Application of homotopy-perturbation method to Klein–Gordon and sine-Gordon equations (original) (raw)
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In this paper, the homotopy analysis method is applied to solve linear fractional problems. Based on this method, a scheme is developed to obtain approximation solution of fractional wave, Burgers, Korteweg-de Vries (KdV), KdV-Burgers, and Klein-Gordon equations with initial conditions, which are introduced by replacing some integer-order time derivatives by fractional derivatives. The fractional derivatives are described in the Caputo sense. So 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 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.
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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.
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In this article, the homotopy analysis method is applied to solve nonlinear fractional partial differential equations.Based on the homotopy analysis method, a scheme is developed to obtain the approximate solution of the nonlinearfractional heat conduction, Kaup–Kupershmidt, Fisher and Huxley equations with initial conditions, introduced byreplacing some integer-order time derivatives by fractional derivatives. The solutions of the studied models are calculatedin the form of convergent series with easily computable components. The results of applying this procedure tothe studied cases show the high accuracy and efficiency of the new technique. The fractional derivative is describedin the Caputo sense. Some illustrative examples are presented to observe some computational results.
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.
Application of homotopy perturbation method for fractional partial differential equations
2014
Fractional partial differential equations arise from many fields of physics and apply a very important role in various branches of science and engineering. Finding accurate and efficient methods for solving partial differential equations of fractional order has become an active research undertaking. In the present paper, the homotopy perturbation method proposed by J-H He has been used to obtain the solution of some fractional partial differential equations with variable coefficients. Exact and/or approximate analytical solutions of these equations are obtained.
Application of Local Fractional Homotopy Perturbation Method in Physical Problems
Advances in Mathematical Physics, 2020
Nonlinear phenomena have important effects on applied mathematics, physics, and issues related to engineering. Most physical phenomena are modeled according to partial differential equations. It is difficult for nonlinear models to obtain the closed form of the solution, and in many cases, only an approximation of the real solution can be obtained. The perturbation method is a wave equation solution using HPM compared with the Fourier series method, and both methods results are good agreement. The percentage of error of ux,t with α=1 and 0.33, t =0.1 sec, between the present research and Yong-Ju Yang study for x≥0.6 is less than 10. Also, the % error for x≥0.5 in α=1 and 0.33, t =0.3 sec, is less than 5, whereas for α=1 and 0.33, t =0.8 and 0.7 sec, the % error for x≥0.4 is less than 8.
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.
Analysis of Homotopy Perturbation Method for Solving Fractional Order Differential Equations
Mathematics
The analysis of Homotopy Perturbation Method (HPM) for the solution of fractional partial differential equations (FPDEs) is presented. A unified convergence theorem is given. In order to validate the theory, the solution of fractional-order Burger-Poisson (FBP) equation is obtained. Furthermore, this work presents the method to find the solution of FPDEs, while the same partial differential equation (PDE) with ordinary derivative i.e., for α = 1 , is not defined in the given domain. Moreover, HPM is applied to a complicated obstacle boundary value problem (BVP) of fractional order.