Conformable Fractional Derivative Research Articles

Wave Solutions in (3 + 1)-Dimensional Generalized Fractional mKdV-ZK Equation utilizing Jacobi Elliptic Functions

Abstract In this research, we investigate the e ects of fractional order on the (3 + 1)-dimensional general-
ized space-time fractional modi ed KdV-Zakharov-Kuznetsov (mKdV-ZK) equation. We approach
the problem by utilizing the conformable fractional derivative. By reducing the mKdV-ZK equation
to an integer order nonlinear ordinary di erential equation, we apply the Jacobi elliptic function
method to nd exact solutions. These solutions are speci cally tailored for the fractional order of
the (3 + 1)-dimensional generalized mKdV-ZK equation, encompassing solitary waves, shock waves,
and periodic waves. We also compare these exact solutions with fractional solutions to gain further
insights. Notably, our approach demonstrates the feasibility of solving nonlinear time-fractional
di erential equations with conformable derivatives. Several diagrams have been included to visually
depict the behavior of the solutions under fractional order when certain special parameter values
are employed.

Non-Markovian dynamics of time-fractional open quantum systems

The Time-Fractional Schrödinger Equation (TFSE) is suitable to describe the non-Markovian dynamics of a quantum system exposed to an environment, which is instructive for understanding and characterizing the time behavior of actual physical systems. Three popular TFSEs, namely Naber’s TFSE I, Naber’s TFSE II, and XGF’s TFSE, have been introduced in the form of ∂β∂tβ with β∈0,1. However, they suffer from the drawbacks. In their respective descriptions, the total probability for finding a particle in the entire space equals one only when β=1, implying that time-fractional quantum mechanics violates quantum mechanical probability conservation. By applying the three TFSEs to a basic single-qubit open system model, we discover that in describing the non-Markovian evolution of the system, the three TFSEs are limited by their own applicable ranges of β. This indicates that the three TFSEs cannot effectively describe the non-Markovian quantum dynamics. To address these issues, we introduce a well-performed TFSE by constructing a new analytic continuation of time without iβ combined with the conformable fractional derivative. In its description, for one thing, the total probability for finding the system in any single-qubit state equals one when β∈0,1. For another, the system evolves correctly in the non-Markovian manner at all values of β. Furthermore, we study the performances of the four TFSEs applying to a two-qubit open system model and show that our TFSE still possesses the above two advantages compared with the other three TFSEs.

Extracting the Ultimate New Soliton Solutions of Some Nonlinear Time Fractional PDEs via the Conformable Fractional Derivative

Nonlinear fractional-order differential equations have an important role in various branches of applied science and fractional engineering. This research paper shows the practical application of three such fractional mathematical models, which are the time-fractional Klein–Gordon equation (KGE), the time-fractional Sharma–Tasso–Olever equation (STOE), and the time-fractional Clannish Random Walker’s Parabolic equation (CRWPE). These models were investigated by using an expansion method for extracting new soliton solutions. Two types of results were found: one was trigonometric and the other one was an exponential form. For a profound explanation of the physical phenomena of the studied fractional models, some results were graphed in 2D, 3D, and contour plots by imposing the distinctive results for some parameters under the oblige conditions. From the numerical investigation, it was noticed that the obtained results referred smooth kink-shaped soliton, ant-kink-shaped soliton, bright kink-shaped soliton, singular periodic solution, and multiple singular periodic solutions. The results also showed that the amplitude of the wave augmented with the pulsation in time, which derived the order of time fractional coefficient, remarkably enhanced the wave propagation, and influenced the nonlinearity impacts.

Dynamical analysis of optical soliton solutions for CGL equation with Kerr law nonlinearity in classical, truncated [formula omitted]-fractional derivative, beta fractional derivative, and conformable fractional derivative types

The study of optical soliton solutions plays a vital role in nonlinear optics. The foremost area of optical solitons research encompasses around optical fiber, telecommunication, meta-surfaces and others related technologies. The aim of this work is to integrate optical soliton solutions of the complex Ginzburg-Landau (CGL) model with Kerr law nonlinearity, also showing the effect of diverse fraction derivative and comparing it with the classical form. Here, the local derivative is used as the conformable wisdom known as the truncated M-fractional derivative, beta fractional derivative, and conformable fraction derivative. We also deliberated on some assets satisfied by the derivative. The CGL model is useful to describe the light propagation in optical communications, optical transmission, and nonlinear optical fiber. Under the right circumstances, the affectionate unified scheme is implemented for the complex Ginzburg-Landau model to generate the optical wave pattern. For α1=2α2, the unified scheme generates the solution of CGL model in terms of hyperbolic, trigonometric, and rational function solutions. This scheme provides some novel optical solitons such as periodic waves, periodic with rogue waves, breather waves, different types of periodic rogue waves, a singular soliton solution, and rogue with the periodic wave for the special value of the free parameters. For α2=-ρ+6α18, the unified scheme generates the solutions of CGL model in terms of hyperbolic, trigonometric, and rational function solutions. This scheme offers some fresh optical solitons such as periodic rogue waves, multi-rogue waves, double periodic waves, and periodic waves. In numerical argument, the wave patterns are offered with 3-D and density plots. To test the stability of the obtained solutions, we show diverse fractional forms such as beta time fractional, and conformable time fractional derivative and compare these fractional derivatives with their classical form in 2D plots. The investigation reveals innovative and explicit solutions, providing insight into the dynamics of the related physical processes. This paper provides a comprehensive examination of the obtained solutions, emphasizing their distinct features and depictions using unified technique. These findings are especially advantageous for specialists in the fields of nonlinear science and mathematical physics, providing significant insights into the behavior and development of nonlinear waves in various physical situations.

The Generalized Fractional-Order Fisher Equation: Stability and Numerical Simulation

This study examines the stability and numerical simulation of the generalized fractional-order Fisher equation. The equation serves as a mathematical model describing population dynamics under the influence of factors such as natural selection and migration. We propose an implicit exponential finite difference method to solve this equation, considering the conformable fractional derivative. Furthermore, we analyze the stability of the method through theoretical considerations. The method involves transforming the problem into systems of nonlinear equations at each time since our method is an implicit method, which is then solved by converting them into linear equations systems using the Newton method. To test the accuracy of the method, we compare the results obtained with exact solutions and with those available in the literature. Additionally, we examine the symmetry of the graphs obtained from the solution to examine the results. The findings of our numerical simulations demonstrate the effectiveness and reliability of the proposed approach in solving the generalized fractional-order Fisher equation.

Bifurcation, phase plane analysis and exact soliton solutions in the nonlinear Schrodinger equation with Atangana's conformable derivative

The nonlinear Schrodinger equation (NLSE) with Atangana's conformable fractional derivative (ACFD) is an equation that describes how the quantum state of a physical system changes in time. This present study examines the exact soliton results (ESRs) and analyze their bifurcations and phase plane (PP) of the NLSE with ACFD through the modified (G′/G)-expansion scheme (MG′/GES). Firstly, ACFD and its properties are included and then by MG′/GES, the exact soliton results and analyze their bifurcations and phase plane of NLSE, displayed with the ACFD, are classified. These obtained ESRs include periodic wave pulse, bright-dark periodic wave pulse, multiple bright-dark soliton pulse, and so many types. We provide 2-D, 3-D and contour diagrams, Hamiltonian function (HF) for phase plane dynamics analysis and bifurcation analysis diagrams to examine the nonlinear effects and the impact of fractional and time parameters on these obtained ESRs. The obtained ESRs illustration the novelty and prominence of the dynamical construction and promulgation performance of the resultant equation and also have practical effects for real world problems. The ESRs obtained with MG′/GES show that this scheme is very humble, well-organized and can be implemented to other the NLSE with ACFD.

Dynamical behavior of water wave phenomena for the 3D fractional WBBM equations using rational sine-Gordon expansion method

To examine the dynamical behavior of travelling wave solutions of the water wave phenomenon for the family of 3D fractional Wazwaz-Benjamin-Bona-Mahony (WBBM) equations, this work employs the rational Sine-Gordon expansion (RSGE) approach based on the conformable fractional derivative. The method generalizes the well-known sine-Gordon expansion using the sine-Gordon equation as an auxiliary equation. In contrast to the conventional sine-Gordon expansion method, it takes a more general approach, a rational function rather than a polynomial one of the solutions of the auxiliary equation. The method described above is used to generate various solutions of the WBBM equations for hyperbolic functions, including soliton, singular soliton, multiple-soliton, kink, cusp, lump-kink, kink double-soliton, etc. The RSGE method contributes to our understanding of nonlinear phenomena, provides exact solutions to nonlinear equations, aids in studying solitons, advances mathematical techniques, and finds applications in various scientific and engineering disciplines. The answers are graphically shown in three-dimensional (3D) surface plots and contour plots using the MATLAB program. The resolutions of the equation, which have appropriate parameters, exhibit the absolute wave configurations in all screens. Furthermore, it can be inferred that the physical characteristics of the discovered solutions and their features may aid in our understanding of the propagation of shallow water waves in nonlinear dynamics.

Travelling Wave Solutions for Some Time-Fractional Nonlinear Differential Equations

This study employs the powerful generalized Kudryashov method to address the challenges posed by fractional differential equations in mathematical physics. The main objective is to obtain new exact solutions for three important equations: the (3+1)-dimensional time fractional Jimbo-Miwa equation, the (3+1)-dimensional time fractional modified KdV-Zakharov-Kuznetsov equation, and the (2+1)-dimensional time fractional Drinfeld-Sokolov-Satsuma-Hirota equation. The generalized Kudryashov method is highly versatile and effective in addressing nonlinear problems, making it a pivotal component in our research. Its adaptability makes it useful in diverse scientific disciplines. The method simplifies complex equations, improving our analytical capabilities and deepening our understanding of system dynamics. Additionally, we define fractional derivatives using the conformable fractional derivative framework, providing a strong foundation for our mathematical investigations. This paper examines the effectiveness of the generalized Kudryashov method in solving complex challenges presented by fractional differential equations and aims to provide guidance for future studies.

Wave propagation analysis in the modified nonlinear time fractional Harry Dym equation: Insights from Khater II method and B-spline schemes

This study aims to investigate the modified nonlinear time fractional Harry Dym equation using analytical and numerical techniques. The modified nonlinear time fractional Harry Dym equation is a generalization of the classical Harry Dym equation, which describes the propagation of nonlinear waves in a variety of physical systems. The conformable fractional derivative is used to define the time fractional derivative in the equation, which provides a natural and straightforward approach. The Khater II method, a powerful analytical technique, is employed to obtain approximate solutions for the equation. Additionally, three numerical schemes, namely, Cubic-B-spline, Quantic-B-spline and Septic-B-spline schemes, are developed and implemented to solve the equation numerically. The numerical results are compared with other numerical solutions to assess the accuracy and efficiency of the proposed schemes. The physical meaning of the modified nonlinear time fractional Harry Dym equation is discussed in detail, and its relation to other nonlinear evolution equations is highlighted. The results of this study provide new insights into the behavior of nonlinear waves in physical systems and contribute to a better understanding of the physical characterizations of the modified nonlinear time fractional Harry Dym equation.

Thenovel numerical solutions for conformable fractional Kuramoto-Sivashinsky equations by using Cq-HATM and CHPETM

The present study used the novel conformable fractional methods to examine the nonlinear Kuramoto-Sivashinsky equations of the conformable fractional derivative. The Kuramoto-Sivashinsky equation is a mathematical model employed to elucidate a diverse range of chemical and physical phenomena, encompassing chemical reaction-diffusion processes, plasma instabilities, viscous flow issues, flame front propagation, and magnetized plasmas. The present inquiry provides an examination of convergence and error for the future scheme. The conformable q-homotopy analysis transform method (Cq-HATM) provides h-curves that illustrate the convergence range of the series solution achieved. In order to validate the effectiveness and suitability of the Cq-HATM, we examine separate instances. In this study, we introduce an application that demonstrates the potential benefits and effectiveness of the proposed approach. Furthermore, an error analysis is performed in order to verify the accuracy of the scheme. Numerical simulations are conducted to validate the precision of the forthcoming approach. The findings obtained from the numerical and graphical analyses are presented in this study. The method provided in this study showcases a notable degree of computational precision and ease in examining and resolving intricate phenomena related to conformable fractional nonlinear partial differential equations within the domains of science and technology.

Novel solution structures localized in fractional integrable coupled nonlinear Schrödinger equations

Fractional calculus highlights its importance and has been expanded to many fields. This paper focuses on fractional integrable systems and their novel structural solutions to comply with the forefront development of nonlinear waves. Specifically, a system of fractional integrable coupled nonlinear Schrödinger (CNLS) equations are first derived from the associated linear spectral problem equipped with space-time conformable fractional derivatives of different orders. Then, by extending Darboux transformation (DT) and its generalized version to the derived fractional CNLS equations, we obtained fractional single-soliton solutions and double- and triple-semirational solutions that obey the dual power laws of independent variables. Under the dominant influences of space-time fractional orders, some novel solution structures are finally revealed based on the obtained solutions. In terms of characteristics, these spatial structures exhibit inclination, symmetry and spikes. Observing from the perspective of propagation, some of these obtained solutions have time-varying velocities and widths. From the perspective of spatiotemporal structures, some solutions form collapsed quadrangular pyramids. All of these novel solution structures cannot be supported by the integer-order CNLS equations.

Existence, uniqueness, and Ulam stability of solutions of fractional conformable Langevin system on the ethane graph

The present study examines the Langevin coupled boundary value problem with the fractional conformable derivative on the ethane graph. The main objective of this research is to establish the existence and uniqueness of solutions with the help of the fixed point theorem and to analyze the Ulam stability of the corresponding problem. Several examples are provided to demonstrate the obtained results.

An Efficient New Technique for Solving Nonlinear Problems Involving the Conformable Fractional Derivatives

In this paper, an efficient new technique is used for solving nonlinear fractional problems that satisfy specific criteria. This technique is referred to as the double conformable fractional Laplace-Elzaki decomposition method (DCFLEDM). This approach combines the double Laplace-Elzaki transform method with the Adomian decomposition method. The fundamental concepts and findings of the recently suggested transformation are presented. For the purpose of assessing the accuracy of our approach, we provide three examples and introduce the series solutions of these equations using DCLEDM. The results show that the proposed strategy is a very effective, reliable, and efficient approach for addressing nonlinear fractional problems using the conformable derivative.

Hybrid cubic and hyperbolic b-spline collocation methods for solving fractional Painlevé and Bagley-Torvik equations in the Conformable, Caputo and Caputo-Fabrizio fractional derivatives

In this paper, we approximate the solution of fractional Painlevé and Bagley-Torvik equations in the Conformable (Co), Caputo (C), and Caputo-Fabrizio (CF) fractional derivatives using hybrid hyperbolic and cubic B-spline collocation methods, which is an extension of the third-degree B-spline function with more smoothness. The hybrid B-spline function is flexible and produces a system of band matrices that can be solved with little computational effort. In this method, three parameters m, η, and λ play an important role in producing accurate results. The proposed methods reduce to the system of linear or nonlinear algebraic equations. The stability and convergence analysis of the methods have been discussed. The numerical examples are presented to illustrate the applications of the methods and compare the computed results with those obtained using other methods.

An insight into the solitonic structure of two distinct fractional nonlinear Schrödinger models in optical fiber using conformable fractional derivative

The goal of this work was to use conformable fractional derivative sense to create some crucial solitary waves for two models of space–time fractional nonlinear Schrödinger equations. We use the unified solver approach to accomplish this goal in a fully unified way. This solution is robust, practical, dependable, and simple to use. The obtained solutions are extremely important for describing critical complicated phenomena in fractional quantum mechanics, optical fiber communications, and energy applications. Some simulations are provided to demonstrate the behavior of the obtained solutions when appropriate physical parameters are used. It was noted that by increasing the fractal factors, the nonlinear wave propagates with a changing phase and wave frequency. Our research may open up new possibilities for optical manipulation in practical applications. Finally, further fractional physical models can be solved using the suggested technique.

Some Properties of Revolution Surfaces in Euclidean 3-space with Conformable Fractional Derivative

Some Properties of Revolution Surfaces in Euclidean 3-space with Conformable Fractional Derivative

A Classical Aspect of the Dirac Equation in the Context of Conformable Fractional Derivative

A Classical Aspect of the Dirac Equation in the Context of Conformable Fractional Derivative

Proportional fractional Dirac dynamic system

<abstract><p>In this study, considering the proportional fractional derivative, which is a generalization of the conformable fractional derivative, we provided some important spectral properties such as the reality of eigenvalues, the orthogonality of eigenfunctions, the self-adjointness of the operator, the asymptotic estimations of eigenfunctions, and Picone's identity for a proportional Dirac system on an arbitrary time scale. We also presented graphics representing the eigenfunctions of the Dirac system on a time scale, produced by taking advantage of the proportional fractional derivative with some special cases. The main purpose of presenting these graphics was to examine the effect of the proportional fractional derivative on the Dirac system on a time scale, as well as the effect of the eigenvalues, which are meaningful for the subject we were studying for the solution functions.</p></abstract>

Analysis of stochastic delay differential equations in the framework of conformable fractional derivatives

<abstract><p>In numerous domains, fractional stochastic delay differential equations are used to model various physical phenomena, and the study of well-posedness ensures that the mathematical models accurately represent physical systems, allowing for meaningful predictions and analysis. A fractional stochastic differential equation is considered well-posed if its solution satisfies the existence, uniqueness, and continuous dependency properties. We established the well-posedness and regularity of solutions of conformable fractional stochastic delay differential equations (CFrSDDEs) of order $ \gamma\in(\frac{1}{2}, 1) $ in $ \mathbb{L}^{\mathrm{p}} $ spaces with $ \mathrm{p}\geq2 $, whose coefficients satisfied a standard Lipschitz condition. More specifically, we first demonstrated the existence and uniqueness of solutions; after that, we demonstrated the continuous dependency of solutions on both the initial values and fractional exponent $ \gamma $. The second section was devoted to examining the regularity of time. As a result, we found that, for each $ \Phi\in(0, \gamma-\frac{1}{2}) $, the solution to the considered problem has a $ \Phi- $H$ \ddot o $lder continuous version. Lastly, two examples that highlighted our findings were provided. The two main elements of the proof were the Burkholder-Davis-Gundy inequality and the weighted norm.</p></abstract>

Intuitionistic Fuzzy Generalized Conformable Fractional Derivative

Intuitionistic Fuzzy Generalized Conformable Fractional Derivative