A family of Newton-type methods for solving nonlinear equations

A family of Newton-type methods free from second and higher order derivatives for solving nonlinear equations is presented. The order of the convergence of this family depends on a function. Under a condition on this function this family converge cubically and by imposing one condition more on this function one can obtain methods of order four. It has been shown that this family covers many of the available iterative methods. From this family two new iterative methods are obtained. Numerical experiments are also included.     

Stefan problem solved by Adomian decomposition method

The solution of the one-phase Stefan problem is presented. This problem consists of finding the distribution of temperature in the domain and the position of the moving interface (freezing front). The proposed solution is based on the Adomian decomposition method and optimalization. The validity of the approach is verified by comparing the results obtained with the analytical solution.     

A convergence analysis of the Adomian decomposition method for an abstract Cauchy problem of a system of first-order nonlinear differential equations

In this article, we study some fundamental results concerning the convergence of the Adomian decomposition method (ADM) for an abstract Cauchy problem of a system of first-order nonlinear differential equations. Under certain conditions, we obtain upper estimates for the norm of solutions of this system. We also obtain results about the error estimates for the approximate solutions by the ADM and discuss their applications.     

Multipoint efficient iterative methods and the dynamics of Ostrowski's method

ABSTRACT

In this work, we introduce a modification into the technique, presented in A. Cordero, J.L. Hueso, E. Martínez, and J.R. Torregrosa [Increasing the convergence order of an iterative method for nonlinear systems, Appl. Math. Lett. 25 (2012), pp. 2369–2374], that increases by two units the convergence order of an iterative method. The main idea is to compose a given iterative method of order p with a modification of Newton's method that introduces just one evaluation of the function, obtaining a new method of order p+2, avoiding the need to compute more than one derivative, so we improve the efficiency index in the scalar case. This procedure can be repeated n times, with the same approximation to the derivative, obtaining new iterative methods of order p+2n. We perform different numerical tests that confirm the theoretical results. By applying this procedure to Newton's method one obtains the well known fourth order Ostrowski's method. We finally analyse its dynamical behaviour on second and third degree real polynomials.     

New recurrence algorithms for the nonclassic Adomian polynomials

In this article, we present new algorithms for the nonclassic Adomian polynomials, which are valuable for solving a wide range of nonlinear functional equations by the Adomian decomposition method, and introduce their symbolic implementation in MATHEMATICA. Beginning with Rach’s new definition of the Adomian polynomials, we derive the explicit expression for each class of the Adomian polynomials, e.g. for the Class II, III and IV Adomian polynomials, where the Z_m,k are called the reduced polynomials. These expressions provide a basis for developing improved algorithmic approaches. By introducing the index vectors, the recurrence algorithms for the reduced polynomials are suitably deduced, which naturally lead to new recurrence algorithms for the Class II and Class III Adomian polynomials. MATHEMATICA programs generating these classes of Adomian polynomials are subsequently presented. Computation shows that for computer generation of the Class III Adomian polynomials, the new algorithm reduces the running times compared with the definitional formula. We also consider the number of summands of these classes of Adomian polynomials and obtain the corresponding formulas. Finally, we demonstrate the versatility of the four classes of Adomian polynomials with several examples, which include the nonlinearity of the form f(t,u), explicitly depending on the argument t.     

Taylor series for the Adomian decomposition method

In this paper, we analyse the exact solutions to scalar partial differential equations obtained, thanks to the summable Taylor series provided by Adomian's decomposition method. We propose a modification of the method which makes the calculations of Taylor coefficients easier and more direct. The difference is essential, for instance, in the case of non-homogenous equations or initial conditions and is illustrated by some examples.     

Adomian decomposition method by Gegenbauer and Jacobi polynomials

In this paper, orthogonal polynomials on [–1,1] interval are used to modify the Adomian decomposition method (ADM). Gegenbauer and Jacobi polynomials are employed to improve the ADM and compared with the method of using Chebyshev and Legendre polynomials. To show the efficiency of the developed method, some linear and nonlinear examples are solved by the proposed method, results are compared with other modifications of the ADM and the exact solutions of the problems.     

Domain decomposition preconditioners for the spectral collocation method

We propose and analyze several block iteration preconditioners for the solution of elliptic problems by spectral collocation methods in a region partitioned into several rectangles. It is shown that convergence is achieved with a rate that does not depend on the polynomial degree of the spectral solution. The iterative methods here presented can be effectively implemented on multiprocessor systems due to their high degree of parallelism.     

Iterative method for fuzzy equations

In this paper, we propose numerical solution for solving a system of fuzzy nonlinear equations based on Fixed point method. The convergence theorem is proved in detail. In this method the algorithm is illustrated by solving several numerical examples.     

Approximate analytical solutions of systems of PDEs by homotopy analysis method

In this paper, the homotopy analysis method (HAM) is applied to obtain series solutions to linear and nonlinear systems of first- and second-order partial differential equations (PDEs). The HAM solutions contain an auxiliary parameter which provides a convenient way of controlling the convergence region of series solutions. It is shown in particular that the solutions obtained by the variational iteration method (VIM) are only special cases of the HAM solutions.     

Adomian拆分法求解胀塑性流体边界层方程近似解

利用Adomian拆分方法及Mathematica软件的符号计算功能求解了一类胀塑性非牛顿流体边界层方程的近似解,对不同的参数给出了壁摩擦力的近似值及解的分布曲线,数值结果表明所得近似解有相当高的精度。     

A class of iterative methods for computing polar decomposition

In this article, there is offered a parametric class of iterative methods for computing the polar decomposition of a matrix. Each iteration of this class needs only one scalar-by-matrix and three matrix-by-matrix multiplications. It is no use computing inversion, so no numerical problems can be created because of ill-conditioning. Some available methods can be included in this class by choosing a suitable value for the parameter. There are obtained conditions under which this class is always quadratically convergent. The numerical comparison performed among six quadratically convergent methods for computing polar decomposition, and a special method of this class, chosen based on a specific value for the parameter, shows that the number of iterations of the special method is considerably near that of a cubically convergent Halley's method. Ten n×n matrices with n=5, 10, 20, 50, 100 were chosen to make this comparison.     

A family of simultaneous zero finding methods

A class of iterative methods with arbitrary high order of convergence for the simultaneous approximation of multiple complex zeros is considered in this paper. A special attention is paid to the fourth order method and its modifications because of their good computational efficiency. The order of convergence of the presented methods is determined. Numerical examples are given.     

即时学习算法在非线性系统迭代学习控制中的应用

运用即时学习算法来解决一类非线性系统的迭代学习控制初值问题。对于任何类型的迭代学习控制算法，即时学习算法都能有效地估计初始控制量，减小了初始输出误差，加快了算法的收敛速度，使得经过有限次迭代后系统输出能严格跟踪理想信号。对机器人系统的仿真结果表明了该方法的有效性。     

Application of homotopy perturbation method to multidimensional partial differential equations

In this paper, an application of homotopy perturbation method (HPM) is applied to solve the kindly of multidimensional partial differential equation such as Helmholtz equation. Comparisons are made between the Adomians decomposition method and HPM. The results reveal that the HPM is very effective and simple and gives the exact solution.     

A method for the approximation of distributions

Le but de cet article est l‘étude d‘une méthode d‘approximation de distributions par des combinaisons linéaires de masses de Dirac. La valeur des coefficients de cette combinaison lineaire est dérminée explidtement, en utilisant des suites de fonctions tendant vers la mesure de Dirac. Nous pr£sentons d‘abord le principe de la méthode puis quelques résultats théoriques de convergence. Enfin, nous donnons des résultats numériques d‘approximation de distributions et nous appliquons la méthode à la résolution numerique d'equations integrates singulieres.

The object of this article is to study a method of approximation of distributions by linear combinations of Dirac masses. The value of the coefficients of this linear combination is explicitly determined by using sequences of functions tending towards the Dirac measure. First, we present the principle of the method, then some theoretical results of convergence. Finally, we give numerical results of approximation of distributions and we apply the method to the numerical solution of singular integral equations.     

Solving inverse eigenvalue problems by a projected Newton method

The inverse eigenvalue problem for symmetric matrices (IEP) can be formulated as a system of two matrix equations. For solving the system a variation of Newton's method is used which has been proposed by Fusco and Zecca [Calcolo XXIII (1986), pp. 285–303] for the simultaneous computation of eigenvalues and eigenvectors of a given symmetric matrix. An iteration step of this method consists of a Newton step followed by an orthonormalization with the consequence that each iterate satisfies one of the given equations. The method is proved to convergence locally quadratically to regular solutions. The algorithm and some numerical examples are presented. In addition, it is shown that the so-called Method III proposed by Friedland, Nocedal, and Overton [SIAM J. Numer. Anal., 24 (1987), pp. 634–667] for solving IEP may be constructed similarly to the method presented here.     

Series solution of nonlinear differential equations by a novel extension of the Laplace transform method

A technique for extending the Laplace transform method to solve nonlinear differential equations is presented. By developing several theorems, which incorporate the Adomian polynomials, the Laplace transformation of nonlinear expressions is made possible. A number of well-known nonlinear equations including the Riccati equation, Clairaut's equation, the Blasius equation and several other ones involving nonlinearities of various types such as exponential and sinusoidal are solved for illustration. The proposed approach is analytical, accurate, and free of integration.     

Numerical solution of non-linear elliptic boundary-value problems by isomorphic iterative methods

New implicit iterative methods are presented for the efficient numerical solution of non-linear elliptic boundary-value problems. Isomorphic iterative schemes in conjunction with preconditioning techniques are used for solving non-linear elliptic equations in two and three-space dimensions. The application of the derived methods on characteristic 2D and 3D non-linear boundary-value problems is discussed and numerical results are given.