On nonlinear fractional Klein-Gordon equation (original) (raw)

Analytical approximate solution for nonlinear space—time fractional Klein—Gordon equation

Chinese Physics B, 2013

The fractional derivatives in the sense of Caputo and the homotopy analysis method are used to construct an approximate solution for the nonlinear space-time fractional derivatives Klein-Gordon equation. The numerical results show that the approaches are easy to implement and accurate when applied to the nonlinear space-time fractional derivatives Klein-Gordon equation. This method introduces a promising tool for solving many space-time fractional partial differential equations. This method is efficient and powerful in solving wide classes of nonlinear evolution fractional order equations.

Application of homotopy-perturbation method to Klein–Gordon and sine-Gordon equations

Chaos, Solitons & Fractals, 2009

Fractional initial-value problems (fIVPs) arise from many fields of physics and play a very important role in various branches of science and engineering. Finding accurate and efficient methods for solving fIVPs has become an active research undertaking. In this paper, both linear and nonlinear fIVPs are considered. Exact and/or approximate analytical solutions of the fIVPs are obtained by the analytic homotopy-perturbation method (HPM). The results of applying this procedure to the studied cases show the high accuracy, simplicity and efficiency of the approach.

Novel Solution for Time-fractional Klein-Gordon Equation with Different Applications

International Journal of Mathematical, Engineering and Management Sciences, 2023

In this paper, for the first time, the Laplace Homotopy Perturbation Method (LHPM), which is the coupling of the Laplace transform and the Homotopy Perturbation Method, is employed to solve non-linear time-fractional Klein-Gordon (TFKG) equations. In other words, for the first time in literature, LHPM is used to solve non-linear TFKG equations. First of all, the procedure of LHPM on TFKG with Caputo fractional derivative is developed. Further, the developed approach of LHPM on TFKG is used for two different examples. This in turn proves the versatile nature of the proposed method. In addition, the validity of the approach is proved by comparing the numerical solutions of both examples with their exact solution. Finally, the comparison of relative errors calculated in each example proves the efficiency and effectiveness of the proposed method on TFKG equations.

The homotopy perturbation method applied to the nonlinear fractional Kolmogorov–Petrovskii–Piskunov equations

Applied Mathematics Letters, 2011

The fractional derivatives in the sense of Caputo, and the homotopy perturbation method are used to construct approximate solutions for nonlinear Kolmogorov-Petrovskii-Piskunov (KPP) equations with respect to time and space fractional derivatives. Also, we apply complex transformation to convert a time and space fractional nonlinear KPP equation to an ordinary differential equation and use the homotopy perturbation method to calculate the approximate solution. This method is efficient and powerful in solving wide classes of nonlinear evolution fractional order equations.

A new analytical method to solve Klein-Gordon equations by using homotopy perturbation Mohand transform method

Malaya Journal of Matematik

In this paper, we will study about Fractional-order partial differential equations in Mathematical Science and we will introduce and analyse fractional calculus with an integral operator that contains the Caputo-Fabrizio's fractionalorder derivative. The advanced method is an appropriate union of the new integral transform named as 'Mohand transform' and the homotopy perturbation method. Some numerical examples are used to communicate the generality and clarity of the proposed method. We will also find the analytical solution of the linear and non-linear Klein-Gordan equation which originate in quantum field theory. The homotopy perturbation Mohand transform method (HPMTM) is a merged form of Mohand transform, homotopy perturbation method, and He's polynomials. Some numerical examples are used to indicate the generality and clarity of the proposed method.

Analysis of nonlinear fractional partial differential equations with the homotopy analysis method

Communications in Nonlinear Science and Numerical Simulation, 2009

In this article, the homotopy analysis method is applied to solve nonlinear fractional partial differential equations. On the basis of the homotopy analysis method, a scheme is developed to obtain the approximate solution of the fractional KdV, K(2, 2), Burgers, BBM-Burgers, cubic Boussinesq, coupled KdV, and Boussinesq-like B(m, n) equations with initial conditions, which are introduced by replacing some integer-order time derivatives by fractional derivatives. The homotopy analysis method for partial differential equations of integer-order is directly extended to derive explicit and numerical solutions of the fractional partial differential equations. The solutions of the studied models are calculated in the form of convergent series with easily computable components. The results of applying this procedure to the studied cases show the high accuracy and efficiency of the new technique.

Modified Homotopy Perturbation Method for Solving Fractional Differential Equations

Journal of Applied Mathematics, 2014

The modified homotopy perturbation method is extended to derive the exact solutions for linear (nonlinear) ordinary (partial) differential equations of fractional order in fluid mechanics. The fractional derivatives are taken in the Caputo sense. This work will present a numerical comparison between the considered method and some other methods through solving various fractional differential equations in applied fields. The obtained results reveal that this method is very effective and simple, accelerates the rapid convergence of the series solution, and reduces the size of work to only one iteration.

Solution of nonlinear fractional differential equations using homotopy analysis method

Applied Mathematical Modelling, 2010

In this article, the homotopy analysis method has been applied to solve nonlinear differential equations of fractional order. The validity of this method has successfully been accomplished by applying it to find the solution of two nonlinear fractional equations. The results obtained by homotopy analysis method have been compared with those exact solutions. The results show that the solution of homotopy analysis method is good agreement with the exact solution.

Fractional Homotopy Perturbation Transform Method for Solving the Time-Fractional KdV , K ( 2 , 2 ) and Burgers Equations

International Journal of Open Problems in Computer Science and Mathematics, 2015

In this paper, the fractional homotopy perturbation transform method (FHPTM) is employed to obtain approximate analytical solutions of the time-fractional KdV, K(2,2) and Burgers equations. The FHPTM can easily be applied to many problems and is capable of reducing the size of computational work. The fractional derivative is described in the Caputo sense. The results show that the FHPTM is an appropriate method for solving nonlinear fractional derivative equation.