5 research outputs found
A New Modification of The HPM for The Duffing Equation With High Nonlinearity
In this work we introduce a new modification of the homotopy perturbation method for solving nonlinear ordinary differential equations. The technique is based on the blending of the Chebyshev pseudo-spectral methods and the homotopy perturbation method (HPM). The method is tested by solving the strongly nonlinear Duffing equation for undamped oscillators. Comparison is made between the proposed technique, the standard HPM, an earlier modification of the HPM and the numerical solutions to demonstrate the high accuracy, applicability and validity of the present approach
Application of homotopy-perturbation method to fractional IVPs
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
Application of homotopy perturbation method for fractional partial differential equations
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
On Spectral-Homotopy Perturbation Method Solution of Nonlinear Differential Equations in Bounded Domains
In this study, a combination of the hybrid Chebyshev spectral technique and the homotopy perturbation method is used to construct an iteration algorithm for solving nonlinear boundary value problems. Test problems are solved in order to demonstrate the efficiency, accuracy and reliability of the new technique and comparisons are made between the obtained results and exact solutions. The results demonstrate that the new spectral homotopy perturbation method is more efficient and converges faster than the standard homotopy analysis method. The methodology presented in the work is useful for solving the BVPs consisting of more than one differential equation in bounded domains.Â
Modified HPMs Inspired by Homotopy Continuation Methods
Nonlinear differential equations have applications in the modelling area for a broad variety of phenomena
and physical processes; having applications for all areas in science and engineering. At the present
time, the homotopy perturbation method (HPM) is amply used to solve in an approximate or exact manner
such nonlinear differential equations. This method has found wide acceptance for its versatility and ease
of use. The origin of the HPM is found in the coupling of homotopy methods with perturbation
methods. Homotopy methods are a well established research area with applications, in particular, an applied
branch of such methods are the homotopy continuation methods, which are employed on the numerical
solution of nonlinear algebraic equation systems. Therefore, this paper presents two modified versions of
standard HPM method inspired in homotopy continuation methods. Both modified HPMs deal
with nonlinearities distribution of the nonlinear differential equation. Besides, we will use a calcium-induced
calcium released mechanism model as study case to test the proposed techniques. Finally, results
will be discussed and possible research lines will be proposed using this work as a starting point