Fractional Space-time Research Articles

On fractional and fractal Einstein’s field equations

In this study, Einstein’s field equations are derived based on two dissimilar frameworks: the first is based on the concepts of “fractional velocity” and “fractal action” motivated by Calcagni’s approach to fractional spacetime while the second is derived based on fractal calculus which is a generalization of ordinary calculus that include fractal sets and curves. The fractional theory displays a breakdown of Lorentz invariance. It was observed that a spatially dependent cosmological constant emerges in the fractional theory. A connection between the fractional order parameter and the dimensionless parameter [Formula: see text] arising in the parameterized post-Newtonian (PPN) formalism is observed. A confrontation with very long-baseline radio interferometry targeting quasars 3C273 and 3C279 is done which proves that the fractional order parameter is within the range [Formula: see text]. Moreover, emergence of quantum Hawking radiation is realized in the theory supporting Hawking’s best calculations that black holes are not black. Nevertheless, based on the fractal calculus approach, there is a conservation of the Lorentz invariance and absence of spatially-dependent cosmological constant. The theory depends on the fractal order [Formula: see text] and gives rise to a fractal Schwarzschild radius of the massive body greater than the conventional radius besides a fractal Hawking’s temperature less than the standard one. However, the confrontation with radio interferometry targeting quasars 3C273 and 3C279 gives [Formula: see text].

Exact Travelling Wave Solution of Fractional Space-time Zakharov–Kuznetsov Equation

Objectives: To find the new exact travelling wave solutions of fractional order space-time Zakharov– Kuznetsov (ZK) equation of ion-acoustic waves. Methods/Statistical: The fractional derivative is defined in the sense of modified Riemann- Liouville derivative to convert the fractional space-time ‎ZK equation to nonlinear ordinary differential equation‎.‎ The two proposed techniques the (ω\'/ω)-expansion and extended (ω\'/ω)-expansion methods are employed for constructing the new exact travelling wave solutions of ZK equation. Findings/Results: The obtained new exact travelling wave solutions of ZK equation include the hyperbolic function, rational function, and trigonometric function. ‎Application: The results reveal that the two used methods here are effective and powerful methods and might be accustomed to ‎establish the other solutions of nonlinear fractional partial differential equations that arising in ‎nonlinear science.‎ Keywords: The (ω\'/ω)-Expansion Method, The Extended (ω\'/ω)-Expansion Method, The Modified Riemann- Liouville Derivative, Homogeneous Balance, The New Exact Travelling Wave Solutions

Modeling anomalous heat diffusion: Comparing fractional derivative and non-linear diffusivity treatments

In the Fourier heat conduction equation, when the flux definition is expressed as the product of a constant diffusivity and the temperature gradient, the characteristic length scale evolves as the square root of time. However, if we replace the 1st order transient and gradient terms in the Fourier equation with fractional derivatives and/or define a non-linear spatially dependent diffusivity, it is possible to generate an anomalous space-time scaling, i.e., a scaling where the time exponent differs from the expected value of 1/2. To compare and contrast the possible consequences of using fractional calculus along with a non-linear flux, we investigate a space-time fractional heat diffusion equation that involves a non-linear diffusivity. Following presentation of the governing non-linear fractional equation, we arrive at a space-time scaling that accounts for the combined anomalous contributions of memory (fractional derivative in time), non-locality (fractional derivative in space), and a non-linear diffusivity. We demonstrate how this scaling can manifest in a physical setting by considering the analytical solution of a non-linear fractional space-time diffusion equation, a limit case Stefan problem related to moisture infiltration into a porous media. A direct physically realizable simulation of this process shows how the anomalous space-time scaling is explicitly related to measures of both the memory and non-linearity in the system. Overall, the findings from this work clearly show how the definition of a non-linear diffusivity might contribute to anomalous diffusion behavior and suggests that, in modeling a particular observation, the roles of fractional derivatives and a suitably defined non-linear diffusivity are interchangeable.

On the fractional order space-time nonlinear equations arising in plasma physics

In this study, the $$\exp (-\phi (\xi ))$$ -expansion function method is considered for solving two classes of space-time fractional partial differential equations of very special interest. The two classes, namely the higher dimensional Kadomtsev–Petviashvili and Boussinesq equations, have a wide range applications in different areas of complex nonlinear physics such as plasma physics, fluid dynamics and nonlinear optics. As a result, the $$\exp (-\phi (\xi ))$$ -expansion function method yields a different class of traveling solutions mapped to trigonometric functions, rational functions and hyperbolic functions. Also, the behavior of these solutions has been significantly affected by changing the values of fractional order where the obtained solutions go back to those obtained previously to the normal case, i.e., $$\alpha =\beta =1$$ . Finally, our finding may be of wide relevance and helpful to better understand the main features and propagation of the nonlinear waves in fractal medium.

The fractional space–time radial diffusion equation in terms of the Fox’s [formula omitted]-function

Based on a generalization of the Hilfer–Katugampola fractional operator, recently introduced, and the Weyl fractional derivative, which are responsible to describe the memory and distance effects, respectively, we investigate the anomalous diffusion in processes in which fractional radial differential equation plays an important and fundamental rule. Similarity solutions for this fractional space–time radial equation are considered. These solutions are presented in terms of the Fox’s H-function. As an application, we present and discuss a special case in fractal Hausdorff dimension.

New optical solitons of space-time conformable fractional perturbed Gerdjikov-Ivanov equation by sine-Gordon equation method

Abstract This paper devotes to conformable fractional space-time perturbed Gerdjikov-Ivanov (GI) equation which appears in nonlinear fiber optics and photonic crystal fibers (PCF). We consider the model with full nonlinearity in order to give a generalized flavor. The sine-Gordon equation approach is carried out to model equation for retrieving the dark, bright, dark-bright, singular and combined singular optical solitons. The constraint conditions are also reported for guaranteeing the existence of these solitons. We also present some graphical simulations of the solutions for better understanding the physical phenomena of the behind the considered model.

Numerical Solution for Fractional Order Space-Time Burger's Equation Using Legendre Wavelet - Chebyshev Wavelet Spectral Collocation Method

In this article, a Legendre wavelet- Chebyshev wavelet spectral collocation method is proposed for solving fractional order space-time Burger's equation with the Legendre wavelet and Chebyshev wavelet operational matrices of fractional derivatives. The fractional derivative is described in the Caputo sense. The proposed method is based on Legendre wavelet-Chebyshev wavelet for space and time variables respectively. This method will reduc the problem under consideration to the solution of nonlinear algebraic equations. In order to confirm the efficiency of the proposed method, two numerical examples are implemented and comparing the numerical solution with the exact one, as well as, of other methods in given literatures, we demonstrate the high accuracy and efficiency of the proposed method.

Fractional space–time nonlinear reaction–convection–diffusion

We comment the fractional space–time reaction–convection–diffusion defined by Grunwald–Letnikov, Riesz and Riesz–Feller fractional derivatives. We find the explicit travelling wave solution, using the similarity reduction. We plot its graphics to illustrate the anomalous behavior and investigate its properties associated with memory and distance effects.

On weighted mixed-norm Sobolev estimates for some basic parabolic equations

Novel global weighted parabolic Sobolev estimates, weighted mixed-norm estimates and a.e. convergence results of singular integrals for evolution equations are obtained. Our results include the classical heat equation \begin{document} $ \partial_tu=\Delta u+f, $ \end{document} the harmonic oscillator evolution equation \begin{document}$\partial_tu=\Delta u-|x|^2u+f, $ \end{document} and their corresponding Cauchy problems. We also show weighted mixed-norm estimates for solutions to degenerate parabolic extension problems arising in connection with the fractional space-time nonlocal equations $(\partial_t-\Delta)^su=f$ and $(\partial_t-\Delta+|x|^2)^su=f$, for $0

On the q-Laplace Transform and Related Special Functions

Motivated by statistical mechanics contexts, we study the properties of the q-Laplace transform, which is an extension of the well-known Laplace transform. In many circumstances, the kernel function to evaluate certain integral forms has been studied. In this article, we establish relationships between q-exponential and other well-known functional forms, such as Mittag–Leffler functions, hypergeometric and H-function, by means of the kernel function of the integral. Traditionally, we have been applying the Laplace transform method to solve differential equations and boundary value problems. Here, we propose an alternative, the q-Laplace transform method, to solve differential equations, such as as the fractional space-time diffusion equation, the generalized kinetic equation and the time fractional heat equation.

Space–time fractional diffusion equation using a derivative with nonsingular and regular kernel

Abstract In this paper, using the fractional operators with Mittag-Leffler kernel in Caputo and Riemann–Liouville sense the space–time fractional diffusion equation is modified, the fractional equation will be examined separately; with fractional spatial derivative and fractional temporal derivative. For the study cases, the order considered is 0 β , γ ≤ 1 respectively. In this alternative representation we introduce the appropriate fractional dimensional parameters which characterize consistently the existence of the fractional space–time derivatives into the fractional diffusion equation, these parameters related to equation results in a fractal space–time geometry provide a new family of solutions for the diffusive processes. The proposed mathematical representation can be useful to understand electrochemical phenomena, propagation of energy in dissipative systems, viscoelastic materials, material heterogeneities and media with different scales.

Midisuperspace quantization: Possibilities for fractional and emergent spacetime dimensions

Recently, motivated by certain loop quantum gravity inspired corrections, it was shown that for spherically symmetric midisuperspace models infinitely many second derivative theories of gravity exist (as revealed by the presence of three arbitrary functions in the corresponding Lagrangian/Hamiltonian) and not just those allowed by spherically symmetric general relativity. This freedom can be interpreted as the freedom to accommodate certain quantum gravity corrections in these models even in the absence of higher curvature terms (at a semi-classical level, at least). For a particular choice of the arbitrary functions it is shown that the new theories map to spherically symmetric general relativity in arbitrary number of (integer) dimensions thus explicitly demonstrating that when working with midisuperspace models, one loses the information about the dimensionality of the full spacetime. In addition, it is shown that these new theories can accommodate scenarios of fractional spacetime dimensions as well as those of emergent spacetime dimensions -- a possibility suggested by various approaches to quantum gravity.

Nonlocal Transport Processes and the Fractional Cattaneo-Vernotte Equation

The Cattaneo-Vernotte equation is a generalization of the heat and particle diffusion equations; this mathematical model combines waves and diffusion with a finite velocity of propagation. In disordered systems the diffusion can be anomalous. In these kinds of systems, the mean-square displacement is proportional to a fractional power of time not equal to one. The anomalous diffusion concept is naturally obtained from diffusion equations using the fractional calculus approach. In this paper we present an alternative representation of the Cattaneo-Vernotte equation using the fractional calculus approach; the spatial-time derivatives of fractional order are approximated using the Caputo-type derivative in the range(0,2]. In this alternative representation we introduce the appropriate fractional dimensional parameters which characterize consistently the existence of the fractional space-time derivatives into the fractional Cattaneo-Vernotte equation. Finally, consider the Dirichlet conditions, the Fourier method was used to find the full solution of the fractional Cattaneo-Vernotte equation in analytic way, and Caputo and Riesz fractional derivatives are considered. The advantage of our representation appears according to the comparison between our model and models presented in the literature, which are not acceptable physically due to the dimensional incompatibility of the solutions. The classical cases are recovered when the fractional derivative exponents are equal to1.

On the Solution of Local Fractional Differential Equations Using Local Fractional Laplace Variational Iteration Method

We present iteration formulae of a fractional space-time telegraph equation using the combination of fractional variational iteration method and local fractional Laplace transform.

Fractional electromagnetic waves in conducting media

We present the fractional wave equation in a conducting material. We used a Maxwell’s equations with the assumptions that the charge density and current density J were zero, and that the permeability and permittivity were constants. The fractional wave equation will be examined separately; with fractional spatial derivative and fractional temporal derivative, finally, consider a Dirichlet conditions, the Fourier method was used to find the full solution of the fractional equation in analytic way. Two auxiliary parameters and are introduced; these parameters characterize consistently the existence of the fractional space-time derivatives into the fractional wave equation. A physical relation between these parameters is reported. The fractional derivative of Caputo type is considered and the corresponding solutions are given in terms of the Mittag-Leffler function show fractal space-time geometry different from the classical integer-order model.

Modeling and simulation of the fractional space-time diffusion equation

In this paper, the space-time fractional diffusion equation related to the electromagnetic transient phenomena in transmission lines is studied, three cases are presented; the diffusion equation with fractional spatial derivative, with fractional temporal derivative and the case with fractional space-time derivatives. For the study cases, the order of the spatial and temporal fractional derivatives are 0 < β, γ ≤ 2, respectively. In this alternative representation we introduce the appropriate fractional dimensional parameters which characterize consistently the existence of the fractional space-time derivatives into the fractional diffusion equation. The general solutions of the proposed equations are expressed in terms of the multivariate Mittag-Leffler functions; these functions depend only on the parameters β and γ and preserve the appropriated physical units for any value of the fractional derivative exponent. Furthermore, an analysis of the fractional time constant was made in order to indicate the change of the medium properties and the presence of dissipation mechanisms. The proposed mathematical representation can be useful to understand electrochemical phenomena, propagation of energy in dissipative systems, irreversible thermodynamics, quantum optics or turbulent diffusion, thermal stresses, models of porous electrodes, the description of gel solvents and anomalous complex processes.

Universal character of the fractional space-time electromagnetic waves in dielectric media

This work presents an alternative solution for the mathematical analysis of the fractional waves in dielectric media. For the fractional wave equation, the Caputo fractional derivative was considered, the order of the spatial and temporal fractional derivatives are , respectively. In this analysis, we introduce the appropriate fractional dimensional parameters which characterize consistently the existence of the fractional space and time derivatives into the fractional wave equation. We will consider source free Maxwell equations in isotropic and homogeneous dielectric medium. The general solutions obtained in our research have been expressed in terms of the multivariate Mittag–Leffler functions, these functions depend only on the parameters and preserving the appropriated physical units according to the system studied.

Fractional and noncommutative spacetimes

We establish a mapping between fractional and noncommutative spacetimes in configuration space. Depending on the scale at which the relation is considered, there arise two possibilities. For a fractional spacetime with log-oscillatory measure, the effective measure near the fundamental scale determining the log-period coincides with the non-rotation-invariant but cyclicity-preserving measure of \kappa-Minkowski. At scales larger than the log-period, the fractional measure is averaged and becomes a power-law with real exponent. This can be also regarded as the cyclicity-inducing measure in a noncommutative spacetime defined by a certain nonlinear algebra of the coordinates, which interpolates between \kappa-Minkowski and canonical spacetime. These results are based upon a braiding formula valid for any nonlinear algebra which can be mapped onto the Heisenberg algebra.

Fractional Dynamics from Einstein Gravity, General Solutions, and Black Holes

We study the fractional gravity for spacetimes with non-integer dimensions. Our constructions are based on a geometric formalism with the fractional Caputo derivative and integral calculus adapted to nonolonomic distributions. This allows us to define a fractional spacetime geometry with fundamental geometric/physical objects and a generalized tensor calculus all being similar to respective integer dimension constructions. Such models of fractional gravity mimic the Einstein gravity theory and various Lagrange-Finsler and Hamilton-Cartan generalizations in nonholonomic variables. The approach suggests a number of new implications for gravity and matter field theories with singular, stochastic, kinetic, fractal, memory etc processes. We prove that the fractional gravitational field equations can be integrated in very general forms following the anholonomic deformation method for constructing exact solutions. Finally, we study some examples of fractional black hole solutions, fractional ellipsoid gravitational configurations and imbedding of such objects in fractional solitonic backgrounds.

On a stochastic fractional partial differential equation with a fractional noise

We will prove the existence, uniqueness and regularity of the solution for a stochastic fractional partial differential equation driven by an additive fractional space–time white noise. Moreover, the absolute continuity of the solution is also obtained.