Fractional Modified Korteweg De Vries Research Articles

New exact soliton and periodic wave solutions of the nonlinear fractional evolution equations with additional term

This paper presents exact solutions for the fractional Korteweg–de Vries equation and the fractional modified Korteweg–de Vries equation with additional term using the functional variable method. The main advantage of the proposed functional variable method over other methods is that it provides more accurate traveling wave solutions with additional free parameters. As a result, soliton and periodic wave solutions are obtained. It is shown that the calculations in the functional variable method are very simple and easy to understand, and this method is very effective for handling nonlinear fractional equations. Exact solutions are great importance in revealing the internal mechanism of physical phenomena. This method presents a wider applicability for handling nonlinear fractional-order wave equations.

The Fractional Analysis of a Nonlinear mKdV Equation with Caputo Operator

In this study, we aim to provide reliable methods for the initial value problem of the fractional modified Korteweg–de Vries (mKdV) equations. Fractional differential equations are essential for more precise simulation of numerous processes. The hybrid Yang transformation decomposition method (YTDM) and Yang homotopy perturbation method (YHPM) are employed in a very simple and straightforward manner to handle the current problems. The derivative of fractional order is displayed in a Caputo form operator. To illustrate the conclusion given from the findings, a few numerical cases are taken into account for their approximate analytical solutions. We looked at two cases and contrasted them with the actual result to validate the methodologies. These techniques create recurrence relations representing the proposed problem’s solution. It is possible to find the series solutions to the given problems, and these solutions have components that converge to precise solutions more quickly. Tables and graphs are used to describe the new results, which demonstrate the present methods’ adequate accuracy. The actual and estimated outcomes are demonstrated in graphs and tables to be quite similar, demonstrating the usefulness of the proposed approaches. The innovation of the current work resides in the application of effective methods that require less calculation and achieve a greater level of accuracy. Additionally, the suggested approaches can be applied in the future to resolve other nonlinear fractional problems, which will be a scientific contribution to the research community.

INVESTIGATION OF THE ELABORATE DYNAMICS OF WEAKLY NONLINEAR FRACTIONAL ION-ACOUSTIC WAVES IN MAGNETIZED ELECTRON–POSITRON PLASMA

This study employs three advanced computational and numerical techniques to solve the nonlinear fractional modified Korteweg–de Vries–Zakharov–Kuznetsov (mKdV–ZK) equation in magnetized plasma. The reductive perturbation approach is utilized to investigate the dynamics of various components, namely isothermal species, immobile background species, and warm adiabatic fluid, in magnetized plasma. Emphasis is placed on unraveling the asymmetrical propagation characteristics of nonlinear electrostatic waves. The model’s solutions encompass diverse types of solitons, including ion-acoustic, dust acoustic, and electron acoustic solitons. Analytical solutions are obtained using a variety of mathematical functions, such as exponents, trigonometry, and hyperbolas. Two- and three-dimensional density graphs illustrate the practical behavior of a single soliton. The primary objective of employing numerical schemes is to assess the accuracy of the derived solutions, and the outcomes demonstrate the efficacy of the analytical method in solving nonlinear mathematical and physical problems. Several techniques are employed to validate the consistency between calculated and estimated results, ensuring the study’s accuracy and reliability. Overall, this investigation underscores the effectiveness of numerical and analytical techniques in tackling complex mathematical models, offering a promising avenue for future research in the field. The findings carry significant implications for comprehending nonlinear phenomena in magnetized plasma and contribute to advancing the field.

Data-driven soliton mappings for integrable fractional nonlinear wave equations via deep learning with Fourier neural operator

In this paper, we firstly extend the Fourier neural operator (FNO) to discovery the mapping between two infinite-dimensional function spaces, where one is the fractional-order index space {ϵ|ϵ∈(0,1)} in the fractional integrable nonlinear wave equations while another denotes the soliton solution in the spatio-temporal function space. In other words, once the FNO network is trained, for any given ϵ∈(0,1), the corresponding soliton solution can be quickly obtained. To be specific, the soliton solutions are learned for the fractional nonlinear Schrödinger (fNLS), fractional Korteweg–de Vries (fKdV), fractional modified Korteweg–de Vries (fmKdV) and fractional sine-Gordon (fsineG) equations. The FNO architecture is utilized to learn the soliton mappings of the above four equations. The data-driven solitons are also compared with exact solutions to illustrate the powerful approximation capability of the FNO. Moreover, we study the influences of several critical factors (e.g., activation functions containing Relu(x), Sigmoid(x), Swish(x) and the new one xtanh(x), channels of fully connected layer) on the performance of the FNO algorithm. As a result, we find that the xtanh(x) and Swish(x) functions perform better than the Relu(x) and Sigmoid(x) functions in the FNO, and the FNO network with a more-channel fully-connected layer performs better as we expect. These results obtained in this paper will be useful to further understand the neural networks for fractional integrable nonlinear wave equations and the mappings between two infinite-dimensional function spaces.

Nonlinear dispersive wave propagation pattern in optical fiber system

Nonlinear fractional-order partial differential equations are an important tool in science and engineering for explaining a wide range of nonlinear processes. We consider the nonlinear space-time fractional modified Korteweg de Vries and sine-Gordon equations in this article and extract diverse sorts of traveling wave as well as soliton solutions using a typical approach, namely the generalized G′/G-expansion approach. These equations are used to model a wide range of nonlinear phenomena, including plasma physics, high-intensity laser-generated plasma, ultra-small electronic devices, optical fibers, control theory, turbulence, acoustics, and others. The fractional derivative is defined in the sense of the beta-derivative established by Atangana and Baleanu. Some standard shapes of waveforms, including kink type, singular-kink type, bell-shaped, periodic-type, single soliton, multiple soliton type, and several other types of solitons, have been established. To validate the physical compatibility of the results, the 3D, contour, and vector plots have been delineated using consistent values of the parameters. The approach used in this study to extract inclusive and standard solutions is approachable, efficient, and speedier in computing.

Correction to: Periodic Waves in the Fractional Modified Korteweg–de Vries Equation

Correction to: Periodic Waves in the Fractional Modified Korteweg–de Vries Equation

Adequate closed form wave solutions to the space–time fractional nonlinear equations in physical sciences

Signal processing, water wave mechanics, waves in electro-hydro-dynamic systems, traffic flows, optical fibers, sound waves, etc. can be superbly illustrated through fractional nonlinear evolution equations, such as the time-fraction Zakharov–Kuznetsov (ZK) equation and the space–time fractional modified Korteweg–de Vries (mKdV) equation. The​ fractional derivative is considered in the sense of beta-derivative. In this study, the compatible and broad-ranging closed form soliton solutions to the stated models by making use of the exp ( − τ η ) -expansion method with the assistance of the traveling wave transformation have been investigated. The physical significance of the obtained solutions for the definite values of the associated parameters is analyzed graphically to contrivance the tangible phenomena. It is established that the exp ( − τ η ) -expansion method is convenient, efficient and provide further advanced solutions that can support to construct scores of wave solutions to diverse models.

Derivation and solution of space fractional modified Korteweg de Vries equation

All applications of fractional calculus to physics propose fractional equations of motion for various complex phenomena. Complex and nonlinear fractional equations are very familiar and often occur in nature. Therefore, the analytical and numerical solutions of this kind of equations are very important and consequential. An important objective in this study is to derive the space fractional modified Korteweg de Vries (SFMKdV) equation and present an efficient modified G′/G -expansion method to solve it. The exact analytical solution of SFMKdV equation is obtained to describe the space fractional parameter effect on the propagation of the ion acoustic (IA) waves. It has been observed that space fractional parameter can modify the height of the IA waves.

New exact solutions to the space–time fractional nonlinear wave equation obtained by the ansatz and functional variable methods

In this paper, the ansatz method and the functional variable method are employed to find new analytic solutions for the space–time nonlinear fractional wave equation, the space–time fractional Kadomtsev–Petviashvili–Benjamin–Bona–Mahony equation and the space–time fractional modified Korteweg–de Vries–Zakharov–Kuznetsov equation. As a result, some exact solutions are obtained in terms of hyperbolic and periodic functions. It is shown that the proposed methods provide a more powerful mathematical tool for constructing exact solutions for many other nonlinear fractional differential equations occurring in nonlinear physical phenomena. We have also presented the numerical simulations for these equations by means of three dimensional plots.

Lie symmetries, symmetry reductions and conservation laws of time fractional modified Korteweg–de Vries (mkdv) equation

In this work, we study Lie symmetry analysis for fractional order differential equations that is one of the applications of symmetries. This study deals with Lie symmetry of fractional order modified Korteweg–de Vries (mKdV) equation. We found Lie symmetries of this equation and then we reduced fractional order modified Korteweg–de Vries (mKdV) equation to fractional order ordinary differential equation with Erdelyi–Kober fractional differential operator. Then we used characteristic method for fractional order differential equations and help of founded these Lie symmetries for finding solutions for given equation. Then we obtained infinite and finite conservation laws of fractional order modified Korteweg–de Vries (mKdV) equation.

A New Simplified Bilinear Method for theN-Soliton Solutions for a Generalized FmKdV Equation with Time-Dependent Variable Coefficients

AbstractIn this paper a generalized fractional modified Korteweg–de Vries (FmKdV) equation with time-dependent variable coefficients, which is a generalized model in nonlinear lattice, plasma physics and ocean dynamics, is investigated. With the aid of a simplified bilinear method, fractional transforms and symbolic computation, the correspondingN-soliton solutions are given and illustrated. The characteristic line method and graphical analysis are applied to discuss the solitonic propagation and collision, including the bidirectional solitons and elastic interactions. Finally, the resonance phenomenon for the equation is examined.

The First Integral Method for Exact Solutions of Nonlinear Fractional Differential Equations

In this paper, we establish exact solutions for some nonlinear fractional differential equations (FDEs). The first integral method with help of the fractional complex transform (FCT) is used to obtain exact solutions for the time fractional modified Korteweg–de Vries (fmKdV) equation and the space–time fractional modified Benjamin–Bona–Mahony (fmBBM) equation. This method is efficient and powerful in solving kind of other nonlinear FDEs.

Analytical solution of nonlinear space–time fractional differential equations using the improved fractional Riccati expansion method

In this paper, the improved fractional Riccati expansion method is proposed to solve fractional differential equations. The method is applied to solve space–time fractional modified Korteweg–de Vries equation, space–time fractional modified regularized long-wave equation, time fractional biological population model, and space–time fractional Klein–Gordon equation. The obtained solutions include generalized trigonometric and hyperbolic functions solutions. Among these solutions, some are found for the first time.

Approximate analytical solution to fractional modified KdV equations

In this paper, the fractional modified Korteweg–de Vries equation (fmKdV) is introduced by replacing the first-order time and space derivatives by fractional derivatives of order α and β with 0 < α , β ≤ 1 . The homotopy-perturbation method (HPM) is applied to derive approximate analytical solutions to fmKdV.