Solution of BVPs for fourth-order integro-differential equations by using homotopy perturbation method

In this study, the homotopy perturbation method proposed by Ji-Huan He is applied to solve both linear and nonlinear boundary value problems for fourth-order integro-differential equations. The analysis is accompanied by numerical examples. The results show that the homotopy perturbation method is of high accuracy, more convenient and efficient for solving integro-differential equations.     

A new analytical method for solving systems of Volterra integral equations

In this paper, a new modified homotopy perturbation method (NHPM) is introduced for solving systems of Volterra integral equations of the second kind. Theorems of existence and uniqueness of the solutions to these equations are presented. Comparison of the results of applying the NHPM with those of the homotopy perturbation method and Adomian's decomposition method leads to significant consequences. Several examples, including the system of linear and nonlinear Volterra integral equations, are given to demonstrate the efficiency of the new method.     

He’s homotopy perturbation method: A strongly promising method for solving non-linear systems of the mixed Volterra–Fredholm integral equations

In this paper, He’s homotopy perturbation method is applied to solve non-linear systems of mixed Volterra–Fredholm integral equations. Two examples are presented to illustrate the ability of the method. Also comparisons are made between the Adomian decomposition method and the homotopy perturbation method. The results reveal that He’s homotopy perturbation method is very effective and simple and in these examples leads to the exact solutions.     

Solution of the differential algebraic equations via homotopy perturbation method and their engineering applications

Differential algebraic equations (DAEs) appear in many fields of physics and have a wide range of applications in various branches of science and engineering. Finding reliable methods to solve DAEs has been the subject of many investigations in recent years. In this paper, the He's homotopy perturbation method is applied for finding the solution of linear and nonlinear DAEs. First, an index reduction technique is implemented for semi-explicit and Hessenberg DAEs, then the obtained problem can be appropriately solved by the homotopy perturbation method. This technique provides a summation of an infinite series with easily computable terms, which converges to the exact solution of the problem. The scheme is tested for some high-index DAEs and the results demonstrate that the method is very straightforward and can be considered as a powerful mathematical tool.     

A general approach to hyperbolic partial differential equations by homotopy perturbation method

The hyperbolic partial differential equations (PDEs) have a wide range of applications in science and engineering. In this article, the exact solutions of some hyperbolic PDEs are presented by means of He's homotopy perturbation method (HPM). The results reveal that the HPM is very effective and convenient in solving nonlinear problems.     

求解非线性期权定价模型的自适应小波同伦摄动技术

Leland 模型是在考虑交易费用的情况下,对 Black - Scholes 模型进行修改得到的非线性期权定价模型. 本文针对 Leland 模型,提出了一种求解非线性动力学模型的自适应多尺度小波同伦摄动法. 该方法首先利用插值小波理论构造了用于逼近连续函数的多尺度小波插值算子,利用该算子可以将非线性期权定价模型方程自适应离散为非线性常微分方程组; 然后将用于求解非线性常微分方程组的同伦摄动技术和小波变换的动态过程相结合,构造了求解 Leland 模型的自适应数值求解方法. 数值模拟结果验证了该方法在数值精度和计算效率方面的优越性.     

The application of homotopy perturbation method for MHD flows of UCM fluids above porous stretching sheets

In this study, by means of homotopy perturbation method (HPM) an approximate analytical solution of the magnetohydrodynamic (MHD) boundary layer flow of an upper-convected Maxwell (UCM) fluid over a porous stretching sheet is obtained. The main feature of the HPM is that it deforms a difficult problem into a set of problems which are easier to solve. HPM produces analytical expressions for the solution of nonlinear differential equations. The obtained analytic solution is in the form of an infinite power series. In this work, the analytical solution obtained by using only two terms from HPM solution. The results reveal that the proposed method is very effective and simple and can be applied to other nonlinear problems. Also it is shown that this method coincides with homotopy analysis method (HAM) for the studied problem.     

Homotopy perturbation technique for solving two-point boundary value problems - comparison with other methods

The two-point boundary value problems occur in a wide variety of problems in engineering and science. In this paper, we implement the homotopy perturbation method for solving the linear and nonlinear two-point boundary value problems. The main aim of this paper is to compare the performance of the homotopy perturbation method with extended Adomian decomposition method and shooting method. As a result, for the same number of terms, the homotopy perturbation method yields relatively more accurate results with rapid convergence than other methods. The computer symbolic systems such as Maple and Mathematica allow us to perform complicated and tedious calculations.     

Solving singular nonlinear boundary value problems by combining the homotopy perturbation method and reproducing kernel Hilbert space method

This paper investigates singular nonlinear boundary value problems (BVPs). The numerical solutions are developed by combining He's homotopy perturbation method (HPM) and reproducing kernel Hilbert space method (RKHSM). He's HPM is based on the use of traditional perturbation method and homotopy technique. The HPM can reduce a nonlinear problem to a sequence of linear problems and generate a rapid convergent series solution in most cases. RKHSM is also an analytical technique, which can solve powerfully singular linear BVPs. Therefore, we solve singular nonlinear BVPs using advantages of these two methods. Three numerical examples are presented to illustrate the strength of the method.     

Homotopy perturbation transform method for nonlinear equations using He’s polynomials

In this paper, a combined form of the Laplace transform method with the homotopy perturbation method is proposed to solve nonlinear equations. This method is called the homotopy perturbation transform method (HPTM). The nonlinear terms can be easily handled by the use of He’s polynomials. The proposed scheme finds the solution without any discretization or restrictive assumptions and avoids the round-off errors. The fact that the proposed technique solves nonlinear problems without using Adomian’s polynomials can be considered as a clear advantage of this algorithm over the decomposition method.     

Solutions of the Cahn–Hilliard equation

In this paper, we found some exact solutions of the Cahn–Hilliard equation and the system of the equations by considering a modified extended tanh function method. A numerical solution to a Cahn–Hilliard equation is obtained using a homotopy perturbation method (HPM) combined with the Adomian decomposition method (ADM). The comparisons are given in the tables.     

Calculating luminosity distance versus redshift in FLRW cosmology via homotopy perturbation method

We propose an efficient analytical method for estimating the luminosity distance in a homogenous Friedmann-Lemaître-Robertson-Walker (FLRW) model of the Universe. This method is based on the homotopy perturbation method (HPM) which has a high accuracy in many nonlinear problems and can be easily implemented. For an analytical calculation of the luminosity distance, we suggest to proceed not from computation of the integral which determines it but from the solution of a certain differential equation with the corresponding initial conditions. Solving this equation by means of HPM, we obtain approximate analytical expressions for the luminosity distance as a function of the redshift for two different types of homotopy. A possible extension of this method to other cosmological models is also discussed.     

An algorithm for solving multi-term diffusion-wave equations of fractional order

In this paper an algorithm, based on a new modified homotopy perturbation method (MHPM), is presented to obtain approximate solutions of multi-term diffusion-wave equations of fractional order. To illustrate the method some examples are provided. The results show the simplicity and the efficiency of the algorithm.     

Numerical study of the solution of the Burgers and coupled Burgers equations by a differential transformation method

In this paper, the Differential Transformation Method (DTM) is employed to obtain the numerical/analytical solutions of the Burgers and coupled Burgers equations. We begin by showing how the differential transformation method applies to the linear and nonlinear parts of any PDE and give some examples to illustrate the sufficiency of the method for solving such nonlinear partial differential equations. We also compare it against three famous methods, namely the homotopy perturbation method, the homotopy analysis method and the variational iteration method. These results show that the technique introduced here is accurate and easy to apply.     

Convergence analysis of the homotopy perturbation method for solving nonlinear ill-posed operator equations

The homotopy perturbation method is used to construct a new iteration algorithm for solving nonlinear ill-posed operator equations. Numerical tests are given, showing that the algorithm is more efficient than the well-known Landweber method.     

An efficient iterated method for mathematical biology model

The purpose of this study is to introduce an efficient iterated homotopy perturbation transform method (IHPTM) for solving a mathematical model of HIV infection of CD4+ T cells. The equations are Laplace transformed, and the nonlinear terms are represented by He’s polynomials. The solutions are obtained in the form of rapidly convergent series with elegantly computable terms. This approach, in contrast to classical perturbation techniques, is valid even for systems without any small/large parameters and therefore can be applied more widely than traditional perturbation techniques, especially when there do not exist any small/large quantities. A good agreement of the novel method solution with the existing solutions is presented graphically and in tabulated forms to study the efficiency and accuracy of IHPTM. This study demonstrates the general validity and the great potential of the IHPTM for solving strongly nonlinear problems.     

PHoMpara – Parallel Implementation of the Polyhedral Homotopy Continuation Method for Polynomial Systems

The polyhedral homotopy continuation method is known to be a successful method for finding all isolated solutions of a system of polynomial equations. PHoM, an implementation of the method in C++, finds all isolated solutions of a polynomial system by constructing a family of modified polyhedral homotopy functions, tracing the solution curves of the homotopy equations, and verifying the obtained solutions. A software package PHoMpara parallelizes PHoM to solve a polynomial system of large size. Many characteristics of the polyhedral homotopy continuation method make parallel implementation efficient and provide excellent scalability. Numerical results include some large polynomial systems that had not been solved.     

Modified homotopy perturbation method for solving the Stokes equations

A new approach is proposed for the stationary Stokes equations. Based on the homotopy perturbation method, some iterative algorithms are constructed, and four kinds of perturbation cases are considered respectively. Numerical experiments show that these algorithms are simple and effective.     

An analytical method for solving linear Fredholm fuzzy integral equations of the second kind

In this paper, we use the parametric form of a fuzzy number and convert a linear fuzzy Fredholm integral equation to two linear systems of integral equations of the second kind in the crisp case. For fuzzy Fredholm integral equations with kernels, the sign of which is difficult to determine, a new parametric form of the fuzzy Fredholm integral equation is introduced. We use the homotopy analysis method to find the approximate solution of the system, and hence, obtain an approximation for fuzzy solutions of the linear fuzzy Fredholm integral equation of the second kind. The proposed method is illustrated by solving some examples. Using the HAM, it is possible to find the exact solution or the approximate solution of the problem in the form of a series.     

The extended homotopy perturbation method for the boundary layer flow due to a stretching sheet with partial slip

A fully analytical solution of the steady, laminar and axisymmetric flow of a Newtonian fluid due to a stretching sheet when there is a partial slip of the fluid past the sheet has been derived using the extended homotopy perturbation method. The solution differs from that obtained by the classical homotopy perturbation method in that it is capable of generating a totally analytical solution up to any desired degree of accuracy and is not limited to the first-order correction terms. For an eight-decimal accuracy, it is sufficient to take 12 terms in the power series in the perturbation parameter, provided that use is made of Shanks’ transformation. Unlike other similar problems involving mass transfer across the sheet and/or the presence of a transverse magnetic field, the solution for the present problem is relatively insensitive to the velocity slip parameter.