On the local error and the local truncation error of linear multistep methods

Two different measures of the local accuracy of a linear multistep method — the local error and the local trunction error — appear in the literature. It is shown that the principal parts of these errors are not identical for general linear multistep methods, but that they are so for a sub-class which contains all methods of Adams type. It is sometimes argued that local error is the more natural measure; this view is challenged.     

A note on stability of polynomial equations

Motivated by the notion of Ulam’s type stability and some recent results of S.-M. Jung, concerning the stability of zeros of polynomials, we prove a stability result for functional equations that have polynomial forms, considerably improving the results in the literature.     

Adaptive methods for piecewise polynomial collocation for ordinary differential equations

Various adaptive methods for the solution of ordinary differential boundary value problems using piecewise polynomial collocation are considered. Five different criteria are compared using both interval subdivision and mesh redistribution. The methods are all based on choosing sub-intervals so that the criterion values have (approximately) equal values in each sub-interval. In addition to the main comparison it is shown by example that at least when accuracy is low then equidistribution may not give a unique solution. The main results that using interval size times maximum residual as criterion gives very much better results than using maximum residual itself. It is also shown that a criterion based on a global error estimate while giving very good results in some cases, is unsatisfactory in other cases. The other criteria considered are that given by De Boor and the last Chebyshev series coefficient. AMS subject classification (2000)  65L10, 65L50, 65L60     

A note on fuzzy differential equations

We study the effect of the forcing term to the solution of a fuzzy differential equation.     

A posteriori error estimation for hp-version time-stepping methods for parabolic partial differential equations

The aim of this paper is to develop an hp-version a posteriori error analysis for the time discretization of parabolic problems by the continuous Galerkin (cG) and the discontinuous Galerkin (dG) time-stepping methods, respectively. The resulting error estimators are fully explicit with respect to the local time-steps and approximation orders. Their performance within an hp-adaptive refinement procedure is illustrated with a series of numerical experiments.     

A note on the arithmetic of differential equations

In this note we give a method for computing the differential Galois group of some linear second-order ordinary differential equations using arithmetic information, namely the p-curvatures.     

A posteriori error estimations for mixed finite-element approximations to the Navier-Stokes equations

A posteriori estimates for mixed finite element discretizations of the Navier-Stokes equations are derived. We show that the task of estimating the error in the evolutionary Navier-Stokes equations can be reduced to the estimation of the error in a steady Stokes problem. As a consequence, any available procedure to estimate the error in a Stokes problem can be used to estimate the error in the nonlinear evolutionary problem. A practical procedure to estimate the error based on the so-called postprocessed approximation is also considered. Both the semidiscrete (in space) and the fully discrete cases are analyzed. Some numerical experiments are provided.     

A note on the oscillatory property for nonlinear difference equations and differential equations

Criteria are given to determine the oscillatory property of solutions of the nonlinear difference equation: Δ^du_n + ∑_{i = 1}^mp_inf_i(u_n, Δu_n,…,Δ^{d ? 1}u_n) = 0, n = 0, 1, 2,…, where d is an arbitrary integer, generalizing results that have been obtained by B. Szmanda (J. Math. Anal. Appl.79 (1981), 90–95) for d = 2. Analogous results are given for the differential equation: u^(d) + ∑_{i = 1}^mp_i(t)f_i(u, u′,…, u^{(d ? 1)}) = 0, t ? t₀, which coincide with the criteria given by 2., 3., 599–602) and 4., 5., 6., 715–719) for the case m = 1.     

Computational bounds on polynomial differential equations

In this paper we study from a computational perspective some properties of the solutions of polynomial ordinary differential equations.We consider elementary (in the sense of Analysis) discrete-time dynamical systems satisfying certain criteria of robustness. We show that those systems can be simulated with elementary and robust continuous-time dynamical systems which can be expanded into fully polynomial ordinary differential equations in Q[π]. This sets a computational lower bound on polynomial ODEs since the former class is large enough to include the dynamics of arbitrary Turing machines.We also apply the previous methods to show that the problem of determining whether the maximal interval of definition of an initial-value problem defined with polynomial ODEs is bounded or not is in general undecidable, even if the parameters of the system are computable and comparable and if the degree of the corresponding polynomial is at most 56.Combined with earlier results on the computability of solutions of polynomial ODEs, one can conclude that there is from a computational point of view a close connection between these systems and Turing machines.     

On the polynomial limit cycles of polynomial differential equations

In this paper we deal with ordinary differential equations of the form dy/dx = P(x, y) where P(x, y) is a real polynomial in the variables x and y, of degree n in the variable y. If y = φ(x) is a solution of this equation defined for x ∈ [0, 1] and which satisfies φ(0) = φ(1), we say that it is a periodic orbit. A limit cycle is an isolated periodic orbit in the set of all periodic orbits. If φ(x) is a polynomial, then φ(x) is called a polynomial solution.     

A posteriori error bound methods for the inclusion of polynomial zeros

Using Carstensen's results from 1991 we state a theorem concerning the localization of polynomial zeros and derive two a posteriori error bound methods with the convergence order 3 and 4. These methods possess useful property of inclusion methods to produce disks containing all simple zeros of a polynomial. We establish computationally verifiable initial conditions that guarantee the convergence of these methods. Some computational aspects and the possibility of implementation on parallel computers are considered, including two numerical examples. A comparison of a posteriori error bound methods with the corresponding circular interval methods, regarding the computational costs and sizes of produced inclusion disks, were given.     

Two sided error bounds for discretisation methods in ordinary differential equations

A wide class of discretisation methods for ordinary differential equations is introduced and a new concept of consistency, called optimal consistency, is defined. This permits convergence of order exactlyp (that is, two sided error bounds) when the method is optimally consistent of orderp. This is then related to the minimal and optimal stability functionals introduced by Spijker and Albrecht, and a new algebraic criterion is given for a discretisation method consistent of orderp to be convergent of orderp + 1. Finally it is shown that the original motivation for the idea of optimal consistency arises from discretisation methods for Volterra integral equations.     

A note on the Gautschi-type method for oscillatory second-order differential equations

The Gautschi-type method has been proposed by Hochbruck and Lubich for oscillatory second-order differential equations. They conjecture that this method allows for a uniform error bound independent of the size of the system. The conjecture is proved in this note.     

Stability and error analysis of one-leg methods for nonlinear delay differential equations

This paper is concerned with the numerical solution of delay differential equations (DDEs). We focus on the stability behaviour and error analysis of one-leg methods with respect to nonlinear DDEs. The new concepts of GR-stability, GAR-stability and weak GAR-stability are introduced. It is proved that a strongly A-stable one-leg method with linear interpolation is GAR-stable, and that an A-stable one-leg method with linear interpolation is GR-stable, weakly GAR-stable and D-convergent of order s, if it is consistent of order s in the classical sense.     

A new Chebyshev polynomial approximation for solving delay differential equations

The purpose of this study is to give a Chebyshev polynomial approximation for the solution of mth-order linear delay differential equations with variable coefficients under the mixed conditions. For this purpose, a new Chebyshev collocation method is introduced. This method is based on taking the truncated Chebyshev expansion of the function in the delay differential equations. Hence, the resulting matrix equation can be solved, and the unknown Chebyshev coefficients can be found approximately. In addition, examples that illustrate the pertinent features of the method are presented, and the results of this investigation are discussed.     

Uniform truncation error for Shannon sampling expansion from local averages

Let Bp, 1≤p≤∞, be the set of all bounded functions in Lp(R) which can be extended to entire functions of exponential type . The uniform bounds for truncation error of Shannon sampling expansion from local averages are obtained for functions f∈Bp with the decay condition A|f(t)| ≤, t = 0,|tδ|where A and δ are positive constants. Furthermore we also establish similar results for non-bandlimit functions in Besov classes with the same decay condition as above.