Application of New Homotopy Perturbation Method for Solving Partial Differential Equations

Homotopy perturbation method (HPM) is a semi-analytical technique for solving linear as well as nonlinear ordinary/partial differential equations. The method may also be used to solve a system of coupled linear and nonlinear differential equations. This chapter first illustrates the basic idea of the HPM. It uses the present method to solve a nonlinear ordinary differential equation, a nonlinear partial differential equation, and a coupled nonlinear system of partial differential equations. It is worth mentioning that the HPM can also be used for handling linear ordinary and partial differential equations. The test problems demonstrated in the chapter confirm that the HPM is an efficient method for solving linear/nonlinear ordinary, partial, and coupled differential equations. The series usually converges with an increase in the number of terms, but one may not expect the compact form solution always.

In the paper, two new analytic methods using the decomposition and linearization technique on nonlinear differential/integral equations are developed, namely, the decomposition–linearization–sequential method (DLSM) and the linearized homotopy perturbation method (LHPM). The DLSM is realized by an integrating factor and the integral of certain function obtained at the previous step for obtaining a sequential analytic solution of nonlinear differential equation, which provides quite accurate analytic solution. Some first- and second-order nonlinear differential equations display the fast convergence and accuracy of the DLSM. An analytic approximation for the Volterra differential–integral equation model of the population growth of a species is obtained by using the LHPM. In addition, the LHPM is also applied to the first-, second-, and third-order nonlinear ordinary differential equations. To reduce the cost of computation of He’s homotopy perturbation method and enhance the accuracy for solving cubically nonlinear jerk equations, the LHPM is implemented by invoking a linearization technique in advance is developed. A generalization of the LHPM to the nth-order nonlinear differential equation is involved, which can greatly simplify the work to find an analytic solution by solving a set of second-order linear differential equations. A remarkable feature of those new analytic methods is that just a few steps and lower-order approximations are sufficient for producing reasonably accurate analytic solutions. For all examples, the second-order analytic solution x2(t) is found to be a good approximation of the real solution. The accuracy of the obtained approximate solutions are identified by the fourth-order Runge–Kutta method. The major objection is to unify the analytic solution methods of different nonlinear differential equations by simply solving a set of first-order or second-order linear differential equations. It is clear that the new technique considerably saves computational costs and converges faster than other analytical solution techniques existing in the literature, including the Picard iteration method. Moreover, the accuracy of the obtained analytic solution is raised.

A semi-analytical method has been planned for the precise solution of the differential equation established on the New Homotopy Perturbation Method (NHPM), and to develop a generalized Burger-Huxley (BH) equation, in this paper. By employing NHPM, two case studies show the precise solution of the BH equation. It is shown that the NHPM is yield solution is convergent from with the easy computability term; NHPM is an effective and easy tool for cracking many real world difficulties. The three-dimension and two dimension graphical solutions of the BH equations are also provided to validate the mathematical models. MATLAB software is used to calculate the series obtained from HPM.

Solution of nonlinear partial differential equations from base equations

Solving linear and nonlinear Abel fuzzy integral equations by homotopy analysis method

PurposeThis paper aims to solve linear and non-linear shallow water wave equations using homotopy perturbation method (HPM). HPM is a straightforward method to handle linear and non-linear differential equations. As such here, one-dimensional shallow water wave equations have been considered to solve those by HPM. Interesting results are reported when the solutions of linear and non-linear equations are compared.Design/methodology/approachHPM was used in this study.FindingsSolution of one-dimensional linear and non-linear shallow water wave equations and comparison of linear and non-linear coupled shallow water waves from the results obtained using present method.Originality/valueCoupled non-linear shallow water wave equations are solved.

The Numerical Solution of Integral Equations on the Half-Line

The integral-differential equations describe many phenomena in different field of mechanical and nuclear engineering, chemistry, astronomy, biology and potential theory. The resolution of boundary problems for these equations are the subject of several papers in which the authors have approached in most cases numerical methods: Monte Carlo, truncated series of Chebyshev polynomials, the fictitious domain method, S N method and the finite element method (1-4). An exact solution was found only in the particular cases. Generally, these are obtained with the help of the methods of mathematical analysis, abstract functional analysis and the spectral methods (5). In recent years, the basic ideas of the homotopy, which is a concept of the topology and differential geometry, were used to obtain the approximate solutions for a wide class of differential, integral and integral-differential equations. We mention here the homotopy perturbation method (HPM) proposed by Ji-Huan He in 1998 and the homotopy analysis method (HAM) proposed by Liao in 1992. The perturbation methods approximate the solution of given problem by a series of small parameters. Unfortunately, the majority of non-linear problems have no small parameters and an unsuitable choice of these parameters can lead to bad effects. The new homotopy perturbation technique (HPM) embeds a parameter p that ranges from zero to one (6,7). When the embedding parameter is zero, we get a linear equation and if it is equal to one, we get the original transport equation. This embedding parameter that belongs to the interval (0, 1) can be considered as a small parameter. The homotopy analysis method (HAM) is one of the most effective methods to obtain the exact and approximate series solutions for the integral-differential equations (8). Being a great freedom in choosing of the initial guess functions and the auxiliary functions, are proposed for these such forms that correspond to the conditions imposed at the ends of the spatial interval and thus the algorithm becomes fast convergent. In the papers (9-11) is shown that the homotopy perturbation method, variational iteration method and Adomian decomposition method are only special cases of the homotopy analysis method solutions. Today, a new homotopy method is added to the above. With this we can solve the multipoint boundary problem for a stationary transport equation in the non-homogeneous media. Comparative studies between the solutions obtained with the help of mentioned algorithms for a numerical example with the appropriate exact solutions illustrate the accuracy and computational efficiency of these new methods.

In this paper an application of the homotopy analysis method for some type of nonlinear and linear integral equations of the second kind is presented. A special case of considered equation is the Volterra- Fredholm integral equation. In homotopy analysis method a series is created. It has shown that if the series is convergent, its sum is the solution of the considered equation. It has been also shown that under proper assumptions the considered equation possesses a unique solution and the series obtained in homotopy analysis method is convergent. The error of the approximate solution was estimated. This approximate solution is obtained when we limit to the partial sum of the series.Application of the method is illustrated with examples.

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).

In this paper, the Laplace transform homotopy analysis method (LHAM) is employed to obtain approximate analytical solutions of the linear and nonlinear differential equations.This method is a combined form of the Laplace transform method and the homotopy analysis method.The proposed scheme finds the solutions without any discretization or restrictive assumptions and is free from round-off errors and therefore, reduces the numerical computations to a great extent.Some illustrative examples are presented and the numerical results show that the solutions of the LHAM are in good agreement with those obtained by exact solution.

In this paper, the nonlinear free vibrations of thin symmetric and non-symmetric cross-ply composite plates subjected to biaxial initial stresses are investigated. Because of their excellent properties such as specific strength and specific stiffness, composite plates have wide applications in aerospace and mechanical structures. Based on Von-Karman's strain-displacement relations and using Galerkin method, the nonlinear differential equation of free vibrations of initially stressed composite plate is obtained. This nonlinear equation is solved using two different analytical perturbation methods, namely method of multiple scales (MTS) and homotopy perturbation method (HPM), to analyze the nonlinear vibrations of initially stressed cross-ply composite plates. Effects of tensile and compressive biaxial initial stresses, initial vibration amplitude, thickness, and aspect ratios of the composite plates on the frequency behavior are investigated. The validity of the results is confirmed by making a comparison with those reported in the literature. According to the results, both analytical solutions show increasing trends for natural frequency parameters by increasing normal initial stresses. Regardless of the value of initial biaxial stresses, for both symmetric and non-symmetric plates, the results of MTS and HPM are in close agreement for the smallest initial amplitude. However, for compressive initial stresses, by increasing initial amplitude ratios, the discrepancies between the results of HPM and MTS increase for symmetric and non-symmetric plates. Although HPM includes less computational effort (smaller length of formulation) than MTS, the linear-to-nonlinear frequency ratios obtained using MTS method become closer to those obtained by HPM as initial vibration amplitude is decreased and initial stress is increased.

In this work, we have studied a class of nonlinear Riccati differential equation which is used as mathematical models in many physically significant fields and applied science. The homotopy perturbation method, has been modified for solving generalized linear and nonlinear Riccati differential equation, which is a second-order ordinary differential equation. Also, we have tested the modified homotopy perturbation method on the solving of different implementations which are show the efficiency and accuracy of the proposed method. The approximated solutions are agreed well with analytical solutions for the tested problems.

In this paper, we introduce fractional-order model of (HTLV-I) infection of CD 4+ T-cells. Homotopy analysis method (HAM) is implemented to give approximate and analytical solutions of the presented problem. The fractional derivatives are described in the Caputo sense. The method introduces a promising tool for solving many linear and nonlinear fractional differential equations. In these schemes, the solution takes the form of a convergent series with easily computable components. Numerical results show that the approach is easy to implement and accurate when applied to ordinary differential equations of fractional order.

In this paper, time-independent Schrödinger equation for a charged particle, in the presence of electric potential and vector potential, has been solved using He's Homotopy Perturbation Method (HPM).HPM is one of the newest analytical methods to solve linear and nonlinear differential equations.In contrast to the traditional perturbation methods, the Homotopy method does not require a small parameter in the equation.In this method, according to the homotopy technique, a Homotopy with an embedding parameter δ ∈ [0, 1] is constructed, and the embedding parameter is considered as a small parameter.Using cylindrical coordinates, it has been found that the z-equation of the charged particle is a one-dimensional harmonic oscillator and the r equation is actually a two-dimensional harmonic oscillator.The obtained results show the evidence of simplicity, usefulness, and effectiveness of the HPM for obtaining approximate analytical solutions for the time-independent Schrödinger equation for a charged particle in parallel electric and magnetic fields.