On the Solvability of General Linear Methods for Dissipative Dynamical Systems

The main purpose of the present paper is to examine the existence and local uniqueness of solutions of the implicit equations arising in the application of a weakly algebraically stable general linear methods to dissipative dynamical systems, and to extend the existing relevant results of Runge-Kutta methods by Humphries and Stuart (1994).     

A note on the local convergence of iterative methods based on Adomian decomposition method and 3-node quadrature rule

Darvishi and Barati [M.T. Darvishi, A. Barati, Super cubic iterative methods to solve systems of nonlinear equations, Appl. Math. Comput., 2006, 10.1016/j.amc.2006.11.022] derived a Super cubic method from the Adomian decomposition method to solve systems of nonlinear equations. The authors showed that the method is third-order convergent using classical Taylor expansion but the numerical experiments conducted by them showed that the method exhibits super cubic convergence. In the present work, using Ostrowski’s technique based on point of attraction, we show that their method is in fact fourth-order convergent. We also prove the local convergence of another fourth-order method from 3-node quadrature rule using point of attraction.     

Componentwise fast convergence in the solution of full-rank systems of nonlinear equations

The asymptotic convergence of parameterized variants of Newton’s method for the solution of nonlinear systems of equations is considered. The original system is perturbed by a term involving the variables and a scalar parameter which is driven to zero as the iteration proceeds. The exact local solutions to the perturbed systems then form a differentiable path leading to a solution of the original system, the scalar parameter determining the progress along the path. A path-following algorithm, which involves an inner iteration in which the perturbed systems are approximately solved, is outlined. It is shown that asymptotically, a single linear system is solved per update of the scalar parameter. It turns out that a componentwise Q-superlinear rate may be attained, both in the direct error and in the residuals, under standard assumptions, and that this rate may be made arbitrarily close to quadratic. Numerical experiments illustrate the results and we discuss the relationships that this method shares with interior methods in constrained optimization. Received: September 8, 2000 / Accepted: September 17, 2001?Published online February 14, 2002     

Semilocal and global convergence of the Newton‐HSS method for systems of nonlinear equations

Newton‐HSS methods, which are variants of inexact Newton methods different from the Newton–Krylov methods, have been shown to be competitive methods for solving large sparse systems of nonlinear equations with positive‐definite Jacobian matrices (J. Comp. Math. 2010; 28 :235–260). In that paper, only local convergence was proved. In this paper, we prove a Kantorovich‐type semilocal convergence. Then we introduce Newton‐HSS methods with a backtracking strategy and analyse their global convergence. Finally, these globally convergent Newton‐HSS methods are shown to work well on several typical examples using different forcing terms to stop the inner iterations. Copyright © 2010 John Wiley & Sons, Ltd.     

Domain of existence for the solution of some IVP's and BVP's by using an efficient ninth-order iterative method

In this paper, we consider the problem of solving initial value problems and boundary value problems through the point of view of its continuous form. It is well known that in most cases these types of problems are solved numerically by performing a discretization and applying the finite difference technique to approximate the derivatives, transforming the equation into a finite-dimensional nonlinear system of equations. However, we would like to focus on the continuous problem, and therefore, we try to set the domain of existence and uniqueness for its analytic solution. For this purpose, we study the semilocal convergence of a Newton-type method with frozen first derivative in Banach spaces. We impose only the assumption that the Fréchet derivative satisfies the Lipschitz continuity condition and that it is bounded in the whole domain in order to obtain appropriate recurrence relations so that we may determine the domains of convergence and uniqueness for the solution. Our final aim is to apply these theoretical results to solve applied problems that come from integral equations, ordinary differential equations, and boundary value problems.     

On non-linear evolution equations of higher order – the introduction and application of a novel computational approach

The purpose of the present paper is to introduce a new computational algebraic procedure that can easily be applied to derive class of solutions of non-linear partial differential equations (nPDE) especially of higher order.The crucial step needs an auxiliary variable satisfying some ordinary differential equations (ODE) of first order containing sine, cosine and their hyperbolic varieties introducing to the first time.General transformations are given to determine class of solutions explicitly.The validity and reliability of the method is tested by its application to some important non-linear evolution equations leading to new class of solutions with physical significance.Nevertheless it should be emphasised that this techniques do not need the solution of complicate nODEs as in the case of similarity reduction.Further, the algorithm works efficiently, is clear structured and can be used in any applications independent of the order of the nPDE. For computational purposes the method is appropriate to rewrite it in any computer languages.Therefore, the given novel algebraic approach is suitable for a wider class of nPDE in order to augment the solution manifold by a straightforward alternative approach.     

On the asymptotic and practical stability of Persidskii-type systems with switching

In the paper, one class of differential systems with nonlinearities satisfying sector constraints is considered. We study the case where some of the sector constraints are given by linear inequalities, and some by nonlinear ones. It is assumed that the coefficients in the system can switch from one set of values to another. Sufficient conditions for the asymptotic and practical stability of the zero solution of the system are investigated using the direct Lyapunov method and the theory of differential inequalities. Restrictions on the switching law that provide a given region of attraction and ultimate bound for solutions of the system are obtained. An approach based on the construction of different differential inequalities for the Lyapunov function in different parts of the phase space is proposed, which makes it possible to improve the results obtained. The results are applied to the analysis of one automatic control system.     

Global analysis of continuous analogues of the Levenberg-Marquardt and Newton-Raphson methods for solving nonlinear equations

Summary Global analyses are given to continuous analogues of the Levenberg-Marquardt methoddx/dt=−(J ^t(x)J(x)+δI)⁻¹J^t(x)g(x), and the Newton-Raphson-Ben-Israel methoddx/dt=−J ⁺(x)g(x), for solving an over- and under-determined systemg(x)=0 of nonlinear equations. The characteristics of both methods are compared. Erros in some literature which dealt with related continous analogue methods are pointed out. The Institute of Statistical Mathematics     

On a family of extremum problems and the properties of an integral

The following extremum problem is studied:

On the convergence rate and optimization of a numerical method with splitting of boundary conditions for the stokes system in a spherical layer in the axisymmetric case: Modification for thick layers

The convergence rate of a fast-converging second-order accurate iterative method with splitting of boundary conditions constructed by the authors for solving an axisymmetric Dirichlet boundary value problem for the Stokes system in a spherical gap is studied numerically. For R/r exceeding about 30, where r and R are the radii of the inner and outer boundary spheres, it is established that the convergence rate of the method is lower (and considerably lower for large R/r) than the convergence rate of its differential version. For this reason, a really simpler, more slowly converging modification of the original method is constructed on the differential level and a finite-element implementation of this modification is built. Numerical experiments have revealed that this modification has the same convergence rate as its differential counterpart for R/r of up to 5 × 10³. When the multigrid method is used to solve the split and auxiliary boundary value problems arising at iterations, the modification is more efficient than the original method starting from R/r ～ 30 and is considerably more efficient for large values of R/r. It is also established that the convergence rates of both methods depend little on the stretching coefficient η of circularly rectangular mesh cells in a range of η that is well sufficient for effective use of the multigrid method for arbitrary values of R/r smaller than ～ 5 × 10³.     

用内禀法求解某些四阶微分方程的有界性和稳定性成果

本文首先对方程（１．１）建立了Ｌｙａｐｕｎｏｖ函数，然后在Ｐ＝０的情况下，证明了平庸解ｘ＝０在大范围内的渐近稳定性，和在ｐ≠０的情况下，研究了（１．１）式的解的有界性问题．这些结论对一些众所周知的成果有所改进．     

On the exponential decay in thermoelasticity without energy dissipation and of type III in presence of an absorbing boundary

We study the asymptotic behavior of the solution of a 3D hyperbolic system arising in the Green-Naghdi models of thermoelasticity of type II and III with a dissipative boundary condition for the displacement and prove that the energy exponentially decays in time.     

On the efficiency of Newton and Broyden numerical methods in the investigation of the regular polygon problem of (N + 1) bodies

Numerical methods of finding the roots of a system of non-linear algebraic equations are treated in this paper. This paper attempts to give an answer to the selection of the most efficient method in a complex problem of Celestial Dynamics, the so-called ring problem of (N + 1) bodies. We apply Newton and Broyden’s method to these problems and we investigate, by means of their use, the planar equilibrium points, the five equilibrium zones, which are symbolized by A₁, A₂, B, C₂, and C₁ (by order of appearance from the center O to the periphery of the imaginary circle on which the primaries lie) [T.J. Kalvouridis, A planar case of the N + 1 body problem: the ring problem. Astrophys. Space Sci. 260 (3) (1999) 309-325], and the attracting regions of the system. The efficiency of these methods is studied through a comparative process. The obtained results are demonstrated in figures and are discussed.