A comparison of the Newton-Krylov method with high order Newton-like methods to solve nonlinear systems

We compare the CPU time and error estimates of some variants of Newton method of the third and fourth-order convergence with those of the Newton-Krylov method used to solve systems of nonlinear equations. By expanding some numerical experiments we show that the use of Newton-Krylov method is better in the cost and accuracy points of view than the use of other high order Newton-like methods when the system is sparse and its size is large.     

Analysis of reaction-diffusion systems by the method of linear determining equations

Exact solutions to two-component systems of reaction-diffusion equations are sought by the method of linear determining equations (LDEs) generalizing the methods of the classical group analysis of differential equations. LDEs are constructed for a system of two second-order evolutionary equations. The results of solving the LDEs are presented for two-component systems of reaction-diffusion equations with polynomial nonlinearities in the diffusion coefficients. Examples of constructing noninvariant solutions are presented for the reaction-diffusion systems that possess invariant manifolds.     

On finite difference approximation of a matrix-vector product in the Jacobian-free Newton-Krylov method

The Jacobian-free Newton-Krylov (JFNK) method is a special kind of Newton-Krylov algorithm, in which the matrix-vector product is approximated by a finite difference scheme. Consequently, it is not necessary to form and store the Jacobian matrix. This can greatly improve the efficiency and enlarge the application area of the Newton-Krylov method. The finite difference scheme has a strong influence on the accuracy and robustness of the JFNK method. In this paper, several methods for approximating the Jacobian-vector product, including the finite difference scheme and the finite difference step size, are analyzed and compared. Numerical results are given to verify the effectiveness of different finite difference methods.     

On the formation of sharp transition layers in two-dimensional reaction-diffusion models

For a singularly perturbed parabolic equation in two dimensions, the formation of a solution with a sharp transition layer from a sufficiently general initial function is considered. An asymptotic analysis is used to estimate the time required for the formation of a contrast structure. Numerical results are presented.     

Diffusion effect and stability analysis of a predator-prey system described by a delayed reaction-diffusion equations

In this paper, we consider a delayed reaction-diffusion equations which describes a two-species predator-prey system with diffusion terms and stage structure. By using the linearization method and the method of upper and lower solutions, we study the local and global stability of the constant equilibria, respectively. The results show that the free diffusion of the delayed reaction-diffusion equations has no effect on the populations when the diffusion is too slow; otherwise, the free diffusion has a certain influence on the populations, however, the influence can be eliminated by improving the parameters to satisfy some suitable conditions.     

Computing the field of values and pseudospectra using the Lanczos method with continuation

The field of values and pseudospectra are useful tools for understanding the behaviour of various matrix processes. To compute these subsets of the complex plane it is necessary to estimate one or two eigenvalues of a large number of parametrized Hermitian matrices; these computations are prohibitively expensive for large, possibly sparse, matrices, if done by use of the QR algorithm. We describe an approach based on the Lanczos method with selective reorthogonalization and Chebyshev acceleration that, when combined with continuation and a shift and invert technique, enables efficient and reliable computation of the field of values and pseudospectra for large matrices. The idea of using the Lanczos method with continuation to compute pseudospectra is not new, but in experiments reported here our algorithm is faster and more accurate than existing algorithms of this type.This work was supported by Engineering and Physical Sciences Research Council grants GR/H/52139 and GR/H/94528.     

The spectral method and numerical continuation algorithm for the von Kármán problem with postbuckling behaviour of solutions

In this paper a spectral method and a numerical continuation algorithm for solving eigenvalue problems for the rectangular von Kármán plate with different boundary conditions (simply supported, partially or totally clamped) and physical parameters are introduced. The solution of these problems has a postbuckling behaviour. The spectral method is based on a variational principle (Galerkin’s approach) with a choice of global basis functions which are combinations of trigonometric functions. Convergence results of this method are proved and the rate of convergence is estimated. The discretized nonlinear model is treated by Newton’s iterative scheme and numerical continuation. Branches of eigenfunctions found by the algorithm are traced. Numerical results of solving the problems for polygonal and ferroconcrete plates are presented. Communicated by A. Zhou.     

On the approximate solutions for system of fractional integro-differential equations using Chebyshev pseudo-spectral method

In this paper, we implement Chebyshev pseudo-spectral method for solving numerically system of linear and non-linear fractional integro-differential equations of Volterra type. The proposed technique is based on the new derived formula of the Caputo fractional derivative. The suggested method reduces this type of systems to the solution of system of linear or non-linear algebraic equations. We give the convergence analysis and derive an upper bound of the error for the derived formula. To demonstrate the validity and applicability of the suggested method, some test examples are given. Also, we present a comparison with the previous work using the homotopy perturbation method.     

Global solutions to the shallow water system with a method of an additional argument

The classical system of shallow water (Saint–Venant) equations describes long surface waves in an inviscid incompressible fluid of a variable depth. Although shock waves are expected in this quasi-linear hyperbolic system for a wide class of initial data, we find a sufficient condition on the initial data that guarantee existence of a global classical solution continued from a local solution. The sufficient conditions can be easily satisfied for the fluid flow propagating in one direction with two characteristic velocities of the same sign and two monotonically increasing Riemann invariants. We prove that these properties persist in the time evolution of the classical solutions to the shallow water equations and provide no shock wave singularities formed in a finite time over a half-line or an infinite line. On a technical side, we develop a novel method of an additional argument, which allows to obtain local and global solutions to the quasi-linear hyperbolic systems in physical rather than characteristic variables.     

Asymptotics of the solution to a singularly perturbed system of first-order partial differential equations with small nonlinearity in the critical case

A formal asymptotic representation of the solution to an initial-boundary value problem for a singularly perturbed system of first-order partial differential equations with small nonlinearity is constructed.     

A new numerical approach for the solution of nonlinear Fredholm integral equations system of second kind by using Bernstein collocation method

This paper presents an efficient numerical method for finding solutions of the nonlinear Fredholm integral equations system of second kind based on Bernstein polynomials basis. The numerical results obtained by the present method have been compared with those obtained by B‐spline wavelet method. This proposed method reduces the system of integral equations to a system of algebraic equations that can be solved easily any of the usual numerical methods. Numerical examples are presented to illustrate the accuracy of the method. Copyright © 2013 John Wiley & Sons, Ltd.     

Numerical analysis of the method of fundamental solution for harmonic problems in annular domains

In this study, we investigate the application of the method of fundamental solutions (MFS) to the Dirichlet problem for Laplace's equation in an annular domain. We examine the properties of the resulting coefficient matrix and its eigenvalues. The convergence of the method is proved for analytic boundary data. An efficient matrix decomposition algorithm using fast Fourier transforms (FFTs) is developed for the computation of the MFS approximation. We also tested the algorithm numerically on several problems confirming the theoretical predictions. © 2005 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2005     

An optimal regularizing algorithm for the recovery of functionals in linear inverse problems with sourcewise represented solution

A linear operator equation with a sourcewise represented exact solution is solved approximately. To this end, the method of extending compacts developed in an earlier work is applied. Based on this method, a new algorithm is proposed for recovering the value of a linear functional at the solution of the linear operator equation. This algorithm is shown to be an optimal regularizing one.     

Decomposition in large system optimization using the method of multipliers

Although the method of multipliers can resolve the dual gaps which will often appear between the true optimum point and the saddle point of the Lagrangian in large system optimization using the Lagrangian approach, it is impossible to decompose the generalized Lagrangian in a straightforward manner because of its quadratic character. A technique using the linear approximation of the perturbed generalized Lagrangian has recently been proposed by Stephanopoulos and Westerberg for the decomposition. In this paper, another attractive decomposition technique which transforms the nonseparable crossproduct terms into the minimum of sum of separable terms is proposed. The computational efforts required for large system optimization can be much reduced by adopting the latter technique in place of the former, as illustrated by application of these two techniques to an optimization problem of a chemical reactor system.The authors would like to acknowledge the valuable comments given by Professor D. G. Luenberger of Stanford University.     

On a system of singularly perturbed second-order quasilinear ordinary differential equations in the critical cases

A singularly perturbed system of second-order quasilinear ordinary differential equations with a small parameter multiplying the second derivatives is examined in the case where the coefficient matrix of the first derivatives is singular and does not depend on the unknown functions.     

Lower and upper bounds of condition number for Vandermonde-wise matrices and method of fundamental solutions using pseudo radial-lines

Consider the method of fundamental solutions (MFS) for 2D Laplace's equation in a bounded simply connected domain $S$. In the standard MFS, the source nodes are located on a closed contour outside the domain boundary $\mathrm{\Gamma }(=\partial S)$, which is called pseudo-boundary. For circular, elliptic, and general closed pseudo-boundaries, analysis and computation have been studied extensively. New locations of source nodes are proposed along two pseudo radial-lines outside $\mathrm{\Gamma }$. Numerical results are very encouraging and promising. Since the success of the MFS mainly depends on stability, our efforts are focused on deriving the lower and upper bounds of condition number (Cond). The study finds stability properties of new Vandermonde-wise matrices on nodes $x_i\in [a,b]$ with . The Vandermonde-wise matrix is called in this article if it can be decomposed into the standard Vandermonde matrix. New lower and upper bounds of Cond are first derived for the standard Vandermonde matrix, and then for new algorithms of the MFS using two pseudo radial-lines. Both lower and upper bounds of Cond are intriguing in the stability study for the MFS. Numerical experiments are carried out to verify the stability analysis made. Since the fundamental solutions (as $\{\ln |\overline{{PQ}_i}|\}$) are the basis functions of the MFS, new Vandermonde-wise matrices are found. Since the nodes $x_i\in [a,b]$ with may come from approximations and interpolations by the Laurent polynomials with singular part, the conclusions in this article are important not only to the MFS but also to matrix analysis.     

Numerical solution of the system of second-order boundary value problems using the local radial basis functions based differential quadrature collocation method

In this research, we propose a numerical scheme to solve the system of second-order boundary value problems. In this way, we use the Local Radial Basis Function Differential Quadrature (LRBFDQ) method for approximating the derivative. The LRBFDQ method approximates the derivatives by Radial Basis Functions (RBFs) interpolation using a small set of nodes in the support domain of any node. So the new scheme needs much less computational work than the globally supported RBFs collocation method. We use two techniques presented by Bayona et al. (2011, 2012) [29], [30] to determine the optimal shape parameter. Some examples are presented to demonstrate the accuracy and easy implementation of the new technique. The results of numerical experiments are compared with the analytical solution, finite difference (FD) method and some published methods to confirm the accuracy and efficiency of the new scheme presented in this paper.     

Gaussian waves in the Fitzhugh-Nagumo equation demonstrate one role of the auxiliary function H(x, t) in the homotopy analysis method

We apply the method of homotopy analysis to study the Fitzhugh-Nagumo equation.

u_t=u_xx+u(u-α)(1-u),