Solving fractional diffusion and wave equations by modified homotopy perturbation method (original) (raw)
Related papers
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.
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.
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.
2014
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.
2015
In this letter, the homotopy perturbation transform method is used to obtain analytical approximate solutions to the systems of nonlinear fractional partial differential equations. The proposed method was derived by combining Laplace transform and homotopy perturbation method. It yields solutions in convergent series forms with easily computable terms. The fractional derivative is described in the Caputo sense. Illustrative examples demonstrate the efficiency of new method. MSC: 26A33 • 34A12 • 35R11