Nonlinear Fractional Klein Gordon Equation Research Articles

A numerical study on the nonlinear fractional Klein–Gordon equation

This article helps to develop a numerical approach based on Fibonacci wavelets for solving fractional Klein-Gordan equations (FKGEs). The FKGEs are solved with Caputo-type fractional derivative. Using the definition of Fibonacci wavelets, we constructed the operational matrices of integration. These operational matrices of integration led to the development of the collocation method called the Fibonacci wavelet collocation method (FWCM). This method transforms the given nonlinear partial differential equation into a system of nonlinear algebraic equations using collocation points to determine the unknown coefficients. By substituting the unknown coefficients in the method, we obtained the numerical solution of the present approach. We furnish the different error norms for the present technique. The obtained results are compared with the Clique polynomial method. These findings demonstrate the computational attractiveness, efficiency, effectiveness, reliability, and robustness of the proposed method for addressing a variety of physical models in science and engineering.

Numerical solutions of nonlinear time fractional Klein-Gordon equation via natural transform decomposition method and iterative Shehu transform method

The time-fractional Klein-Gordon equation was analyzed in this article using the natural transform decomposition method (NTDM) and the iterative Shehu transform method (ISTM) with a singular kernel derivative. We employed natural transform (NT) and Shehu transform (ST), followed by inverse natural transform and inverse Shehu transform respectively, to obtain the solution. The numerical simulations are provided to ensure that the methods under consideration are efficient. The current framework captures the behaviour of the obtained findings for various fractional orders. The findings of this study indicate that the proposed methods are effective and reliable in analyzing fractional differential equations.

Two Fibonacci operational matrix pseudo-spectral schemes for nonlinear fractional Klein–Gordon equation

This paper is devoted to developing spectral solutions for the nonlinear fractional Klein–Gordon equation. The typical collocation method and the tau method are employed for obtaining the desired numerical solutions. In order to do this, a new operational matrix of fractional derivatives of Fibonacci polynomials is established. The idea behind the derivation of this matrix is based on utilizing the connection formula between the Fibonacci and Chebyshev polynomials. The introduced operational matrix is used along with the weighted residual quadrature spectral method and the collocation method to convert the nonlinear fractional Klein–Gordon equation into a system of algebraic equations. By solving the resulting system, we obtain a semi-analytic solution. The convergence and error analysis of the method are discussed. Some numerical results and discussions are presented aiming to illustrate the wide applicability and accuracy of the proposed algorithms.

An Extension of the Double G ′ / G , 1 / G -Expansion Method for Conformable Fractional Differential Equations

The phenomena, molecular path in a liquid or a gas, fluctuating price stoke, fission and fusion, quantum field theory, relativistic wave motion, etc., are modeled through the nonlinear time fractional clannish random Walker’s parabolic (CRWP) equation, nonlinear time fractional SharmaTassoOlver (STO) equation, and the nonlinear space-time fractional KleinGordon equation. The fractional derivative is described in the sense of conformable derivative. From there, the G ′ / G , 1 / G -expansion method is found to be ensuing, effective, and capable to provide functional solutions to nonlinear models concerning physical and engineering problems. In this study, an extension of the G ′ / G , 1 / G -expansion method has been introduced. This enhancement establishes broad-ranging and adequate fresh solutions. In addition, some existing solutions attainable in the literature also confirm the validity of the suggested extension. We believe that the extension might be added to the literature as a reliable and efficient technique to examine a wide variety of nonlinear fractional systems with parameters including solitary and periodic wave solutions to nonlinear FDEs.

Exact Solution for Nonlinear Local Fractional Partial Differential Equations

In this work, we extend the existing local fractional Sumudu decomposition method to solve the nonlinear local fractional partial differential equations. Then, we apply this new algorithm to resolve the nonlinear local fractional gas dynamics equation and nonlinear local fractional Klein-Gordon equation, so we get the desired non-differentiable exact solutions. The steps to solve the examples and the results obtained, showed the flexibility of applying this algorithm, and therefore, it can be applied to similar examples.

Exact Travelling Wave Solutions for Space-Time Fractional Klein-Gordon Equation and (2+1)-Dimensional Time-Fractional Zoomeron Equation via Auxiliary Equation Method

AbstractIn this paper, the new exact solutions of nonlinear conformable fractional partial differential equations(CFPDEs) are achieved by using auxiliary equation method for the nonlinear space-time fractional Klein-Gordon equation and the (2+1)-dimensional time-fractional Zoomeron equation. The technique is easily applicable which can be applied successfully to get the solutions for different types of nonlinear CFPDEs. The conformable fractional derivative(CFD) definitions are used to cope with the fractional derivatives.

A Fully Discrete Spectral Method for the Nonlinear Time Fractional Klein-Gordon Equation

The numerical approximation of the nonlinear time fractional Klein-Gordon equation in a bounded domain is considered. The time fractional derivative is described in the Caputo sense with the order $\gamma$ ($1 < \gamma < 2$). A fully discrete spectral scheme is proposed on the basis of finite difference discretization in time and Legendre spectral approximation in space. The stability and convergence of the fully discrete scheme are rigorously established. The convergence rate of the fully discrete scheme in $H^1$ norm is $\mathrm{O}(\tau^{3-\gamma} + N^{1-m})$, where $\tau$, $N$ and $m$ are the time-step size, polynomial degree and regularity in the space variable of the exact solution, respectively. Numerical examples are presented to support the theoretical results.

Auxiliary Equation Method for Fractional Differential Equations with Modified Riemann–Liouville Derivative

Abstract:In this work, the auxiliary equation method is applied to derive exact solutions of nonlinear fractional Klein–Gordon equation and space-time fractional Symmetric Regularized Long Wave equation. Consequently, some exact solutions of these equations are successfully obtained. These solutions are formed in fractional complex transform to convert fractional differential equations into ordinary differential equations. The fractional derivatives are described in Jumarie’s modified Riemann–Liouville sense. The exact solutions founded by the suggested method indicate that the approach is easy to implement and powerful.

The modified simple equation method for solving some fractional-order nonlinear equations

Nonlinear fractional differential equations are encountered in various fields of mathematics, physics, chemistry, biology, engineering and in numerous other applications. Exact solutions of these equations play a crucial role in the proper understanding of the qualitative features of many phenomena and processes in various areas of natural science. Thus, many effective and powerful methods have been established and improved. In this study, we establish exact solutions of the time fractional biological population model equation and nonlinear fractional Klein–Gordon equation by using the modified simple equation method.

A compact difference scheme for a two dimensional nonlinear fractional Klein–Gordon equation in polar coordinates

In this paper, a high order finite difference scheme for a two dimensional nonlinear fractional Klein–Gordon equation in polar coordinates is proposed. The difficulty caused by the variable coefficient and the nonlinear term in the equation is carefully handled. Stability and convergence are analyzed using the matrix form of the scheme. Numerical examples are provided to demonstrate the theoretical results.

Analytical and numerical validation for solving the fractional Klein-Gordon equation using the fractional complex transform and variational iteration methods

AbstractIn this paper, we implement the fractional complex transform method to convert the nonlinear fractional Klein-Gordon equation (FKGE) to an ordinary differential equation. We use the variational iteration method (VIM) to solve the resulting ODE. The fractional derivatives are presented in terms of the Caputo sense. Some numerical examples are presented to validate the proposed techniques. Finally, a comparison with the numerical solution using Runge-Kutta of order four is given.

Approximate Series Solution of Nonlinear, Fractional Klein-Gordon Equations Using Fractional Reduced Differential Transform Method

This analysis proposes an analytical-numerical approach for providing solutions of a class of nonlinear fractional Klein-Gordon equation subjected to appropriate initial conditions in Caputo sense by using the Fractional Reduced Differential Transform Method (FRDTM). This technique provides the solutions very accurately and efficiently in convergent series formula with easily computable coefficients. The behavior of the approximate series solution for different values of fractional-order "a" is shown graphically. A comparative study is presented between the FRDTM and Implicit Runge-Kutta approach to illustrate the efficiency and reliability of the proposed technique. Our numerical investigations indicate that the FRDTM is simple, powerful mathematical tool and fully compatible with the complexity of such problems.

On the solutions of the nonlinear fractional differential equations via the modified trial equation method

In this study, the nonlinear fractional partial differential equations have been defined by the modified Riemann–Liouville fractional derivative. By using this fractional derivative and traveling wave transformation, the nonlinear fractional partial differential equations have been converted into nonlinear ordinary differential equations. The modified trial equation method is implemented to obtain exact solutions of the nonlinear fractional Klein–Gordon equation and fractional clannish random walker's parabolic equation. As a result, some exact solutions including single kink solution and periodic and rational function solutions of these equations have been successfully obtained. Copyright © 2015 John Wiley &amp; Sons, Ltd.

A Riccati-Bernoulli sub-ODE method for nonlinear partial differential equations and its application

The Riccati-Bernoulli sub-ODE method is firstly proposed to construct exact traveling wave solutions, solitary wave solutions, and peaked wave solutions for nonlinear partial differential equations. A Backlund transformation of the Riccati-Bernoulli equation is given. By using a traveling wave transformation and the Riccati-Bernoulli equation, nonlinear partial differential equations can be converted into a set of algebraic equations. Exact solutions of nonlinear partial differential equations can be obtained by solving a set of algebraic equations. By applying the Riccati-Bernoulli sub-ODE method to the Eckhaus equation, the nonlinear fractional Klein-Gordon equation, the generalized Ostrovsky equation, and the generalized Zakharov-Kuznetsov-Burgers equation, traveling solutions, solitary wave solutions, and peaked wave solutions are obtained directly. Applying a Backlund transformation of the Riccati-Bernoulli equation, an infinite sequence of solutions of the above equations is obtained. The proposed method provides a powerful and simple mathematical tool for solving some nonlinear partial differential equations in mathematical physics.

Approximate Solution of Non-Linear Fractional Klein-Gordon Equation Using Spectral Collocation Method

In this paper, we implement the spectral collocation method with the help of the Legendre poly-nomials for solving the non-linear Fractional (Caputo sense) Klein-Gordon Equation (FKGE). We present an approximate formula of the fractional derivative. The Legendre collocation method is used to reduce FKGE to the solution of system of ODEs which is solved by using finite difference method. The results of applying the proposed method to the non-linear FKGE show the simplicity and the efficiency of the proposed method.

Topological and non-topological soliton solutions to some time-fractional differential equations

This paper investigates, for the first time, the applicability and effectiveness of He’s semi-inverse variational principle method and the ansatz method on systems of nonlinear fractional partial differential equations. He’s semi-inverse variational principle method and the ansatz method are used to construct exact solutions of nonlinear fractional Klein–Gordon equation and generalized Hirota–Satsuma coupled KdV system. These equations have been widely applied in many branches of nonlinear sciences such as nonlinear optics, plasma physics, superconductivity and quantum mechanics. So, finding exact solutions of such equations are very helpful in the theoretical and numerical studies.

A high‐order compact scheme for the nonlinear fractional Klein–Gordon equation

In this article, a high‐order finite difference scheme for a kind of nonlinear fractional Klein–Gordon equation is derived. The time fractional derivative is described in the Caputo sense. The solvability of the difference system is discussed by the Leray–Schauder fixed point theorem, while the stability and L∞ convergence of the finite difference scheme are proved by the energy method. Numerical examples are provided to demonstrate the theoretical results. © 2014 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 31: 706–722, 2015

Two-Dimensional Differential Transform Method and Modified Differential Transform Method for Solving Nonlinear Fractional Klein–Gordon Equation

In this paper, two-dimensional differential transform method and modified differential transform method applied successfully for finding the approximate analytical solution of the nonlinear fractional Klein–Gordon equation. The plotted graph illustrates the behavior of the solution for different values of fractional order α. Three test examples are given to demonstrate the ability of both the methods for solving fractional nonlinear equation.

Application of the simplest equation method to some time-fractional partial differential equations

In this paper, we establish exact solutions for some time fractional differential equations. The simplest equation method is used to construct the exact solutions of nonlinear fractional Klein–Gordon equation, Generalized Hirota–Satsuma coupled KdV system of time fractional order and nonlinear fractional Sharma–Tasso–Olever equation. The simplest equation method presents a wide applicability to handling nonlinear wave equations.

On nonlinear fractional Klein–Gordon equation

Numerical methods are used to find exact solution for the nonlinear differential equations. In the last decades Iterative methods have been used for solving fractional differential equations. In this paper, the Homotopy perturbation method has been successively applied for finding approximate analytical solutions of the fractional nonlinear Klein–Gordon equation can be used as numerical algorithm. The behavior of solutions and the effects of different values of fractional order α are shown graphically. Some examples are given to show ability of the method for solving the fractional nonlinear equation.