A MAPLE package of new ADM-Padé approximate solution for nonlinear problems

Based on the new definition of four classes of Adomian polynomials proposed by Rach in 2008, a MAPLE package of new Adomian-Padé approximate solution for solving nonlinear problems is presented. This package combines the merits of the Adomian decomposition method and the diagonal Padé technique, and may give more accurate solutions of nonlinear problems with strong nonlinearity. Besides, the package is user-friendly and efficient, one only needs to input the initial conditions, governing equation and four optional parameters, then our package will output the analytic approximate solution within a few seconds, where the equation is decomposed into three parts, they are the linear term R, nonlinear term NN and source function g, which are all in functional form. Meanwhile, several graphs generated from the above solutions are displayed and demonstrate a favorable comparison. In this paper, several different types of examples are given to illustrate the validity and promising flexibility of the package. This package provides us with a convenient and useful tool for dealing with nonlinear problems, as well as its electronic version is free to download via the journal website.     

A reliable algorithm for solving fourth-order boundary value problems

In this paper, we present an efficient numerical algorithm for approximate solutions of fourth-order boundary values problems with twopoint boundary conditions. The Adomian decomposition method and a modified form of this method are applied to construct the numerical solution. The scheme is tested on one linear problem and two nonlinear problems. The obtained results demonstrate the applicability and efficiency of the proposed scheme.     

Near-field and far-field approximations by the Adomian and asymptotic decomposition methods

The Adomian decomposition method and the asymptotic decomposition method give the near-field approximate solution and far-field approximate solution, respectively, for linear and nonlinear differential equations. The Padé approximants give solution continuation of series solutions, but the continuation is usually effective only on some finite domain, and it can not always give the asymptotic behavior as the independent variables approach infinity. We investigate the global approximate solution by matching the near-field approximation derived from the Adomian decomposition method with the far-field approximation derived from the asymptotic decomposition method for linear and nonlinear differential equations. For several examples we find that there exists an overlap between the near-field approximation and the far-field approximation, so we can match them to obtain a global approximate solution. For other nonlinear examples where the series solution from the Adomian decomposition method has a finite convergent domain, we can match the Padé approximant of the near-field approximation with the far-field approximation to obtain a global approximate solution representing the true, entire solution over an infinite domain.     

Numerical comparison of methods for solving linear differential equations of fractional order

In this article, we implement relatively new analytical techniques, the variational iteration method and the Adomian decomposition method, for solving linear differential equations of fractional order. The two methods in applied mathematics can be used as alternative methods for obtaining analytic and approximate solutions for different types of fractional differential equations. In these schemes, the solution takes the form of a convergent series with easily computable components. This paper will present a numerical comparison between the two methods and a conventional method such as the fractional difference method for solving linear differential equations of fractional order. The numerical results demonstrates that the new methods are quite accurate and readily implemented.     

Comparison of the Adomian decomposition method with homotopy perturbation method for the solutions of seventh order boundary value problems

The aim of this paper is to compare the Adomian decomposition method and the homotopy perturbation method for solving the linear and nonlinear seventh order boundary value problems. The approximate solutions of the problems obtained with a small amount of computation in both methods. Two numerical examples have been considered to illustrate the accuracy and implementation of the methods.     

Rational approximation solution of the fractional Sharma–Tasso–Olever equation

In the paper, we implement relatively new analytical techniques, the variational iteration method, the Adomian decomposition method and the homotopy perturbation method, for obtaining a rational approximation solution of the fractional Sharma–Tasso–Olever equation. The three methods in applied mathematics can be used as alternative methods for obtaining an analytic and approximate solution for different types of differential equations. In these schemes, the solution takes the form of a convergent series with easily computable components. The numerical results demonstrate the significant features, efficiency and reliability of the three approaches.     

A nonlinear fractional model to describe the population dynamics of two interacting species

In this paper, the approximate analytical solutions of Lotka–Volterra model with fractional derivative have been obtained by using hybrid analytic approach. This approach is amalgamation of homotopy analysis method, Laplace transform, and homotopy polynomials. First, we present an alternative framework of the method that can be used simply and effectively to handle nonlinear problems arising in several physical phenomena. Then, existence and uniqueness of solutions for the fractional Lotka–Volterra equations are discussed. We also carry out a detailed analysis on the stability of equilibrium. Further, we have derived the approximate solutions of predator and prey populations for different particular cases by using initial values. The numerical simulations of the result are depicted through different graphical representations showing that this hybrid analytic method is reliable and powerful method to solve linear and nonlinear fractional models arising in science and engineering. Copyright © 2017 John Wiley & Sons, Ltd.     

Combining approximate solutions for linear discrete ill-posed problems

Linear discrete ill-posed problems of small to medium size are commonly solved by first computing the singular value decomposition of the matrix and then determining an approximate solution by one of several available numerical methods, such as the truncated singular value decomposition or Tikhonov regularization. The determination of an approximate solution is relatively inexpensive once the singular value decomposition is available. This paper proposes to compute several approximate solutions by standard methods and then extract a new candidate solution from the linear subspace spanned by the available approximate solutions. We also describe how the method may be used for large-scale problems.     

A New Analytical Approach to Solve Magnetohydrodynamics Flow Over a Nonlinear Porous Stretching Sheet by Laplace Padé Decomposition Method

The aspire of this article is to bring in a new approximate method, that is to say the Laplace Padé decomposition method which is a mixture of Laplace decomposition and Padé approximation to offer an analytical approximate way out to magnetohydrodynamics flow over a nonlinear porous stretching sheet. This new iteration approach provides us with a convenient way to approximate solution. A closed agreement between the obtained solution and some well-known results has been established. The proposed procedure can be applied to handle other nonlinear problems.     

Numerical methods for nonlinear partial differential equations of fractional order

In this article, we implement relatively new analytical techniques, the variational iteration method and the Adomian decomposition method, for solving nonlinear partial differential equations of fractional order. The fractional derivatives are described in the Caputo sense. The two methods in applied mathematics can be used as alternative methods for obtaining analytic and approximate solutions for different types of fractional differential equations. In these schemes, the solution takes the form of a convergent series with easily computable components. Numerical results show that the two approaches are easy to implement and accurate when applied to partial differential equations of fractional order.     

Exp-Function Based Solution of Nonlinear Radhakrishnan,Kundu and Laskshmanan (RKL) Equation

In this work, we implement a relatively new analytical technique, Exp-Function method, for solving special form of generalized nonlinear Radhakrishnan, Kundu and Laskshmanan equation (RKL), which may contain highly nonlinear terms. This method can be used as an alternative to obtain analytic and approximate solutions of different types of differential equations applied in engineering mathematics. Some numerical examples are presented to illustrate the efficiency and reliability of Exp-Function method. It is predicted that Exp-Function method can be found widely applicable in engineering problems.     

An effective approach for numerical solution of linear and nonlinear singular boundary value problems

In this study, an effective approach is presented to obtain a numerical solution of linear and nonlinear singular boundary value problems. The proposed method is constructed by combining reproducing kernel and Legendre polynomials. Legendre basis functions are used to get the kernel function, and then the approximate solution is obtained as a finite series sum. Comparison of numerical results is made with the results obtained by other methods available in the literature. Furthermore, efficiency and accuracy of the method are demonstrated in tabulated results and plotted graphs. The numerical outcomes demonstrate that our method is very effective, applicable, and convenient.     

Numerical method for solving the nonlinear four-point boundary value problems

In this paper, a new reproducing kernel space is constructed skillfully in order to solve a class of nonlinear four-point boundary value problems. The exact solution of the linear problem can be expressed in the form of series and the approximate solution of the nonlinear problem is given by the iterative formula. Compared with known investigations, the advantages of our method are that the representation of exact solution is obtained in a new reproducing kernel Hilbert space and accuracy of numerical computation is higher. Meanwhile we present the convergent theorem, complexity analysis and error estimation. The performance of the new method is illustrated with several numerical examples.     

A trust-region method with a conic model for unconstrained optimization

In this paper, we propose and analyze a new conic trust-region algorithm for solving the unconstrained optimization problems. A new strategy is proposed to construct the conic model and the relevant conic trust-region subproblems are solved by an approximate solution method. This approximate solution method is not only easy to implement but also preserves the strong convergence properties of the exact solution methods. Under reasonable conditions, the locally linear and superlinear convergence of the proposed algorithm is established. The numerical experiments show that this algorithm is both feasible and efficient. Copyright © 2008 John Wiley & Sons, Ltd.     

New higher-order numerical one-step methods based on the Adomian and the modified decomposition methods

We develop new, higher-order numerical one-step methods and apply them to several examples to investigate approximate discrete solutions of nonlinear differential equations. These new algorithms are derived from the Adomian decomposition method (ADM) and the Rach-Adomian-Meyers modified decomposition method (MDM) to present an alternative to such classic schemes as the explicit Runge-Kutta methods for engineering models, which require high accuracy with low computational costs for repetitive simulations in contrast to a one-size-fits-all philosophy. This new approach incorporates the notion of analytic continuation, which extends the region of convergence without resort to other techniques that are also used to accelerate the rate of convergence such as the diagonal Padé approximants or the iterated Shanks transforms. Hence global solutions instead of only local solutions are directly realized albeit in a discretized representation. We observe that one of the difficulties in applying explicit Runge-Kutta one-step methods is that there is no general procedure to generate higher-order numeric methods. It becomes a time-consuming, tedious endeavor to generate higher-order explicit Runge-Kutta formulas, because it is constrained by the traditional Picard formalism as used to represent nonlinear differential equations. The ADM and the MDM rely instead upon Adomian’s representation and the Adomian polynomials to permit a straightforward universal procedure to generate higher-order numeric methods at will such as a 12th-order or 24th-order one-step method, if need be. Another key advantage is that we can easily estimate the maximum step-size prior to computing data sets representing the discretized solution, because we can approximate the radius of convergence from the solution approximants unlike the Runge-Kutta approach with its intrinsic linearization between computed data points. We propose new variable step-size, variable order and variable step-size, variable order algorithms for automatic step-size control to increase the computational efficiency and reduce the computational costs even further for critical engineering models.     

Solving the 2-D elliptic Monge-Ampère equation by a Kansa’s method

In this paper, a Kansa’s method is designed to solve numerically the Monge-Ampère equation. The primitive Kansa’s method is a meshfree method which applying the combination of some radial basis functions (such as Hardy’s MQ) to approximate the solution of the linear parabolic, hyperbolic and elliptic problems. But this method is deteriorated when is used to solve nonlinear partial differential equations. We approximate the solution in some local triangular subdomains by using the combination of some cubic polynomials. Then the given problems can be computed in each subdomains independently. We prove the stability and convergence of the new method for the elliptic Monge-Ampère equation. Finally, some numerical experiments are presented to demonstrate the theoretical results.     

The dual reciprocity boundary integral equation technique to solve a class of the linear and nonlinear fractional partial differential equations

In this paper, we apply the boundary integral equation technique and the dual reciprocity boundary elements method (DRBEM) for the numerical solution of linear and nonlinear time‐fractional partial differential equations (TFPDEs). The main aim of the present paper is to examine the applicability and efficiency of DRBEM for solving TFPDEs. We employ the time‐stepping scheme to approximate the time derivative, and the method of linear radial basis functions is also used in the DRBEM technique. This method is improved by using a predictor–corrector scheme to overcome the nonlinearity that appears in the nonlinear problems under consideration. To confirm the accuracy of the new approach, several examples are presented. The convergence of the DRBEM is studied numerically by comparing the exact solutions of the problems under investigation. Copyright © 2015 John Wiley & Sons, Ltd.     

基于近似惯性流形的后验Galerkin的方法

１．引言 为提高用数值方法解非线性发展方程及非线性椭圆边值问题的逼近阶,许多学者例如Ｊ．Ｎｏｖｏ和 Ｅ．Ｔｉｔｉ［４］, Ｍａｒｉｏｎ和 Ｔｅｍａｎ［６］,Ｊ．Ｘｕ［７］以及 Ｗ．Ｌａｙｔｏｎ［９］等人,提出了后验Ｇａｌｅｒｋｉｎ方法、近似惯性流形方法、非线性Ｇａｌｅｒｋｉｎ方法、各种区域分裂法、多重网格法等等．本文根据［１］提出了一种新的高精度的后验 Ｇａｌｅｒｋｉｎ方法．它的逼近阶是经典 Ｇａｌｅｒｋｉｎ方法逼近阶的两倍． 考虑非线性椭圆边值问题这里ｎ是按ｄ＝２,３）上具有分段光滑边界ｒ的有界区域,…     

A finite difference scheme for nonlinear ultra-parabolic equations

In this paper, our aim is to study a numerical method for an ultraparabolic equation with nonlinear source function. Mathematically, the bibliography on initial–boundary value problems for ultraparabolic equations is not extensive although the problems have many applications related to option pricing, multi-parameter Brownian motion, population dynamics and so forth. In this work, we present the approximate solution by virtue of finite difference scheme and Fourier series. For the nonlinear case, we use an iterative scheme by linear approximation to get the approximate solution and obtain error estimates. A numerical example is given to justify the theoretical analysis.     

地下结构和岩土介质弹塑性耦合分析的摄动半解析法*

本文研究了一个用于物理非线性相互作用分析的有效的数值方法。结构和介质耦合分析的弹塑性问题可用摄动法转化为几个线性问题,然后对相应的线性问题分别用有限条和有限层法分析地下结构和岩土介质以达到简化计算的目的。这种方法用了两次半解析技术——摄动和半解析解函数——将三维非线性耦合问题化为一维的数值问题。此外,本法是半解析法结合解析的摄动法应用于非线性问题的新进展,同时也是近年来发展的摄动数值法的一个分支。