Time-fractional Korteweg-de Vries Research Articles

Numerical simulation methods and analysis for the dynamics of the time-fractional KdV equation

In this paper, two classes of efficient difference schemes for the simulation of the time-fractional Korteweg–de Vries equation are carried out. The temporal derivative is approximated with the help of the uniform/nonuniform L1 formula and the uniform L2-1σ formula, respectively. The spatial derivative is done with the uniform centered difference scheme. Unique solvability, boundedness and convergence of the corresponding difference schemes are rigorously proved at length. As far as we know, the proposed schemes achieve the highest convergence order in space among all the difference methods for the time-fractional KdV equation. Several numerical examples confirm the theoretical results and demonstrate the dynamic behavior.

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.

Homotopy time-fractional enhancement soliton solution for six component complex cometary plasma with thermal heavy ions

A dusty plasma system composed of three inertial species and three Kappa-distributed inertial-less species is considered. The hydrodynamic fluid equation used to describe investigated plasma system. The plasma system under consideration is a very rich system, so in this paper the effective Debye length and effective plasma frequency are considered to normalize all physical quantities. The Kortweg de-Veris equation is derived using the reductive perturbation method. Two envelope solitons (bright and dark) are propagating, and their appearance depends mainly on the values thermal parameter of heavy ions (oxygen ions) as well as the mass ratio for dust plasma. The homotopy perturbed transform method (HPTM) is used to solve the reduced time-fractional Korteweg-de Vries (TFKdV) equation. The effects of fractional parameters on the solution solutions are examined. Our results show that the shape of the solitary pulse is highly controlled by the fractional order, as well as the plasma parameters in addition to that the HPTM is very useful method, where few terms approximately three terms are enough to get successful approximate solution for fractional type equations.

Traveling Wave Solutions of Fractional Differential Equations Arising in Warm Plasma

This paper aims to study the fractional differential systems arising in warm plasma, which exhibits traveling wave-type solutions. Time-fractional Korteweg-De Vries (KdV) and time-fractional Kawahara equations are used to analyze cold collision-free plasma, which exhibits magnet-acoustic waves and shock wave formation respectively. The decomposition method is used to solve the proposed equations. Also, the convergence and uniqueness of the obtained solution are discussed. To illuminate the effectiveness of the presented method, the solutions of these equations are obtained and compared with the exact solution. Furthermore, solutions are obtained for different values of time-fractional order and represented graphically.

Analytical approach for travelling wave solution of non-linear fifth-order time-fractional Korteweg–De Vries equation

Analytical approach for travelling wave solution of non-linear fifth-order time-fractional Korteweg–De Vries equation

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.

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.

BOUNDARY CONTROLLABILITY FOR THE TIME-FRACTIONAL NONLINEAR KORTEWEG-DE VRIES (KDV) EQUATION

In this paper, we study the time-fractional nonlinear Korteweg-de Vries (KdV) equation. By using the theory of semigroups, we prove the well-posedness of the time-fractional nonlinear KdV equation. Moreover, we present the boundary controllability result for the problem.

Numerical Solution of Time Fractional Stochastic Korteweg–de Vries Equation via Implicit Meshless Approach

In this article, a numerical scheme is implemented to approximate the solution of time fractional stochastic Korteweg–de Vries (KdV) equation. The structure of the proposed method is such that it first employ finite difference technique to transform the time fractional stochastic KdV equation into elliptic stochastic differential equations (SDEs). Then resulting elliptic SDEs has been estimated via a meshless method based on radial basis functions (RBFs). It should be mentioned that every RBFs with sufficient smoothness can be applied. In this paper, we use Gaussian RBFs which are infinitely smoothness to approximate the functions in the resulting elliptic SDEs. The most important advantage of proposed method respect to traditional numerical method is that the noise terms are directly simulated at the collocation points in each step time. Finally, some test problems are presented to investigate the performance and accuracy of the new method.

Fractional Soliton Dynamics and Spectral Transform of Time-Fractional Nonlinear Systems: A Concrete Example

In this paper, the spectral transform with the reputation of nonlinear Fourier transform is extended for the first time to a local time-fractional Korteweg-de vries (tfKdV) equation. More specifically, a linear spectral problem associated with the KdV equation of integer order is first equipped with local time-fractional derivative. Based on the spectral problem with the equipped local time-fractional derivative, the local tfKdV equation with Lax integrability is then derived and solved by extending the spectral transform. As a result, a formula of exact solution with Mittag-Leffler functions is obtained. Finally, in the case of reflectionless potential the obtained exact solution is reduced to fractional n-soliton solution. In order to gain more insights into the fractional n-soliton dynamics, the dynamical evolutions of the reduced fractional one-, two-, and three-soliton solutions are simulated. It is shown that the velocities of the reduced fractional one-, two-, and three-soliton solutions change with the fractional order.

Solutions of time-fractional third- and fifth-order Korteweg–de-Vries equations using homotopy perturbation transform method

PurposeThis study aims to find the solution of time-fractional Korteweg–de-Vries (tfKdV) equations which may be used for modeling various wave phenomena using homotopy perturbation transform method (HPTM).Design/methodology/approachHPTM, which consists of mainly two parts, the first part is the application of Laplace transform to the differential equation and the second part is finding the convergent series-type solution using homotopy perturbation method (HPM), based on He’s polynomials.FindingsThe study obtained the solution of tfKdV equations. An existing result “as the fractional order of KdV equation given in the first example decreases the wave bifurcates into two peaks” is confirmed with present results by HPTM. A worth mentioning point may be noted from the results is that the number of terms required for acquiring the convergent solution may not be the same for different time-fractional orders.Originality/valueAlthough third-order tfKdV and mKdV equations have already been solved by ADM and HPM, respectively, the fifth-order tfKdV equation has not been solved yet. Accordingly, here HPTM is applied to two tfKdV equations of order three and five which are used for modeling various wave phenomena. The results of third-order KdV and KdV equations are compared with existing results.

Comparison of Caputo and conformable derivatives for time-fractional Korteweg–de Vries equation via the finite difference method

In this study, we investigate the nonlinear time-fractional Korteweg–de Vries (KdV) equation by using the (1/G[Formula: see text])-expansion method and the finite forward difference method. We first obtain the exact wave solutions of the nonlinear time-fractional KdV equation. In addition, we used the finite-forward difference method to obtain numerical solutions in this equations. When these solutions are obtained, the indexed forms of both Caputo and conformable derivatives are used. By using indexing technique, it is shown that the numerical results of the nonlinear time-fractional KdV equation approaches the exact solution. The two- and three-dimensional surfaces of the obtained analytical solutions are plotted. The von Neumann stability analysis of the used numerical scheme with the studied equation is carried out. The L2and L[Formula: see text] error norms are computed. The exact solutions and numerical approximations are compared by supporting with graphical plots and tables.

New fractional derivatives applied to the Korteweg–de Vries and Korteweg–de Vries–Burger’s equations

In this paper, we extend the model of the Korteweg–de Vries (KDV) and Korteweg–de Vries–Burger’s (KDVB) to new model time fractional Korteweg–de Vries (TFKDV) and time fractional Korteweg–de Vries-Burger’s (TFKDVB) with Liouville–Caputo (LC), Caputo–Fabrizio (CF), and Atangana-Baleanu (AB) fractional time derivative equations, respectively. We utilize the q-homotopy analysis transform method (q-HATM) to compute the approximate solutions of TFKDV and TFKDVB using LC, CF and AB in Liouville–Caputo sense. We study the convergence analysis of q-HATM by computing the Residual Error Function (REF) and finding the interval of the convergence through the h-curves. Also, we find the optimal values of h so that, we assure the convergence of the approximate solutions. The results are very effective and accurate in solving the TFKDV and TFKDVB.

Analytical solutions of conformable time, space, and time-space fractional KdV equations

In this study, we consider the Korteweg-de Vries (KdV) equation for solitary waves in the domain of conformable fractional calculus. By means of this fractional theory, we obtain exact solutions for time, space, and time-space fractional KdV equations and demonstrate our results graphically according to the fractional order of the related equations. Furthermore, we report that the fractional order in the solution of the time fractional KdV equation is associated with viscosity of the medium by comparing three-dimensional picture presentations of solutions of time fractional KdV and Burgers-KdV equations.

Finite Difference/Collocation Method for a Generalized Time-Fractional KdV Equation

In this paper, we studied the numerical solution of a time-fractional Korteweg–de Vries (KdV) equation with new generalized fractional derivative proposed recently. The fractional derivative employed in this paper was defined in Caputo sense and contained a scale function and a weight function. A finite difference/collocation scheme based on Jacobi–Gauss–Lobatto (JGL) nodes was applied to solve this equation and the corresponding stability was analyzed theoretically, while the convergence was verified numerically. Furthermore, we investigated the behavior of solution of the generalized KdV equation depending on its parameter δ , scale function z ( t ) in fractional derivative. We found that the full discrete scheme was effective to obtain a numerical solution of the new KdV equation with different conditions. The wave number δ in front of the third order space derivative term played a significant role in splitting a soliton wave into multiple small pieces.

Invariant Subspace Method and Exact Solutions of Certain Nonlinear Time Fractional Partial Differential Equations

Abstract We show, using invariant subspace method, how to derive exact solutions to the time fractional Korteweg-de Vries (KdV) equation, potential KdV equation with absorption term, KdV-Burgers equation and a time fractional partial differential equation with quadratic nonlinearity. Also we extend the invariant subspace method to nonlinear time fractional differential-difference equations and derive exact solutions of the time fractional discrete KdV and Toda lattice equations.

Ion-acoustic waves in unmagnetized collisionless weakly relativistic plasma of warm-ion and isothermal-electron using time-fractional KdV equation

Collisionless unmagnetized plasma consisting of a mixture of warm ion-fluid and isothermal-electron is considered, assuming that the ion flow velocity has a weak relativistic effect. The reductive perturbation method has been employed to derive the Korteweg–de Vries (KdV) equation for small – but finite-amplitude electrostatic ion-acoustic waves in this plasma. The semi-inverse method and Agrawal’s method lead to the Euler–Lagrange equation that leads to the time fractional KdV equation. The variational-iteration method given by He is used to solve the derived time fractional KdV equation. The calculations show that the fractional order may play the same rule of higher order dissipation in KdV equation to modulate the soliton wave amplitude in the plasma system. The results of the present investigation may be applicable to some plasma environments, such as space-plasmas, laser-plasma interaction, plasma sheet boundary layer of the earth’s magnetosphere, solar atmosphere and interplanetary space.