Analysis of the new homotopy perturbation method for linear and nonlinear problems (original) (raw)
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An elegant and powerful technique is Homotopy Perturbation Method (HPM) to solve linear and nonlinear ordinary and partial differential equations. The method, which is a coupling of the traditional perturbation method and homotopy in topology, deforms continuously to a simple problem which can be solved easily. The method does not depend upon a small parameter in the equation. Using the initial conditions this method provides an analytical or exact solution. From the calculation and its graphical representation it is clear that how the solution of the original equation and its behavior depends on the initial conditions. Therefore there have been attempts to develop new techniques for obtaining analytical solutions which reasonably approximate the exact solutions. Many problems in natural and engineering sciences are modeled by nonlinear partial differential equations (NPDEs). The theory of nonlinear problem has recently undergone much study. Nonlinear phenomena have important applications in applied mathematics, physics, and issues related to engineering. In this paper we have applied this method to Burger's equation and an example of highly nonlinear partial differential equation to get the most accurate solutions. The final results tell us that the proposed method is more efficient and easier to handle when is compared with the exact solutions or Adomian Decomposition Method (ADM).
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In this work, we have studied a general class of linear second-order partial differential equations which is used as mathematical models in many physically significant fields and applied science. The homotopy perturbation method (HPM) has been used for solving generalized linear second-order partial differential equation. Also, we have tested the HPM on the solving of different implementations which show the efficiency and accuracy of the method. The approximated solutions are agree well with analytical solutions for the tested problems Moreover, the approximated solutions proved that the proposed method to be efficient and high accurate. AMS subject classification:
2009
In this paper homotopy perturbation method (HPM) is employed to solve two kinds of differential equations: one dimensional non homogeneous parabolic partial differential equation and non linear differen- tial equation. Using the HPM, an exact analytical solution to non homogeneous parabolic partial differential equation and an approximate explicit solution for a non linear differential equation were obtained. The re- sults obtained by HPM for the non linear differential equation were compared with those results obtained by the exact analytical solution. The comparison shows a complete agreement between results and also shows that this new method may be applicable for solving engineering problem because it needs less computations efforts and is easier than others.
Comparison between the homotopy analysis method and homotopy perturbation method
Applied Mathematics and Computation, 2005
In this paper, we show that the so-called ''homotopy perturbation method'' is only a special case of the homotopy analysis method. Both methods are in principle based on Taylor series with respect to an embedding parameter. Besides, both can give very good approximations by means of a few terms, if initial guess and auxiliary linear operator are good enough. The difference is that, ''the homotopy perturbation method'' had to use a good enough initial guess, but this is not absolutely necessary for the homotopy analysis method. This is mainly because the homotopy analysis method contains the auxiliary parameter ⁄, which provides us with a simple way to adjust and control the convergence region and rate of solution series. So, the homotopy analysis method is more general. Besides, the update of the concept of the ''analytical solution'' is discussed.
Study of convergence of homotopy perturbation method for systems of partial differential equations
Computers & Mathematics with Applications, 2009
The aim of this paper is convergence study of homotopy perturbation method for systems of nonlinear partial differential equations. The sufficient condition for convergence of the method is addressed. Since mathematical modeling of numerous scientific and engineering experiments lead to Brusselator and Burgers' system of equations, it is worth trying new methods to solve these systems. We construct a new efficient recurrent relation to solve nonlinear Burgers' and Brusselator systems of equations. Comparison of the results obtained by homotopy perturbation method with those of Adomian's decomposition method and dual-reciprocity boundary element method leads to significant consequences. Two standard problems are used to validate the method.