Modified Riemann Liouville Derivative Research Articles

Exploring Solitons Solutions of a (3+1)-Dimensional Fractional mKdV-ZK Equation

This study presents the application of the ϕ6 model expansion technique to find exact solutions for the (3+1)-dimensional space-time fractional modified KdV-Zakharov-Kuznetsov equation under Jumarie’s modified Riemann–Liouville derivative (JMRLD). The suggested method captures dark, periodic, traveling, and singular soliton solutions, providing deep insights into wave behavior. Clear graphics demonstrate that the solutions are greatly affected by changes in the fractional order, deepening our understanding and revealing the hidden dynamics of wave propagation. The considered equation has several applications in fluid dynamics, plasma physics, and nonlinear optics.

Construction of Soliton Solutions of Time-Fractional Caudrey–Dodd–Gibbon–Sawada–Kotera Equation with Painlevé Analysis in Plasma Physics

Fractional calculus with symmetric kernels is a fast-growing field of mathematics with many applications in all branches of science and engineering, notably electromagnetic, biology, optics, viscoelasticity, fluid mechanics, electrochemistry, and signals processing. With the use of the Sardar sub-equation and the Bernoulli sub-ODE methods, new trigonometric and hyperbolic solutions to the time-fractional Caudrey–Dodd–Gibbon–Sawada–Kotera equation have been constructed in this paper. Notably, the definition of our fractional derivative is based on the Jumarie’s modified Riemann–Liouville derivative, which offers a strong basis for our mathematical explorations. This equation is widely utilized to report a variety of fascinating physical events in the domains of classical mechanics, plasma physics, fluid dynamics, heat transfer, and acoustics. It is presumed that the acquired outcomes have not been documented in earlier research. Numerous standard wave profiles, such as kink, smooth bell-shaped and anti-bell-shaped soliton, W-shaped, M-shaped, multi-wave, periodic, bright singular and dark singular soliton, and combined dark and bright soliton, are illustrated in order to thoroughly analyze the wave nature of the solutions. Painlevé analysis of the proposed study is also part of this work. To illustrate how the fractional derivative affects the precise solutions of the equation via 2D and 3D plots.

Extended Fan’s sub-ODE technique and its application to a fractional nonlinear coupled network including multicomponents — LC blocks

The present work deal with the improved Fan’s sub-ordinary differential equation (ODE) method and peruse abundant soliton solutions for a fractional nonlinear coupled network including multicomponents — LC blocks. The extended Fan’s sub-ODE technique is applied on a fractional nonlinear coupled electrical transmission network in addition — LC blocks and through some graphical representations we exhibit novelties. The integer version of the electrical circuit explored here is recently investigated but its fractional partner is weakly scrutinized. Then, we pay attention on the fractional nonlinear coupled electrical network including multicomponents — LC blocks and peruse peculiar voltage waves from the voltage wave equation governing their dynamics. The voltage waves dynamics equation is obtained by means of the Kirchhoff laws’ including the modified Riemann–Liouville derivatives. Through the fractional complex transform, we derive a nonlinear ordinary differential equation that is tackled by the suggested technique. The main gaps of the present investigation based not only on the use of the fractional version of the electrical model, but, also in the capacitor charge nonlinearity’s (the integer version used Qn,mt=C0Vn,m+b1Vn,m2, the fractional partner explored now Qn,mt=C0Vn,m+b2Vn,m3) and the proposed straightforward implemented. In addition, the technique performed here led us to discover new kind of voltage waves by using five auxiliary equation’s key parameters, more than two or three ones commonly involved into such straightforward method reported in the literature. We also prove the existence of our findings by examining the fixed points, phase portraits and related potentials according accurate electrical parameters ranges, coupling elements values wisely chosen. The current technique is reported in the present investigation in manner to be replicable to many other physical systems (electrical, mechanical, optics, acoustics to cite just a few) described by nonlinear evolution equations fractional or not.

Soliton solutions of the space–time fractional nonlocal nonlinear Schrödinger equation

In this paper, the space–time fractional nonlocal nonlinear Schrödinger equations under Jumarie modified Riemann–Liouville derivative is studied. The soliton solutions are formally derived by using the extended tanh method of fractional Riccati subequation and the ansatz method combined with the Hermite–Gaussian function and Mittag-Leffler function. Different types of solutions are obtained and the necessary conditions for the existence of solutions are given. With the help of Maple, we plot figures exhibit that solitons are forming as fractional order λ increasing, and also, we noticed that at different order n of the Hermite–Gaussian function Hn(y), mainly affects the amplitude of soliton. Furthermore, we have discussed the new soliton solutions in the fractional case. It is of great significance to understand complex physical phenomena described by fractional nonlocal nonlinear equations.

Analytical solutions and mathematical simulation of traveling wave solutions to fractional order nonlinear equations

We investigate the tanh–coth method’s usefulness in understanding space–time fractional nonlinear evolution problems. To use this method and then examine their plots, we take into consideration three significant equations: the space–time fractional Burgers equation, the space–time fractional regularized long wave equation, and the space–time fractional Boussinesq equation. The modified Riemann–Liouville derivatives are used to define the fractional derivative. The approach provides a linearization- or small perturbation-free analytical solution in the form of a convergent series with easily calculable components. For the aforementioned nonlinear fractional equations, we have discovered their precise solutions. We utilize a generalized fractional complex transform to translate these to ordinary differential equations which subsequently resulted into number of exact solutions. The method provides us abundant analytical traveling wave solutions containing fewer number parameters with the aid of Mathematica and MATLAB.

CONSTRUCTION OF CLOSED FORM SOLITON SOLUTIONS TO THE SPACE-TIME FRACTIONAL SYMMETRIC REGULARIZED LONG WAVE EQUATION USING TWO RELIABLE METHODS

The employment of nonlinear fractional partial differential equations (NLFPDEs) is not limited to branches of mathematics entirely but also applicable in other science fields such as biology, physics and engineering. This paper derives some solitary wave solutions for the space-time fractional symmetric regularized long wave (SRLW) equation by means of the improved [Formula: see text]-expansion approach and the [Formula: see text]-expansion method. We use the definition of the Jumarie’s modified Riemann–Liouville derivative to handle the fractional derivatives appearing in this equation. Diverse types of soliton solutions are successfully expressed on the form of rational, hyperbolic, trigonometric, and complex functions. We extract kink wave, internal solitary wave, and solitary wave solutions. The performances of the proposed methods are compared with each other. Moreover, we compare the constructed results with some published solutions. The long behaviors of the obtained solutions are plotted in 2D and 3D figures. The resulting outcomes point out that the used techniques promise to empower us to deal with more NLFPDEs arising in mathematical physics.

A Study of the Soliton Solutions with an Intrinsic Fractional Discrete Nonlinear Electrical Transmission Line

This study aims to identify soliton structures as an inherent fractional discrete nonlinear electrical transmission lattice. Here, the analysis is founded on the idea that the electrical properties of a capacitor typically contain a non-integer-order time derivative in a realistic system. We construct a non-integer order nonlinear partial differential equation of such voltage dynamics using Kirchhoff’s principles for the model under study. It was discovered that the behavior for newly generated soliton solutions is impacted by both the non-integer-order time derivative and connected parameters. Regardless of structure, the fractional-order alters the propagation velocity of such a voltage wave, thus bringing up a localized framework under low coupling coefficient values. The generalized auxiliary equation method drove us to these solitary structures while employing the modified Riemann–Liouville derivatives and the non-integer order complex transform. As well as addressing sensitivity testing, we also investigate how our model’s altered dynamical framework shows quasi-periodic properties. Some randomly selected solutions are shown graphically for physical interpretation, and conclusions are held at the end.

The study of new optical soliton solutions to the time-space fractional nonlinear dynamical model with novel mechanisms

In this work, we successfully employ two constructive mathematical tools such as the extended rational sine-cosine/sinh-cosh and novel Φ6-expansion model techniques, for obtaining different types of optical solitons and other solutions to the (2+1)-dimensional time-space fractional nonlinear Schrodinger (NLS) equation within the modified Riemann–Liouville derivatives. Solitary wave, dark, singular, and their combined solutions are retrieved. Moreover, hyperbolic, trigonometric, plane wave periodic wave, and Jacobi elliptic function solutions are also derived. The novelty of the obtained solutions is revealed by comparing them with the previous finding. By selecting suitable parametric values, numerical simulation and physical interpretations of the achieved outcomes are manifested with interesting figures presenting the meaning of the acquired results. The novel approaches are used through the symbolic computations impart a dynamical and potent mathematical implement for solving various benevolent nonlinear wave problems. The results suggest that these constructing techniques are concise, direct, efficient, and straightforward. The resulting solutions are novel, intriguing, and potentially helpful for understanding energy transit and diffusion processes in mathematical models of several disciplines of interest, including telecommunication, engineering, mathematical biology, mathematical physics, and nonlinear optics etc.

Note on the fractional Mittag-Leffler functions by applying the modified Riemann-Liouville derivatives

In this article, the fractional derivatives in the sense of the modified Riemann-Liouville derivative is employed for constructing some results related to Mittag-Leffler functions and established a number of important relationships between the Mittag-Leffler functions and Wright function.

Novel Analytical Approach for the Space-Time Fractional (2+1)-Dimensional Breaking Soliton Equation via Mathematical Methods

The aim of this work is to build novel analytical wave solutions of the nonlinear space-time fractional (2+1)-dimensional breaking soliton equations, with regards to the modified Riemann–Liouville derivative, by employing mathematical schemes, namely, the improved simple equation and modified F-expansion methods. We used the fractional complex transformation of the concern fractional differential equation to convert it for the solvable integer order differential equation. After the successful implementation of the presented methods, a comprehensive class of novel and broad-ranging exact and solitary travelling wave solutions were discovered, in terms of trigonometric, rational and hyperbolic functions. Hence, the present methods are reliable and efficient for solving nonlinear fractional problems in mathematics physics.

New optical soliton solutions of fractional perturbed nonlinear Schrödinger equation in nanofibers

In this article, the space-time fractional perturbed nonlinear Schrödinger equation (NLSE) in nanofibers is studied using the improved [Formula: see text] expansion method (ITEM) to explore new exact solutions. The perturbed nonlinear Schrodinger equation is a nonlinear model that occurs in nanofibers. The ITEM is an efficient method to obtain the exact solutions for nonlinear differential equations. With the help of the modified Riemann–Liouville derivative, an equivalent ordinary differential equation has been obtained from the nonlinear fractional differential equation. Several new exact solutions to the fractional perturbed NLSE have been devised using the ITEM, which is the latest proficient method for analyzing nonlinear partial differential models. The proposed method may be applied for searching exact travelling wave solutions of other nonlinear fractional partial differential equations that appear in engineering and physics fields. Furthermore, the obtained soliton solutions are depicted in some 3D graphs to observe the behaviour of these solutions.

On the solutions of the space–time fractional coupled Jaulent–Miodek equation associated with energy-dependent Schrödinger potential

The objective of this paper is to construct the entirely analytical solutions of the space–time fractional coupled Jaulent–Miodek equation involving Jumarie’s modified Riemann–Liouville derivative in mathematical physics. Via the improved fractional sub-equation method and the (G(α)∕G)-expansion method, we obtain new types of the exact solutions of the space–time fractional coupled Jaulent–Miodek equation.

On the soliton solutions for an intrinsic fractional discrete nonlinear electrical transmission line

This paper focuses on finding soliton solutions for an intrinsic fractional discrete nonlinear electrical transmission lattice. Our investigation is based on the fact that for a realistic system, the electrical characteristics of a capacitor (and an inductor via skin effect) should include a fractional-order time derivative. In this respect for the model under consideration, we derive a fractional nonlinear partial differential equation for the voltage dynamics by applying the Kirchhoff’s laws. It is realized that the behavior of new soliton solutions obtained is influenced by the fractional-order time derivative as well as the coupling values. The fractional order also modifies the propagation velocity of the voltage wave notwithstanding their structure and tends to set up localized structure for low coupling parameter values. However, for a high value of the coupling parameter, the fractional order is less seen on the shapes of the new solitary solutions that are analytically derived. Several methods such as the Kudryashov method, the $$(G'/G)$$ -expansion method, the Jacobi elliptical functions method and the Weierstrass elliptic function expansion method led us to derive these solitary solutions while using the modified Riemann–Liouville derivatives in addition to the fractional complex transform. An insight into the overall dynamics of our network is provided through the analysis of the phase portraits.

Optical singular and dark solitons to the (2 + 1)-dimensional time–space fractional nonlinear Schrödinger equation

In this study, the (2+1)-dimensional nonlinear Schrödinger equation with fractional-order derivative in both time and space is considered within the modified Riemann–Liouville derivatives. This fractional-order equation is newly derived by using the fractional variational principle and the semi-inverse method. By introducing the fractional complex transform, the (G′∕G)-expansion method, the tanh–coth method, and the modified simple equation method are used to obtain the explicit analytical solutions of this equation. As a result, optical singular and dark soliton solutions are obtained. The behaviours of the soliton solutions are expressed by 2D and 3D graphs.

Analytical solutions of linear fractional partial differential equations using fractional Fourier transform

This paper discusses the analytical solutions of fractional partial differential equations using Integral Transform method. The fractional derivatives are considered with reference to modified Riemann–Liouville derivatives. Fractional Fourier transform (FrFT) is applied to solve fractional heat diffusion, fractional wave, fractional telegraph and fractional kinetic equations. The method proposed here is effective enough to work on these equations efficiently.

Study on Fractional Differential Equations with Modified Riemann–Liouville Derivative via Kudryashov Method

Abstract In this work, the Kudryashov method is handled to find exact solutions of nonlinear fractional partial differential equations in the sense of the modified Riemann–Liouville derivative as given by Guy Jumarie. Firstly, these fractional equations can be turned into another nonlinear ordinary differential equations by fractional complex transformation. Then, the method is applied to solve the space-time fractional Symmetric Regularized Long Wave equation and the space-time fractional generalized Hirota–Satsuma coupled KdV equation. The obtained solutions include generalized hyperbolic functions solutions.

Bifurcation of traveling waves and exact solutions of Kadomtsev–Petviashvili modified equal width equation with fractional temporal evolution

The article is devoted in the study of the Kadomtsev–Petviashvili modified equal width equation with fractional temporal evolution in which modified Riemann–Liouville derivative is considered. We discuss the dynamical behavior of the system using bifurcation theory in different parametric regions. We also depict the phase portraits of the traveling wave solutions and obtain explicit traveling waves including solitary wave and breaking wave solution. We apply the $$(G'/G)$$ -expansion method and the F-expansion method along with fractional complex transformation, and obtain a large variety of exact traveling wave solutions which includes solitary wave, kink-type wave, and periodic wave solutions of the equation. Finally, we demonstrate remarkable features of the traveling wave solutions via interesting figures and phase portraits.

Multi-solitary wave solutions to the general time fractional Sharma–Tasso–Olver equation and the time fractional Cahn–Allen equation

The space time fractional Sharma–Tasso–Olver (STO) equation and the time fractional Cahn–Allen (CA) equation are well-designated to fission and fusion phenomena including those for solitons, electromagnetic interactions, quantum relativistic atom theory, phase isolation in several components bass system, the relativistic energy-momentum relation. In this article, we have exerted the exact solution to STO and CA equation in the light of fractional derivative (e.g. modified Riemann–Liouville derivative). By using wave transformation the ODE of integer order is generated by the fractional partial differential equation (FPDE). We investigated the travelling wave solution to these equations through the recently established double-expansion method. The results obtained in view of hyperbolic, trigonometric and rational functions containing parameters. We have demonstrated that this method is convenient, effective and powerful tools for solving nonlinear fractional differential equations (NLFDEs).

New solitary wave solutions of (2+1)-dimensional space–time fractional Burgers equation and Korteweg–de Vries equation

In this article, via the improved fractional subequation method, the fully analytical solutions of the (2+1)-dimensional space–time fractional Burgers equation and Korteweg–de Vries equation involving Jumarie’s modified Riemann–Liouville derivative have been derived. As a result, with the aid of symbolic computation, many types of new analytical solutions are obtained, which include new solitary wave, periodic wave and rational wave solutions. The graphical representations show that these gained solutions have abundant structures.

Soliton Solutions of Space-Time Fractional-Order Modified Extended Zakharov-Kuznetsov Equation in Plasma Physics

The aim of this article is to calculate the soliton solutions of space-time fractional-order modified extended Zakharov-Kuznetsov equation which is modeled to investigate the waves in magnetized plasma physics. Fractional derivatives in the form of modified Riemann-Liouville derivatives are used. Complex fractional transformation is applied to convert the original nonlinear partial differential equation into another nonlinear ordinary differential equation. Then, soliton solutions are obtained by using (1/G')-expansion method. Bright and dark soliton solutions are also obtain with ansatz method. These solutions may be of significant importance in plasma physics where this equation is modeled for some special physical phenomenon.