Fractional Korteweg-de Vries Equation Research Articles

Some finite difference methods for solving linear fractional KdV equation

The time-fractional Korteweg de Vries equation can be viewed as a generalization of the classical KdV equation. The KdV equations can be applied in modeling tsunami propagation, coastal wave dynamics, and oceanic wave interactions. In this study, we construct two standard finite difference methods using finite difference methods with conformable and Caputo approximations to solve a time-fractional Korteweg-de Vries (KdV) equation. These two methods are named as FDMCA and FDMCO. FDMCA utilizes Caputo's derivative and a finite-forward difference approach for discretization, while FDMCO employs conformable discretization. To study the stability, we use the Von Neumann Stability Analysis for some fractional parameter values. We perform error analysis using L1 & L∞ norms and relative errors, and we present results through graphical representations and tables. Our obtained results demonstrate strong agreement between numerical and exact solutions when the fractional operator is close to 1.0 for both methods. Generally, this study enhances our comprehension of the capabilities and constraints of FDMCO and FDMCA when used to solve such types of partial differential equations laying some ground for further research.

Bifurcation analysis of dynamical systems with fractional order differential equations via the modified Riemann-Liouville derivative

In this manuscript, the solutions of linear dynamical systems with fractional differential equations via themodified Riemann-Liouville derivative is derived. By using Jumarie type of derivative (JRL), we stated and provedthe Existence and uniqueness theorems of the dynamical systems with fractional order equations. Also a novel stability analysis of fractional dynamical systems by Jumarie type derivative is established and some important stability conditions are determined. The achieved results have various applications in mathematics, plasma physics and almost all branches of physics that have non-conservative forces. Finally, we investigated interesting application of nonlinear space-time fractional Korteweg-de Vries (STFKdV) equation in Saturn F-ring’s region. Moreover, our investigation could be basic interest to explain and interpret the effects of fractional and modification parameters on STFKdV equation. This is novel study on this model by dynamical system (DS) to describe the behavior of nonlinear waves without solve this system.

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.

Fuzzy solutions of some variants of the fractional order Korteweg-de-Vries equations via an analytical method

Under uncertainty, the analytical behaviour of fractional partial differential equations is frequently puzzling and challenging to predict. Therefore, in order to address these problems, it is essential to create a proper, extensive, and highly effective theory. The theory of fuzzy fractional partial differential equations is a relatively new idea with applications in applied mathematics and engineering. In this article, we examined the different types of fuzzy fractional-order Korteweg-de-Vries (KdV) equations along with their analytical fuzzy solutions. To construct a series-type solution under a fuzzy notion for the relevant research, we utilise the fractional reduced differential transform method (FRDTM) in the Caputo sense. The solutions of various kinds of fuzzy fractional KdV equations that have been developed are more broadly applicable. The fuzzy concept is used to overrule the uncertainty in physical models based on the parametric form of the fuzzy number. The numerical and graphical presentation demonstrates the symmetry between the lower and upper cut representations of the fuzzy solutions and may be helpful in improving understanding of automatic control models, measure theory, physics, biology, computer science, quantum optics, economics, artificial intelligence, and other domains, as well as non-financial analysis.

Collisional Solitons Described by Two-Sided Beta Time Fractional Korteweg-de Vries Equations in Fluid-Filled Elastic Tubes

This article deals with the basic features of collisional radial displacements in a prestressed thin elastic tube filled having inviscid fluid with the presence of nonlocal operator. By implementing the extended Poincare–Lighthill–Kuo method and a variational approach, the new two-sided beta time fractional Korteweg-de-Vries (BTF-KdV) equations are derived based on the concept of beta fractional derivative (BFD). Additionally, the BTF-KdV equations are suggested to observe the effect of related parameters on the local and nonlocal coherent head-on collision phenomena for the considered system. It is observed that the proposed equations along with their new solutions not only applicable with the presence of locality but also nonlocality to study the resonance wave phenomena in fluid-filled elastic tube. The outcomes reveal that the BFD and other physical parameters related to tube and fluid have a significant impact on the propagation of pressure wave structures.

Analytical Solution of the Local Fractional KdV Equation

This research work is dedicated to solving the n-generalized Korteweg–de Vries (KdV) equation in a fractional sense. The method is a combination of the Sumudu transform and the Adomian decomposition method. This method has significant advantages for solving differential equations that are both linear and nonlinear. It is easy to find the solutions to fractional-order PDEs with less computing labor.

Approximate analytical solutions of generalized fractional Korteweg-de Vries equation

In this paper, a generalized Korteweg-de Vries equation involving a temporal fractional derivative and a spatial fractal derivative is studied. The temporal fractional derivative can describe the non-local property and memory property, while the spatial fractal derivative can model the space discontinuity. Its approximate analytical solution is presented using He?s variational iteration method, which is extremely effective for the fractal-fractional differential equations.

A Reliable Way to Deal with the Coupled Fractional Korteweg-De Vries Equations within the Caputo Operator

The development of numeric-analytic solutions and the construction of fractional order mathematical models for practical issues are of the highest concern in a variety of physics, applied mathematics, and engineering applications. The nonlinear Kersten–Krasil’shchik-coupled Korteweg–de Vries-modified Korteweg–de Vries (KdV-mKdV) system is treated analytically in this paper using a unique method, known as the Laplace residual power series (LRPS) approach to find some approximate solutions. The RPS methodology and the Laplace transform operator are combined in the LRPS method. We provide a detailed introduction to the proposed method for dealing with fractional Kersten–Krasil’shchik-linked KdV-mKdV models. When compared to exact solutions, the approach provides analytical solutions with good accuracy. We demonstrate the effectiveness of the current strategy compared to alternative methods for solving nonlinear equations using an illustrative example. The LRPS technique’s results show and highlight that the method may be used for a variety of time-fractional models of physical processes with simplicity and computing effectiveness.

The Efficient Techniques for Non-Linear Fractional View Analysis of the KdV Equation

The solutions to fractional differentials equations are very difficult to investigate. In particular, the solutions of fractional partial differential equations are challenging tasks for mathematicians. In the present article, an extension to this idea is presented to obtain the solutions of non-linear fractional Korteweg–de Vries equations. The solutions comparison of the proposed problems is done via two analytical procedures, which are known as the Residual power series method (RPSM) and q-HATM, respectively. The graphical and tabular analysis are presented to show the reliability and competency of the suggested techniques. The comparison has shown the greater contact between exact, RPSM, and q-HATM solutions. The fractional solutions are in good control and provide many important dynamics of the given problems.

Soliton solutions for nonlinear variable-order fractional Korteweg–de Vries (KdV) equation arising in shallow water waves

Nonlinear fractional differential equations provide suitable models to describe real-world phenomena and many fractional derivatives are varying with time and space. The present study considers the advanced and broad spectrum of the nonlinear (NL) variable-order fractional differential equation (VO-FDE) in sense of VO Caputo fractional derivative (CFD) to describe the physical models. The VO-FDE transforms into an ordinary differential equation (ODE) and then solving by the modified (G′/G)-expansion method. For accuracy, the space-time VO fractional Korteweg-de Vries (KdV) equation is solved by the proposed method and obtained some new types of periodic wave, singular, and Kink exact solutions. The newly obtained solutions confirmed that the proposed method is well-ordered and capable implement to find a class of NL-VO equations. The VO non-integer performance of the solutions is studied broadly by using 2D and 3D graphical representation. The results revealed that the NL VO-FDEs are highly active, functional and convenient in explaining the problems in scientific physics.

Comparison of modified ADM and classical finite difference method for some third-order and fifth-order KdV equations

Abstract The KdV equation, which appears as an asymptotic model in physical systems ranging from water waves to plasma physics, has been studied. In this paper, we are concerned with dispersive nonlinear KdV equations by using two reliable methods: Shehu Adomian decomposition method (STADM) and the classical finite difference method for solving three numerical experiments. STADM is constructed by combining Shehu’s transform and Adomian decomposition method, and the nonlinear terms can be easily handled using Adomian’s polynomials. The Shehu transform is used to accelerate the convergence of the solution series in most cases and to overcome the deficiency that is mainly caused by unsatisfied conditions in other analytical techniques. We compare the approximate and numerical results with the exact solution for the two numerical experiments. The third numerical experiment does not have an exact solution and we compare profiles from the two methods vs the space domain at some values of time. This study provides us with information about which of the two methods are effective based on the numerical experiment chosen. Knowledge acquired will enable us to construct methods for other related partial differential equations such as stochastic Korteweg-de Vries (KdV), KdV-Burgers, and fractional KdV equations.

Symmetry analysis of the generalized space and time fractional Korteweg–de Vries equation

In this paper, we studied the generalized space and time fractional Korteweg–de Vries (KdV) equation in the sense of the Riemann–Liouville fractional derivative. Initially, the symmetry of this considered equation through the symmetry analysis method was obtained. Next, a one-parameter Lie group of point transformation was yielded. Then, this considered fractional model can be translated into an ordinary differential equation of fractional order via the Erdélyi–Kober fractional differential operator and the Erdélyi–Kober fractional integral operator. Finally, with the help of the nonlinear self-adjointness, conservation laws of the generalized space and time fractional KdV equation can be found. This approach can provide us with a new scheme for studying space and time differential equations of fractional derivative.

Green’s Function for the Fractional KDV Equation on the Periodic Domain via Mittag-Leffler Function

Abstract The linear operator c + (−Δ) α/2, where c > 0 and (−Δ) α/2 is the fractional Laplacian on the periodic domain, arises in the existence of periodic travelling waves in the fractional Korteweg–de Vries equation. We establish a relation of the Green function of this linear operator with the Mittag–Leffler function, which was previously used in the context of the Riemann–Liouville and Caputo fractional derivatives. By using this relation, we prove that the Green function is strictly positive and single-lobe (monotonically decreasing away from the maximum point) for every c > 0 and every α ∈ (0, 2]. On the other hand, we argue from numerical approximations that in the case of α ∈ (2, 4], the Green function is positive and single-lobe for small c and non-positive and non-single lobe for large c.

The Soliton Solutions for Some Nonlinear Fractional Differential Equations with Beta-Derivative

Nonlinear fractional differential equations have gained a significant place in mathematical physics. Finding the solutions to these equations has emerged as a field of study that has attracted a lot of attention lately. In this work, He’s semi-inverse variation method and the ansatz method have been applied to find the soliton solutions for fractional Korteweg–de Vries equation, fractional equal width equation, and fractional modified equal width equation defined by Atangana’s conformable derivative (beta-derivative). These two methods are effective methods employed to get the soliton solutions of these nonlinear equations. All of the calculations in this work have been obtained using the Maple program and the solutions have been replaced in the equations and their accuracy has been confirmed. In addition, graphics of some of the solutions are also included. The found solutions in this study have the potential to be useful in mathematical physics and engineering.

New traveling solutions of the fractional nonlinear KdV and ZKBBM equations with 𝒜ℬℛ fractional operator

This research paper investigates novel explicit wave solutions of the fractional Korteweg–de Vries (KdV) equation and the fractional Zakharov–Kuznetsov–Benjamin–Bona–Mahony (ZKBBM) equation. These models are used as gravity models in water and an interaction model between the long waves. The Atangana–Baleanu ([Formula: see text]) fractional operator is utilized for the first time to convert the fractional form of both models into nonlinear partial differential equations with an integer order. The extended simplest equation method is employed to construct some distinct types of solitary wave solutions such as exponential, rational, hyperbolic and trigonometric functions. For more illustration of our obtained solutions, some figures for them are given. The power and practical properties of the used method are tested.

Nonclassical Lie symmetry and conservation laws of the nonlinear time-fractional Korteweg–de Vries equation

In this paper, we use the symmetry of the Lie group analysis as one of the powerful tools that deals with the wide class of fractional order differential equations in the Riemann–Liouville concept. In this study, first, we employ the classical and nonclassical Lie symmetries (LS) to acquire similarity reductions of the nonlinear fractional far field Korteweg–de Vries (KdV) equation, and second, we find the related exact solutions for the derived generators. Finally, according to the LS generators acquired, we construct conservation laws for related classical and nonclassical vector fields of the fractional far field KdV equation.

On persistence properties in weighted spaces for solutions of the fractional Korteweg–de Vries equation

Persistence problems in weighted spaces have been studied for different dispersive models involving non-local operators. Generally, these models do not propagate polynomial weights of arbitrary magnitude, and the maximum decay rate is associated with the dispersive part of the equation. Altogether, this analysis is complemented by unique continuation principles that determine optimal spatial decay. This work is intended to establish the above questions for a weakly dispersive perturbation of the inviscid Burgers equation. More precisely, we consider the fractional Korteweg–de Vries equation, which comprises the Burgers–Hilbert equation and dispersive effects weaker than those of the Benjamin–Ono equation.

Long time behavior of the fractional Korteweg-de Vries equation with cubic nonlinearity

<p style='text-indent:20px;'>We prove global existence and modified scattering for the solutions of the Cauchy problem to the fractional Korteweg-de Vries equation with cubic nonlinearity for small, smooth and localized initial data.

Averaging Principle for Multiscale Stochastic Fractional Schrödinger–Korteweg-de Vries System

This work concerns the averaging principle for multiscale stochastic fractional Schrodinger–Korteweg-de Vries system. Stochastic averaging principle is a powerful tool for studying qualitative analysis of stochastic dynamical systems with different time-scales. To be more precise, under suitable conditions, the multiscale system can be reduced to a single stochastic fractional Korteweg-de Vries equation (averaged equation) with a modified coefficient, the slow component of multiscale system towards to the solution of the averaged equation in moment. In order to prove the averaging principle, there are two key points: uniform bounds for the solution and Holder continuity of time variable for the slow component. For this, we need a crucial property—smoothing effect of the fractional Korteweg-de Vries semigroup, but the traditional method can’t help us obtain this property. Such analysis does not follow in a straightforward manner from results already available in the mathematical literature. On the contrary, it requires the introduction of some new ideas and techniques. We try to overcome this difficulty by the duality method, the interpolation arguments, the energy estimate method and refined inequality technique.

On the propagation of regularity for solutions of the fractional Korteweg-de Vries equation

We consider the initial value problem (IVP) for the fractional Korteweg-de Vries equation (fKdV)(0.1){∂tu−Dxα∂xu+u∂xu=0,x,t∈R,0<α<1,u(x,0)=u0(x). It has been shown that the solutions to certain dispersive equations satisfy the propagation of regularity phenomena. More precisely, it deals in determine whether regularity of the initial data on the right hand side of the real line is propagated to the left hand side by the flow solution. This property was found originally in solutions of Korteweg-de Vries (KdV) equation and it has been also verified in other dispersive equations as the Benjamin-Ono (BO) equation.Recently, it has been shown that the solutions of the dispersive generalized Benjamin-Ono (DGBO) equation, this is α∈(2,3) in (0.1); also satisfy the propagation of regularity phenomena. This is achieved by introducing a commutator decomposition to handle the dispersive part in the equation. Following the approach used in the DGBO, we prove that the solutions of the fKdV also satisfies the propagation of regularity phenomena. Consequently, this type of regularity travels with infinite speed to its left as time evolves.