Application of He’s homotopy perturbation method for Laplace transform

In this paper, an application of He’s homotopy perturbation method is proposed to compute Laplace transform. The results reveal that the method is very effective and simple.     

Approximate solution for system of differential-difference equations by means of the homotopy analysis method

In this paper, the homotopy analysis method is successfully applied to solve approximate solutions of the differential-difference system. The solution of another relativistic Toda lattice system is considered. Comparisons made between the results of the proposed method and exact solutions reveal that the homotopy analysis method is very effective in solving differential-difference system.     

The numerical solutions of differential transform method and the Laplace transform method for a system of differential equations

The differential transform method is one of the approximate methods which can be easily applied to many linear and nonlinear problems and is capable of reducing the size of computational work. Exact solutions can also be achieved by the known forms of the series solutions. In this paper, we present the definition and operation of the one-dimensional differential transform and investigate the particular exact solutions of system of ordinary differential equations that usually arise in mathematical biology by a one-dimensional differential transform method. The numerical results of the present method are presented and compared with the exact solutions that are calculated by the Laplace transform method.     

Iteratively weighted thresholding homotopy method for the sparse solution of underdetermined linear equations

Recently, iteratively reweighted methods have attracted much interest in compressed sensing, outperforming their unweighted counterparts in most cases. In these methods, decision variables and weights are optimized alternatingly, or decision variables are optimized under heuristically chosen weights. In this paper,we present a novel weighted l1-norm minimization problem for the sparsest solution of underdetermined linear equations. We propose an iteratively weighted thresholding method for this problem, wherein decision variables and weights are optimized simultaneously. Furthermore, we prove that the iteration process will converge eventually. Using the homotopy technique, we enhance the performance of the iteratively weighted thresholding method. Finally, extensive computational experiments show that our method performs better in terms of both running time and recovery accuracy compared with some state-of-the-art methods.     

A new modified Laplace decomposition method for higher order boundary value problems

In this article, we suggest modified Laplace decomposition method for analytical solution of eighth-order boundary value problems (BVPs). The numerical application indicates the effectiveness and stability of the proposed algorithm. The efficiency of proposed method is examined with the help of linear and nonlinear problems.     

A new technique of using homotopy analysis method for solving high‐order nonlinear differential equations

In this paper, a new technique of homotopy analysis method (HAM) is proposed for solving high‐order nonlinear initial value problems. This method improves the convergence of the series solution, eliminates the unneeded terms and reduces time consuming in the standard homotopy analysis method (HAM) by transform the nth‐order nonlinear differential equation to a system of n first‐order equations. Second‐ and third‐ order problems are solved as illustration examples of the proposed method. Copyright © 2010 John Wiley & Sons, Ltd.     

Optimal homotopy analysis method for nonlinear differential equations in the boundary layer

Optimal homotopy analysis method is a powerful tool for nonlinear differential equations. In this method, the convergence of the series solutions is controlled by one or more parameters which can be determined by minimizing a certain function. There are several approaches to determine the optimal values of these parameters, which can be divided into two categories, i.e. global optimization approach and step-by-step optimization approach. In the global optimization approach, all the parameters are optimized simultaneously at the last order of approximation. However, this process leads to a system of coupled, nonlinear algebraic equations in multiple variables which are very difficult to solve. In the step-by-step approach, the optimal values of these parameters are determined sequentially, that is, they are determined one by one at different orders of approximation. In this way, the computational efficiency is significantly improved, especially when high order of approximation is needed. In this paper, we provide extensive examples arising in similarity and non-similarity boundary layer theory to investigate the performance of these approaches. The results reveal that with the step-by-step approach, convergent solutions of high order of approximation can be obtained within much less CPU time, compared with the global approach and the traditional HAM.     

A one-step optimal homotopy analysis method for nonlinear differential equations

In this paper, a one-step optimal approach is proposed to improve the computational efficiency of the homotopy analysis method (HAM) for nonlinear problems. A generalized homotopy equation is first expressed by means of a unknown embedding function in Taylor series, whose coefficient is then determined one by one by minimizing the square residual error of the governing equation. Since at each order of approximation, only one algebraic equation with one unknown variable is solved, the computational efficiency is significantly improved, especially for high-order approximations. Some examples are used to illustrate the validity of this one-step optimal approach, which indicate that convergent series solution can be obtained by the optimal homotopy analysis method with much less CPU time. Using this one-step optimal approach, the homotopy analysis method might be applied to solve rather complicated differential equations with strong nonlinearity.     

Two-parameter homotopy method for nonlinear equations

We introduce a second auxiliary parameter into the zero-order deformation equation and propose a generalization of the homotopy analysis method. This includes the derivation of a general solution in terms of the Bell polynomials for nonlinear equations. Numerical examples show that the proposed zero-order deformation equation improves the convergence region and rate of the series solution and allows greater freedom in the selection of auxiliary operators. This facilitates the development of a homotopy iteration scheme for nonlinear equations with discontinuous or zero derivatives that are not amenable to Newton-type iteration schemes. The homotopy iteration scheme represents a generalization of conventional iteration schemes and additional examples demonstrate its applicability for a wider range of nonlinear problems.     

On the homotopy analysis method for backward/forward-backward stochastic differential equations

In this paper, an analytic approximation method for highly nonlinear equations, namely the homotopy analysis method (HAM), is employed to solve some backward stochastic differential equations (BSDEs) and forward-backward stochastic differential equations (FBSDEs), including one with high dimensionality (up to 12 dimensions). By means of the HAM, convergent series solutions can be quickly obtained with high accuracy for a FBSDE in a 6-dimensional case, within less than 1 % CPU time used by a currently reported numerical method for the same case [34]. Especially, as dimensionality enlarges, the increase of computational complexity for the HAM is not as dramatic as this numerical method. All of these demonstrate the validity and high efficiency of the HAM for the backward/forward-backward stochastic differential equations in science, engineering, and finance.     

The combined Laplace transform and new homotopy perturbation methods for stiff systems of ODEs

In this paper, we propose new technique for solving stiff system of ordinary differential equations. This algorithm is based on Laplace transform and homotopy perturbation methods. The new technique is applied to solving two mathematical models of stiff problem. We show that the present approach is relatively easy, efficient and highly accurate.     

A regularization method for approximating the inverse Laplace transform

A new method for approximating the inerse Laplace transform is presented. We first change our Laplace transform equation into a convolution type integral equation, where Tikhonov regularization techniques and the Fourier transformation are easily applied. We finally obtain a regularized approximation to the inverse Laplace transform as finite sum     

On the coupling of the homotopy perturbation method and Laplace transformation

In this paper, a Laplace homotopy perturbation method is employed for solving one-dimensional non-homogeneous partial differential equations with a variable coefficient. This method is a combination of the Laplace transform and the Homotopy Perturbation Method (LHPM). LHPM presents an accurate methodology to solve non-homogeneous partial differential equations with a variable coefficient. The aim of using the Laplace transform is to overcome the deficiency that is mainly caused by unsatisfied conditions in other semi-analytical methods such as HPM, VIM, and ADM. The approximate solutions obtained by means of LHPM in a wide range of the problem’s domain were compared with those results obtained from the actual solutions, the Homotopy Perturbation Method (HPM) and the finite element method. The comparison shows a precise agreement between the results, and introduces this new method as an applicable one which it needs fewer computations and is much easier and more convenient than others, so it can be widely used in engineering too.     

Solution of nonlinear fractional differential equations using homotopy analysis method

In this article, the homotopy analysis method has been applied to solve nonlinear differential equations of fractional order. The validity of this method has successfully been accomplished by applying it to find the solution of two nonlinear fractional equations. The results obtained by homotopy analysis method have been compared with those exact solutions. The results show that the solution of homotopy analysis method is good agreement with the exact solution.