Fractional Klein-Gordon Research Articles

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.

An Efficient Computational Method for the Time-Space Fractional Klein-Gordon Equation

In this paper, we present a computational method to solve fractional Klein-Gordon equation (FKGE). The proposed technique is the grouping of orthogonal polynomial matrices and collocation method. The benefit of the computational method is that it reduces the FKGE into a system of algebraic equations which makes the problem straightforward and easy to solve. The main reason for using this technique is high accuracy and low computational cost compared to some other methods. The main solution behaviours of these equations are due to fractional orders which are explained graphically.Numerical results obtained by proposed computational method are also compared with the exact solution. The results obtained by suggested technique reveals that the method is very useful for solving FKGE.

Numerical investigation of fractional nonlinear sine-Gordon and Klein-Gordon models arising in relativistic quantum mechanics

This paper presents a method for the approximate solution of the time-fractional nonlinear sine-Gordon and the Klein-Gordon models described in Caputo sense and with the order 1 < γ < 2. This method discretizes the unknown solution in two main steps. First, the temporal discretization of the governing problems is obtained by means of the finite difference scheme. Second, the spatial terms are expanded using local radial basis functions, where each basis function is approximated by a weighted linear summation of function values. The spatial discretization achieved through the local radial basis functions and finite difference (LRBF-FD) is highly accurate, since in local collocation techniques we only consider the nodes located in the subdomain around the collocation node surrounding the local collocation point. In fact, the new approach avoids the ill-conditioning problem resulting from the adoption of the global collocation and leads to sparse systems. The numerical stability and convergence are examined and confirmed numerically to support the theoretical formulation. Numerical experiments assess the effectiveness and capability of the algorithm and demonstrate its good computational performance on irregular domains.

A linearized high-order Galerkin finite element approach for two-dimensional nonlinear time fractional Klein-Gordon equations

In this paper, we propose a linearized finite element method for solving two-dimensional fractional Klein-Gordon equations with a cubic nonlinear term. The employed time discretization is a weighted combination of the L2 − 1σ formula introduced recently by Lyu and Vong (Numer. Algorithms 78(2):485–511, 2018), Galerkin finite element method is used for the spatial discretization, and the cubic nonlinear term is handled explicitly. Using mathematical induction, we prove that the numerical solution is bounded and the fully discrete scheme is convergent with second-order accuracy in time. In numerical experiments, some problems with both smooth and non-smooth exact solutions are considered.

A new analysis for Klein-Gordon model with local fractional derivative

This work adopts Yang’s local fractional derivative to define the fractional Klein-Gordon equation in a fractal space or microgravity space. The variational principle of local fractional Klein-Gordon equation is successfully established via fractional semi-inverse transform method and the classical He’s variational iteration method (HVIM) is used to obtain its approximate analytical solution.

An efficient numerical technique for variable order time fractional nonlinear Klein-Gordon equation

This paper studies the variable order time fractional nonlinear Klein-Gordon equation (V-TFNKGE). An optimization method using the Caputo type definition and a set of basis functions, namely the generalized polynomials (GP), is proposed. The solution is expanded in terms of the GP with unknown free coefficients and control parameters. First, a new variable order time fractional operational matrix of the Caputo type for the GP is derived. Then, based on a operational matrix and the Lagrange multipliers, an optimization process achieves the approximate solution. Additionally, the convergence of the proposed method is analyzed and two numerical examples illustrate its good performance.

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.

Numerical solution of the linear time fractional Klein-Gordon equation using transform based localized RBF method and quadrature

This work aims to approximate the solution of the linear time-fractional Klein-Gordon equations in Caputo's sense. The Laplace transform is applied to linear time fractional Klein-Gordon equation to eliminate the time variable and avoid the time stepping procedure. Application of the Laplace transform avoids the time instability issues which commonly occurs in time stepping methods and reduces the computational cost. The transform problem is then solved using local RBFs and finally the solution is obtained by the inverse Laplace transform. The solution is represented as an integral along a smooth curve in the complex plane which is then approximated by quadrature rule. The proposed method is capable of solving linear time fractional partial differential equations. The stability and convergence of the method are discussed. The efficiency of the method is demonstrated with the help of numerical experiments.

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.

Exact solutions of nonlinear fractional order partial differential equations via singular manifold method

Singular manifold method is applied to solve nonlinear fractional partial differential equations (NLFPDEs). Fractional complex transformation is firstly used to reduce the equations to ordinary differential equations (ODEs). Using Bäcklund transformation the Schwarzian derivatives for the Eigen functions are obtained leading to a new variety of exact solutions. The method is applied to nonlinear time fractional Klein-Gordon, Cahn-Hilliard, Burger and Cahn-Allen equations. Some of the resulting solutions are graphically illustrated showing periodic kink, multi soliton and kink solutions.

Analytical Solutions of Fractional Klein-Gordon and Gas Dynamics Equations, via the (G′/G)-Expansion Method

In this article, the ( G ′ / G ) -expansion method is used for the analytical solutions of fractional-order Klein-Gordon and Gas Dynamics equations. The fractional derivatives are defined in the term of Jumarie’s operator. The proposed method is based on certain variable transformation, which transforms the given problems into ordinary differential equations. The solution of resultant ordinary differential equation can be expressed by a polynomial in ( G ′ / G ) , where G = G ( ξ ) satisfies a second order linear ordinary differential equation. In this paper, ( G ′ / G ) -expansion method will represent, the travelling wave solutions of fractional-order Klein-Gordon and Gas Dynamics equations in the term of trigonometric, hyperbolic and rational functions.

A Reliable Numerical Algorithm for the Fractional Klein-Gordon Equation

The key purpose of the present work is to introduce a numerical algorithm for the solution of the fractional Klein-Gordon equation (FKGE). The numerical algorithm is based on the applications of the operational matrices of the Legendre scaling functions. The main advantage of the numerical algorithm is that it reduces the FKGE into Sylvester form of algebraic equations which significantly simplify the problem. Numerical results derived by using suggested numerical scheme are compared with the exact solution. The results show that the suggested algorithm is very user friendly for solving FKGE and accurate.

GALERKIN-FEM FOR OBTAINING THE NUMERICAL SOLUTION OF THE LINEAR FRACTIONAL KLEIN-GORDON EQUATION

In this paper, an efficient numerical method for solving the linear fractional Klein-Gordon equation (LFKGE) is introduced. The proposed method depends on the Galerkin finite element method (GFEM) using quadratic B-spline base functions and replaces the Caputo fractional derivative using $ L2 $ discretization formula. The introduced technique reduces LFKGE to a system of algebraic equations, which solved using conjugate gradient method. The study the stability analysis to the approximation obtained by the proposed scheme is given. To test the accuracy of the proposed method we evaluated the error norm $ L_{2} $. It is shown that the presented scheme is unconditionally stable. Numerical example is given to show the validity and the accuracy of the introduced algorithm.

Conformable fractional derivative and its application to fractional Klein-Gordon equation

This paper adopts conformable fractional derivative to describe the fractional Klein-Gordon equations. The conformable fractional derivative is a new simple well-behaved definition. The fractional complex transform and variational iteration method are used to solve the fractional equation. The result shows that the proposed technology is very powerful and efficient for fractional differential equations.

Variational iteration method combined with new transform to solve fractional partial differential equations

The aim of this paper is to combined the variational iteration method with Aboodh transform method to solve linear and nonlinear fractional partial differential equations. Some illustrative examples are given as the linear and nonlinear fractional Klein-Gordon equations and the time fractional diffusion equation. The results reveal that this method is very effective, simple and can be applied to other physical differential equations with fractional order. The fractional derivative is taken in the Caputo sense.

A New Perspective on The Numerical Solution for Fractional Klein Gordon Equation

In the present manuscript, a new numerical scheme is presented for solving the time fractional nonlinear Klein-Gordon equation. The approximate solutions of the fractional equation are based on cubic B-spline collocation finite element method and L2 algorithm. The fractional derivative in the given equation is handled in terms of Caputo sense. Using the methods, fractional differential equation is converted into algebraic equation system that are appropriate for computer coding. Then, two model problems are considered and their error norms are calculated to demonstrate the reliability and efficiency of the proposed method. The newly calculated error norms show that numerical results are in a good agreement with the exact solutions.

Analytic Solution of Spatial-Temporal Fractional Klein-Gordon Equation Arising in Physical Models

Making use of bivariate (α, β)-fractional power series expansion scheme, we provide an analytical solution to the spatial–temporal fractional Klein–Gordon equation that has many applications in physics such as quantum mechanics, solitons, and condensed matter physics. The twofold fractional derivative orders, namely α, β ∈ (0, 1], are described in the Caputo sense. Some examples are given to show the ability and efficiency of the proposed method for solving (non)linear (non)homogeneous Klein–Gordon equation in fractal space. The obtained closed-form solutions are found to be in a consensus with the exact solutions exist in the literature when α, β → 1, which demonstrates the feasibility and validity of the proposed method.

Analytic solution for the space-time fractional Klein-Gordon and coupled conformable Boussinesq equations

Abstract In this paper, we constructed a travelling wave solution for space-time fractional nonlinear partial differential equations by using the modified extended Tanh method with Riccati equation. The method is used to obtain analytic solutions for the space-time fractional Klein-Gordon and coupled conformable space-time fractional Boussinesq equations. The fractional complex transforms and the properties of modified Riemann-Liouville derivative have been used to convert these equations into nonlinear ordinary differential equations.

Time-fractional Cahn–Allen and time-fractional Klein–Gordon equations: Lie symmetry analysis, explicit solutions and convergence analysis

This research analyzes the symmetry analysis, explicit solutions and convergence analysis to the time fractional Cahn–Allen (CA) and time-fractional Klein–Gordon (KG) equations with Riemann–Liouville (RL) derivative. The time fractional CA and time fractional KG are reduced to respective nonlinear ordinary differential equation of fractional order. We solve the reduced fractional ODEs using an explicit power series method. The convergence analysis for the obtained explicit solutions are investigated. Some figures for the obtained explicit solutions are also presented.

A linearized second-order scheme for nonlinear time fractional Klein-Gordon type equations

We consider difference schemes for nonlinear time fractional Klein-Gordon type equations in this paper. A linearized scheme is proposed to solve the problem. As a result, iterative method need not be employed. One of the main difficulties for the analysis is that certain weight averages of the approximated solutions are considered in the discretization and standard energy estimates cannot be applied directly. By introducing a new grid function, which further approximates the solution, and using ideas in some recent studies, we show that the method converges with second-order accuracy in time.