Fractional Space-time Research Articles

Exploring Nonlinear Fractional (2+1)-Dimensional KP–Burgers Model: Derivation and Ion Acoustic Solitary Wave Solution

The purpose of this study is to conduct a complete examination into the fractional (2+1)-dimensional nonlinear KP-Burgers (KP-B) model in the setting of quantum plasma. The main goal is to present a mathematical explanation and derivation for the fractional (2+1)-dimensional nonlinear KP-B model. We used the reductive perturbation technique to obtain the fractional (2+1)-dimensional ion acoustic solitary wave, which leads to the nonlinear KP-B model. To solve the fractional space-time KP-B model, we used the modified sub-equation technique and the extended hyperbolic function methodology. This methodology covers rational, periodic wave, and hyperbolic function solutions. Ion acoustic waves were explored in relation to the effects of ion pressure and an external electric field. The effects of density and fractional order on the properties of a single solution are examined. The plasma system is defined as an unmagnetized epi plasma system composed of relativistic ions, positrons, and nonextensive electrons that can be found in a range of astrophysical and cosmological environments. The solution variables include electron-positron and ion-electron temperature ratios, electron and positron nonextensivity strength, ion kinematic viscosity, positron concentration, and the weakly relativistic streaming factor. The fractional order and associated plasma properties have a considerable influence on the phase velocity of ion acoustic waves. When the fractional order equals one, the obtained results are consistent with known outcomes. This work makes a substantial contribution to our understanding of nonlinear events in quantum plasmas, particularly ion acoustic waves. It provides a solid approach for solution development and analysis, with implications for a variety of astrophysical and cosmological scenarios.

Is Yang-Mills theory unitary in fractional spacetime dimensions?

Is Yang-Mills theory unitary in fractional spacetime dimensions?

Noether’s currents for conformable fractional scalar field theories

The construction of fractional derivatives with the right properties for use in field theory is reputed to be a difficult task, essentially because of the absence of a unique definition and uniform properties. The conformable fractional derivative introduced in 2014 by Khalil et al. in their seminal paper is a novel and well-behaved definition of fractional derivative for a function that is derivable in the usual sense. In this paper, we investigate the consistency of the Euler–Lagrange formalism for a field theory defined on such a fractional space–time. We especially focus on the relation between symmetries and conservation laws (Noether’s currents), about the symmetry group introduced to construct the Lagrangian of the field. In particular, we show that the use of the conformable derivative induces additional terms in the calculation of the action variation. We also investigate the conservation of the Noether current and show that this property only takes place on condition that the equations of motion are verified with a new definition of the conserved law.

Investigation of space-time dynamics of perturbed and unperturbed Chen-Lee-Liu equation: Unveiling bifurcations and chaotic structures

In this paper, the fractional space-time nonlinear Chen-Lee-Liu equation has been considered using various methods. The investigation of the transition from periodic to quasi-periodic behavior has been conducted using a saddle-node bifurcation approach. The paper reports the conditions of multi-dimensional bifurcations of dynamical solutions. Additionally, a direct algebraic method has been used to calculate various 2D and 3D solitonic structures of the equation, and an analysis of their accuracy and effectiveness has been conducted. Furthermore, the Galilean transformation has been used to convert the equation into a planar dynamical system, which is further utilized to obtain bifurcations and chaotic structures. Chaotic structures of perturbed dynamical system are observed and detected through chaos detecting tools such as 2D-phase portrait, 3D-phase portrait, time series analysis, multistability and Lyapunov exponents over time. Further, sensitivity behavior for a range of initial conditions, both perturbed and unperturbed. The results suggest that the investigated equation exhibits a higher degree of multi-stability.

Analytical insights into solitary wave solutions of the fractional Estevez-Mansfield-Clarkson equation

<abstract><p>This study delved into the dynamics of wave solutions within the Estevez-Mansfield-Clarkson equation in fractional nonlinear space-time. Utilizing conformable fractional derivatives, the equation governing shallow water phenomena and fluid dynamics was transformed into a nonlinear ordinary differential equation. Applying the Riccati Bernoulli sub-ODE approach yielded a finite series representation. Notably, our findings revealed novel solitary wave solutions characterized by kink, anti-kink, periodic, and shock functions. Visualized through 3D and contour graphs, kink and periodic waves emerged as distinct observable manifestations. Intriguingly, the diversity of results surpassed previous results, contributing to a deeper understanding of the intricate dynamics inherent in the system.</p></abstract>

On the fractional space-time Fourier transforms

ABSTRACT The notion of a fractional space-time Fourier transform (FSFT) is outlined in this paper, and the properties of invertibility, linearity, Plancherel and others are derived. By establishing the relationship between the FSFT and space-time Fourier transform, a directional uncertainty principle (UP) and its specialization to coordinates are proved. Moreover, the Hardy UP of the FSFT is also obtained. The fractional Mustard convolution for space-time valued signals is introduced and written in terms of the standard convolution. Finally, the FSFT is employed in solving a partial differential equation in space-time analysis.

Distinct exact solutions for the conformable fractional derivative Gerdjikov-Ivanov equation via three credible methods

This paper reveals the conformable fractional space-time perturbed Gerdjikov-Ivanov (GI) equation, which is applied to nonlinear fibre optics together with photonic crystal fibres. The foremost intent of our operation is adopting the unified method, the modified F-expansion method and the modified Kudryashov method to hunt for definite solutions to the equation. In addition, by selecting fairish values, fluctuation behaviours of the solutions are drafted.

Numeric Fem’s Solution for Space-Time Diffusion Partial Differential Equations with Caputo–Fabrizion and Riemann–Liouville Fractional Order’s Derivatives

Abstract In this paper, we use the finite element method to solve the fractional space-time diffusion equation over finite fields. This equation is obtained from the standard diffusion equation by replacing the first temporal derivative with the new fractional derivative recently introduced by Caputo and Fabrizion and the second spatial derivative with the Riemann–Liouville fractional derivative. The existence and uniqueness of the numerical solution and the result of error estimation are given. Numerical examples are used to support the theoretical results.

Compatible extension of the [formula omitted]-expansion approach for equations with conformable derivative

In this study, the compatible extensions of the (G′/G)-expansion approach and the generalized (G′/G)-expansion scheme are proposed to generate scores of radical closed-form solutions of nonlinear fractional evolution equations. The originality and improvements of the extensions are confirmed by their application to the fractional space-time paired Burgers equations. The application of the proposed extensions highlights their effectiveness by providing dissimilar solutions for assorted physical forms in nonlinear science. In order to explain some of the wave solutions geometrically, we represent them as two- and three-dimensional graphs. The results demonstrate that the techniques presented in this study are effective and straightforward ways to address a variety of equations in mathematical physics with conformable derivative.

BSDEs generated by fractional space-time noise and related SPDEs

This paper is concerned with the backward stochastic differential equations whose generator is a weighted fractional Brownian field: Yt=ξ+∫tTYsW(ds,Bs)−∫tTZsdBs, 0≤t≤T, where W is a (d+1)-parameter weighted fractional Brownian field of Hurst parameter H=(H0,H1,⋯,Hd), which provide probabilistic interpretations (Feynman-Kac formulas) for certain linear stochastic partial differential equations with colored space-time noise. Conditions on the Hurst parameter H and on the decay rate of the weight are given to ensure the existence and uniqueness of the solution pair. Moreover, the explicit expression for both components Y and Z of the solution pair is given.

Propagation of Electromagnetic Waves in Fractional Space Time Dimensions

Propagation of Electromagnetic Waves in Fractional Space Time Dimensions

Study of a fractional stochastic heat equation

In this article, we study a d-dimensional stochastic nonlinear heat equation (SNLH) with a quadratic nonlinearity, forced by a fractional space-time white noise: ∂ t u − ∆u = ρ 2 u 2 +Ḃ , t ∈ [0, T ] , x ∈ R d , u 0 = φ. Two types of regimes are exhibited, depending on the ranges of the Hurst index H = (H 0 , ..., H d) ∈ (0, 1) d+1. In particular, we show that the local well-posedness of (SNLH) resulting from the Da Prato-Debussche trick, is easily obtained when 2H 0 + d i=1 H i > d. On the contrary, (SNLH) is much more difficult to handle when 2H 0 + d i=1 H i ≤ d. In this case, the model has to be interpreted in the Wick sense, thanks to a time-dependent renormalization. Helped with the regularising effect of the heat semigroup, we establish local well-posedness results for (SNLH) for all dimension d ≥ 1. Contents

The Inviscid Limit of Viscous Burgers at Nondegenerate Shock Formation

We study the vanishing viscosity limit of the one-dimensional Burgers equation near nondegenerate shock formation. We develop a matched asymptotic expansion that describes small-viscosity solutions to arbitrary order up to the moment the first shock forms. The inner part of this expansion has a novel structure based on a fractional spacetime Taylor series for the inviscid solution. We obtain sharp vanishing viscosity rates in a variety of norms, including \(L^\infty \). Comparable prior results break down in the vicinity of shock formation. We partially fill this gap.

A Comparative Analysis of Fractional Space-Time Advection-Dispersion Equation via Semi-Analytical Methods

The approximate solutions of the time fractional advection-dispersion equation are presented in this article. The nonlocal nature of solute movement and the nonuniformity of fluid flow velocity in the advection-dispersion process lead to the formation of a heterogeneous system, which can be modeled using a fractional advection-dispersion equation, which generalizes the classical advection-dispersion equation and replaces the time derivative with the fractional Caputo derivative. Researchers use a variety of numerical techniques to study such fractional models, but the nonlocality of the derivative having fractional order leads to high computation complexity and complex calculations, so the task is to find an efficient technique that requires less computation and provides greater accuracy when numerically solving such models. A innovative techniques, homotopy perturbation method and new iteration method, are used in connection with the Elzaki transform to solve the “fractional advection-dispersion equation” which provides the solution in the convergent series form. When the homotopy perturbation method is used with the Elzaki transform, fast convergent series solutions can be obtained with less computation. By solving some cases of time-fractional advection-dispersion equation with varied initial conditions with the help of new iterative transform method and homotopy perturbation transform method demonstrates the usefulness of the proposed methods.

Analytical soliton solutions for cold bosonic atoms (CBA) in a zigzag optical lattice model employing efficient methods

This research finds an equation in a continuous domain and a discrete equation governing the system of cold bosonic atoms (CBA) in a zigzag optical lattice using a continuum approximation. Many solutions to the equation were obtained using two distinct methods: the three-wave approach (multi-wave interaction, rational solutions, and rational solution interaction) and the extended sub-equation method. These analytical approaches are more effective, consistent, and comprehensive mathematical tools for obtaining various exact closed-form solutions for a wide range of fractional space-time nonlinear evolution equations encountered in optical physics, condensed matter physics, and plasma physics. The solutions generated are in the form of hyperbolic and trigonometric solutions, and other-form solutions are obtained. Three-dimensional graphics and contour plots are often used to depict the graphical representations of the combined soliton solutions. These findings will aid our understanding of the dynamics of the zigzag optical grids and many other structures formed by colder bosonic atoms. The applied approaches are more simple, efficient, and straightforward to obtain the closed-form solutions for various nonlinear evolution equations in the fields of nonlinear sciences and physical engineering.

Dynamical behaviors of various optical soliton solutions for the Fokas–Lenells equation

In this work, the new optical soliton solutions and interaction solutions for the space-time fractional Fokas–Lenells equation with fractional [Formula: see text]-derivatives are constructed via three mathematical analytical techniques, namely the extended SE method, unified solver method, and three-wave methods. The results have proved the efficiency of the suggested techniques for obtaining abundant optical soliton solutions to nonlinear evolution equations (NLEEs) and closed-form solutions in the forms of rational function solutions; hyperbolic and trigonometric function solutions and multi-wave interaction solutions are obtained. These techniques are more efficient, robust, and powerful mathematical tools for acquiring several optical soliton solutions for many other fractional space-time NLEEs that arise in optical physics and plasma physics. The graphical representations of the combined optical solitons are demonstrated using three- and two-dimensional graphics.

A plentiful supply of soliton solutions for DNA Peyrard–Bishop equation by means of a new auxiliary equation strategy

In this paper, we present a work on dynamic equation of Deoxyribonucleic acid (DNA) derived from the Peyrard–Bishop (PB) model oscillator chain for various dynamical solitary wave solutions. In order to construct novel soliton solutions in the DNA dynamic PB model with beta-derivative, the efficiency of the newly developed algorithms is being investigated, which could include a new auxiliary equation strategy (NAES). Some precise soliton solutions comprising dark, light and other forms of multi-wave soliton solutions are achieved via the proposed methods. Furthermore, mathematical models demonstrate the singularity of our work in comparison to current literary materials and even describe some results using the classic Peyrard–Bishop model. All the established results contribute to the possibility of extending the approach to solve other nonlinear equations of fractional space–time derivatives in nonlinear sciences. The strategy that has been proposed recently is specific and is being employed to produce novel closed-form solutions for all many other FNLEEs.

New optical soliton solutions via two distinctive schemes for the DNA Peyrard–Bishop equation in fractal order

The deoxyribonucleic acid (DNA) dynamical equation, which emerges from the oscillator chain known as the Peyrard–Bishop (PB) model for abundant optical soliton solutions, is presented, along with a novel fractional derivative operator. The Kudryashov expansion method and the extended hyperbolic function (HF) method are used to construct novel abundant exact soliton solutions, including light, dark, and other special solutions that can be directly evaluated. These newly formed soliton solutions acquired here lead one to ask whether the analytical approach could be extended to deal with other nonlinear evolution equations with fractional space–time derivatives arising in engineering physics and nonlinear sciences. It is noted that the newly proposed methods’ performance is most reliable and efficient, and they will be used to construct new generalized expressions of exact closed-form solutions for any other NPDEs of fractional order.

Geometrical study and solutions for family of burgers-like equation with fractional order space time

Oneofthebasichydrodynamicmodelsisthefamous Burgersformula. Thisdescribesthe creation ofthestickyparticlevelocityfieldwhichinteractsonlywhenitcollides. In the present paper, we are going to give a systematic formulation of a new family of Burgers-like equations with fractional space–time order. Exact solutions are constructed by using the direct method. A geometrical study for the fractional space is presented and applied to the obtained solutions. The present solutions are the same as the integer one and revealed that the arbitrariness of fraction orders makes systems even richer.

Plenty of soliton solutions to the DNA Peyrard-Bishop equation via two distinctive strategies

Here, the Deoxyribo-Nucleic Acid (DNA) dynamic equation that arises from the oscillator chain named the Peyrard-Bishop model for plenty of solitary wave solutions is presented. The efficacy of newly designed algorithms are investigated, namely, the extended Auxiliary equation method and Kudryashov expansion method for constructing the new solitary wave solutions of the DNAdynamic Peyrard-Bishop model with beta-derivative. Here, the proposed methods contribute to a range of accurate solutions for soliton, including light, dark, and other solutions are obtained. In addition, some results are also clarified by computer simulations demonstrating the uniqueness of our work relative to the existing literature on the classic Peyrard-Bishop model. These solutions lead to the issue of the possibility to expand the method to deal with other non-linear equations of fractional space-time derivatives in non-linear science. It is noted that the newly proposed approach is accurate and is used to create new general closed-form solutions for all other fractional NPDEs.