,Verhulst growth,and global regulation: Exact time-dependent solutions

A family of evolution equations describing a power-law nonlinear diffusion process coupled with a local Verhulst-like growth dynamics, and incorporating a global regulation mechanism, is considered. These equations admit an interpretation in terms of population dynamics, and are related to the so-called conserved Fisher equation. Exact time-dependent solutions exhibiting a maximum nonextensive q$q$-entropy shape are obtained.     

Modulational instability of electron-acoustic waves in a plasma with a q-nonextensive electron velocity distribution

The amplitude modulation of electron-acoustic waves (EAWs) propagating in space plasmas whose constituents are inertial cold electrons, hot nonextensive q-distributed electrons, and stationary ions is presented theoretically. The nonlinear Schrödinger equation (NLSE) which governs the modulational instability of the EAWs is obtained using reductive perturbation method (RPM). The presence of the hot nonextensive q-distributed electrons is shown to influence the modulational instability of the waves. Further, the nondimensional parameter α=ne0/nc0, which is the equilibrium density ratio of the hot to cold electron component, is shown to play a vital role in the formation of both bright and dark solitons.     

Analysis of a time fractional wave-like equation with the homotopy analysis method

The time fractional wave-like differential equation with a variable coefficient is studied analytically. By using a simple transformation, the governing equation is reduced to two fractional ordinary differential equations. Then the homotopy analysis method is employed to derive the solutions of these equations. The accurate series solutions are obtained. Especially, when ?f=?g=−1, these solutions are exactly the same as those results given by the Adomian decomposition method. The present work shows the validity and great potential of the homotopy analysis method for solving nonlinear fractional differential equations. The basic idea described in this Letter is expected to be further employed to solve other similar nonlinear problems in fractional calculus.     

Finite difference approximations for a fractional advection diffusion problem

The use of the conventional advection diffusion equation in many physical situations has been questioned by many investigators in recent years and alternative diffusion models have been proposed. Fractional space derivatives are used to model anomalous diffusion or dispersion, where a particle plume spreads at a rate inconsistent with the classical Brownian motion model. When a fractional derivative replaces the second derivative in a diffusion or dispersion model, it leads to enhanced diffusion, also called superdiffusion. We consider a one-dimensional advection–diffusion model, where the usual second-order derivative gives place to a fractional derivative of order α$\alpha$, with 1<α?2$1<\alpha ?2$. We derive explicit finite difference schemes which can be seen as generalizations of already existing schemes in the literature for the advection–diffusion equation. We present the order of accuracy of the schemes and in order to show its convergence we prove they are stable under certain conditions. In the end we present a test problem.     

Analytical solutions to time-fractional partial differential equations in a two-dimensional multilayer annulus

In this paper, analytical solutions to time-fractional partial differential equations in a multi-layer annulus are presented. The final solutions are obtained in terms of Mittag-Leffler function by using the finite integral transform technique and Laplace transform technique. In addition, the classical diffusion equation (α=1$\alpha =1$), the Helmholtz equation (α→0$\alpha \to 0$) and the wave equation (α=2$\alpha =2$) are discussed as special cases. Finally, an illustrative example problem for the three-layer semi-circular annular region is solved and numerical results are presented graphically for various kind of order of fractional derivative.     

Exact solutions for nonlinear fractional anomalous diffusion equations

This work is devoted to investigating exact solutions of generalized nonlinear fractional diffusion equations with external force and absorption. We first investigate the nonlinear anomalous diffusion equations with one-fractional derivative and then multi-fractional ones. In both situations, we obtain the corresponding exact solution, its diffusive behavior, and the sufficient and necessary conditions for solutions satisfying the boundary condition W(±∞,t)=0 and the sharp initial condition W(x,0)=δ(x).     

Jeans' gravitational instability of a thermally conducting plasma

The gravitational instability of an infinitely extending homogenous plasma endowed with several physical mechanisms, namely Hall currents, finite conductivity, ion viscosity and thermal conductivity is considered. The main result is that the various parameters play different physical roles in the perturbed problem. Jeans' criterion is analyzed in the framework of Tsallis' statistics for possible modifications due to the presence of nonextensive effects. A simple generalization of the Jeans' criterion is obtained and the standard values are obtained in the limiting case q=1, q being the nonextensive parameter.     

The operator method for solving the fractional Fokker-Planck equation

The operator method has been used to solve the fractional Fokker-Planck equation (FPE) which recently formulated as a model for the anomalous transport process. Two classes of special interest of fractional F-P equations coming from plasma physics and charged particle transport problem has been considered. It is shown that the mean square-displacement 〈x²(t)〉 satisfy the universal power law characterized the anomalous time evolution i.e. .     

Traveling wave solutions of conformable time-fractional Zakharov–Kuznetsov and Zoomeron equations

To model physical phenomena more accurately, fractional order differential equations have been widely used. Investigating exact solutions of the fractional differential equations have become more important because of the applications in applied mathematics, mathematical physics, and other areas. In this work, by means of the trial solution method and complete discrimination system, exact traveling wave solutions of the conformable time-fractional Zakharov–Kuznetsov equation and conformable time-fractional Zoomeron equation have been obtained and also solutions have been illustrated. Finding exact solutions of these equations that are encountered in plasma physics, nonlinear optics, fluid mechanics, and laser physics can help to understand nature of the complex phenomena.     

Fokker-Planck equation with fractional coordinate derivatives

Using the generalized Kolmogorov-Feller equation with long-range interaction, we obtain kinetic equations with fractional derivatives with respect to coordinates. The method of successive approximations, with averaging with respect to a fast variable, is used. The main assumption is that the correlation function of probability densities of particles to make a step has a power-law dependence. As a result, we obtain a Fokker-Planck equation with fractional coordinate derivative of order 1<α<2.     

X-ray binary systems and nonextensivity

We study the x-ray intensity variations obtained from the time series of 155 light curves of x-ray binary systems collected by the instrument All Sky Monitor on board the satellite Rossi X-Ray Timing Explorer. These intensity distributions are adequately fitted by q-Gaussian distributions which maximize the Tsallis entropy and in turn satisfy a nonlinear Fokker-Planck equation, indicating their nonextensive and nonequilibrium behavior. From the values of the entropic index q obtained, we give a physical interpretation of the dynamics in x-ray binary systems based on the kinetic foundation of generalized thermostatistics. The present findings indicate that the binary systems display a nonextensive and turbulent behavior.     

Lack of anomalous diffusion in linear translationally-invariant systems determined by only one initial condition

It is shown that as far as the linear diffusion equation meets both time- and space-translational invariance, the time dependence of a moment of degree α is a polynomial of degree at most equal to α, while all connected moments are at most linear functions of time. As a special case, the variance is an at most linear function of time.     

Solutions of the space-time fractional Cattaneo diffusion equation

The object of this paper is to present the exact solution of the fractional Cattaneo equation for describing anomalous diffusion. The classical Cattaneo model has been generalised to the space-time fractional Cattaneo model. The method of the joint Laplace and Fourier transform is used in deriving the solution. The solutions of the fractional Cattaneo equation are obtained under integral and series forms in terms of the H-functions. Finally, the fractional order moments are also investigated.     

On the solution of the fractional nonlinear Schrödinger equation

We present the nonlinear Schrödinger (NLS) equation of fractional order. The fractional derivatives are described in the Caputo sense. The Adomian decomposition method (ADM) in applied mathematics can be used as alternative methods for obtaining analytic and approximate solutions for different types of fractional differential equations. In these schemes, the solution constructed in power series with easily computable components.     

Solutions for a non-Markovian diffusion equation

Solutions for a non-Markovian diffusion equation are investigated. For this equation, we consider a spatial and time dependent diffusion coefficient and the presence of an absorbent term. The solutions exhibit an anomalous behavior which may be related to the solutions of fractional diffusion equations and anomalous diffusion.     

Cumulative Diminuations with Fibonacci Approach,Golden Section and Physics

In this study, physical quantities of a nonequilibrium system in the stages of its orientation towards equilibrium has been formulated by a simple cumulative diminuation mechanism and Fibonacci recursion approximation. Fibonacci p-numbers are obtained in power law forms and generalized diminuation sections are related to diminuation percents. The consequences of the fractal structure of space and the memory effects are concretely established by a simple mechanism. Thus, the reality why nature prefers power laws rather than exponentials ones is explained. It has been introduced that, Fibonacci p-numbers are elements of a Generalized Cantor set. The fractal dimensions of the Generalized Cantor sets have been obtained by different methods. The generalized golden section which was used by M.S. El Naschie in his works on high energy physics is evaluated in this frame.