Fractional Korteweg-de Vries Research Articles

Fully Discrete Finite Difference Schemes for the Fractional Korteweg-de Vries Equation

Fully Discrete Finite Difference Schemes for the Fractional Korteweg-de Vries Equation

Stability and Convergence analysis of a Crank–Nicolson Galerkin scheme for the fractional Korteweg-de Vries equation

Stability and Convergence analysis of a Crank–Nicolson Galerkin scheme for the fractional Korteweg-de Vries equation

Local fractional Laplace transform iterative method for solving Korteweg–de Vries equation with the local fractional derivative

The local fractional Korteweg–de Vries equations were considered in this paper. Series solutions for the linear and nonlinear case were examined using the local fractional Laplace transform iterative method. This proposed method is the coupling of the local fractional Laplace transform with the new iterative method. The existence and uniqueness of the solutions is considered using the principle of mathematical induction. Furthermore, illustrative examples were considered and the graphs are shown. The results obtained in this study reveal the benefit of this method and provide valuable insight into the behavior of complex phenomena in a precise and efficient manner with less computational work and implementation ease.

Various wave solutions to the nonlinear fractional Korteweg-de Vries Zakharov-Kuznetsov equation by a new approach to the two-variable expansion scheme

Solving nonlinear differential equations is crucial for the design, optimization, and characterization of engineering systems. However, investigating such equations poses significant challenges and requires the development of new mathematical and computational techniques. In this article, we extend the two-variable (G′/G,1/G) -expansion approach and establish scores of general analytical solutions, consisting of different functions and arbitrary parameters, to the time-fractional Korteweg–de Vries Zakharov-Kuznetsov (KdV-ZK) equation. The KdV-ZK equation provides a powerful model for modulating nonlinear waves in an assortment of physical systems, including plasma physics, atmospheric science, fluid dynamics, soliton theory, and optical fibers. The proposed technique allows extracting several novel solutions to the KdV-ZK equation, which is essential for gaining new insights into physical problems. In addition, the suggested extension shortens the calculation process and enhances computational efficiency. The obtained solitons have potential applications in several scientific and technological fields. The beta fractional derivative and the consistent wave variable are considered to restructure the stated fractional nonlinear equation. This study investigates the merits and demerits of the fractional-order derivative β on the dynamics of the system through the representation of two-dimensional graphs, which vary according to different values of β. The enhanced computational efficiency might assist the researchers in exploring a wider range of phenomena, facilitating further extensive investigations, and contributing to overall progress in the fields of science and engineering.

Highest waves for fractional Korteweg–De Vries and Degasperis–Procesi equations

Highest waves for fractional Korteweg–De Vries and Degasperis–Procesi equations

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.

Decay of solitary waves of fractional Korteweg-de Vries type equations

We study the solitary waves of fractional Korteweg-de Vries type equations, that are related to the 1-dimensional semi-linear fractional equations:|D|αu+u−f(u)=0, with α∈(0,2), a prescribed coefficient p⁎(α), and a non-linearity f(u)=|u|p−1u for p∈(1,p⁎(α)), or f(u)=up with an integer p∈[2;p⁎(α)). Asymptotic developments of order 1 at infinity of solutions are given, as well as second order developments for positive solutions, in terms of the coefficient of dispersion α and of the non-linearity p. The main tools are the kernel formulation introduced by Bona and Li, and an accurate description of the kernel by complex analysis theory.

Traveling wave solutions of generalized seventh-order time-fractional KdV models through He-Laplace algorithm

Non-linear evolution equations play a prominent role in describing a wide range of phenomena in optical fibers, fluid dynamics, electromagnetic radiation, plasma and solid state physics. An important category of non-linear evolution models that characterizes shallow wave phenomena are the Korteweg-de Vries (KdV) models. In this regard, time-fractional Korteweg-de Vries models of seventh order are the main focus of this research. A general KdV seventh-order equation is considered with different coefficients to form Lax, Kaup–Kuperschimdt and Sawada–Kotera–Ito KdV models. An efficient semi-analytical algorithm named as He-Laplace (HLM) is applied for the solution of these models. In this algorithm, Laplace transform is hybrid with homotopy perturbation method (HPM). This study provides important results as non-linear evolution seventh-order models in fractional sense have not been captured through HLM in current literature. Absolute errors are computed and compared with already existing results to confirm the superiority of proposed algorithm over other existing techniques. Numerical and graphical investigations are conducted to evaluate the approximate series form solutions. The dynamic behavior of fractional parameter is observed by calculating residual errors and plotting two dimensional diagrams throughout the fractional-domain. Analysis confirms that the proposed methodology provides an effective and convenient way for solving fractional KdV models.

Spatio-temporal fractional shock waves solution for fractional Korteweg-de Vries burgers equations

ABSTRACT We examine the nonlinear characteristics of localized spatial (temporal) fractional shock excitations in a viscous fluid. By applying the natural transform decomposition method and with the aid of standard Caputo's operators, the stationary shock solutions for the spatial (temporal) fractional Korteweg-de Vries Burgers equations are derived. The results reveal that the temporal (Spatial) fractional index significantly modifies the localized solutions for the shocks. Moreover, The Banach contraction analysis shows that the wave solutions converge at , where stands for the temporal (Spatial) variables. Numerical analysis confirms that the index α effect nonlinear waves coupling and therefore causes the steeping of the fractional shocks. Variations in γ on the other hand reduce the wave dispersion and in turn localized the shock potentials. This study is important to understand the nonlinear evolution of shock waves in collisional plasma where the spatial/temporal fractional shocks suffer modifications in the pulse amplitude as well as in the spatial extensions.

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.

Asymptotic $N$-soliton-like solutions of the fractional Korteweg–de Vries equation

We construct N -soliton solutions for the fractional Korteweg–de Vries (fKdV) equation \partial_t u - \partial_x(|D|^{\alpha}u - u^2 )=0, in the whole sub-critical range \alpha \in(1/2,2) . More precisely, if Q_c denotes the ground state solution associated to (fKdV) evolving with velocity c , then, given 0<c_1< \cdots < c_N , we prove the existence of a solution U of (fKdV) satisfying \lim_{t\to\infty} \Big\lVert U(t,\cdot) - \sum_{j=1}^N Q_{c_j}(x-\rho_j(t)) \Big\|_{H^{{\alpha}/2}}=0, where \rho'_{j}(t) \sim c_j as t \to +\infty . The proof adapts the construction of Martel in the generalized KdV setting [Amer. J. Math. 127 (2005), pp. 1103–1140] to the fractional case. The main new difficulties are the polynomial decay of the ground state Q_c and the use of local techniques (monotonicity properties for a portion of the mass and the energy) for a non-local equation. To bypass these difficulties, we use symmetric and non-symmetric weighted commutator estimates. The symmetric ones were proved by Kenig, Martel and Robbiano [Annales de l’IHP Analyse Non Linéaire 28 (2011), pp. 853–887], while the non-symmetric ones seem to be new.

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.

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.

Fractional integrable and related discrete nonlinear Schrödinger equations

Integrable fractional equations such as the fractional Korteweg-deVries and nonlinear Schrödinger equations are key to the intersection of nonlinear dynamics and fractional calculus. In this manuscript, the first discrete/differential difference equation of this type is found, the fractional integrable discrete nonlinear Schrödinger equation. This equation is linearized; special soliton solutions are found whose peak velocities exhibit more complicated behavior than other previously obtained fractional integrable equations. This equation is compared with the closely related fractional averaged discrete nonlinear Schrödinger equation which has simpler structure than the integrable case. For positive fractional parameter and small amplitude waves, the soliton solutions of the integrable and averaged equations have similar behavior.

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.