A posteriori error estimates for linear equations

Summary This paper describes upper and lowerp-norm error bounds for approximate solutions of the linear system of equationsAx=b. These bounds imply that the error is proportional to the quantity wherer is the residual andq is the conjugate index top. The constant of proportionality is larger than 1 and lies in a specified range. Similar results are obtained for approximations toA ^–1 and solutions of nonsingular linear equations on general spaces.Research was partially supported by NSF Grant DMS8901477     

A posteriori estimates of inverse operators for initial value problems in linear ordinary differential equations

We present constructive a posteriori estimates of inverse operators for initial value problems in linear ordinary differential equations (ODEs) on a bounded interval. Here, “constructive” indicates that we can obtain bounds of the operator norm in which all constants are explicitly given or are represented in a numerically computable form. In general, it is difficult to estimate these inverse operators a priori. We, therefore, propose a technique for obtaining a posteriori estimates by using Galerkin approximation of inverse operators. This type of estimation will play an important role in the numerical verification of solutions for initial value problems in nonlinear ODEs as well as for parabolic initial boundary value problems.     

Piecewise polynomial collocation for linear boundary value problems of fractional differential equations

We consider a class of boundary value problems for linear multi-term fractional differential equations which involve Caputo-type fractional derivatives. Using an integral equation reformulation of the boundary value problem, some regularity properties of the exact solution are derived. Based on these properties, the numerical solution of boundary value problems by piecewise polynomial collocation methods is discussed. In particular, we study the attainable order of convergence of proposed algorithms and show how the convergence rate depends on the choice of the grid and collocation points. Theoretical results are verified by two numerical examples.     

Integral equations and initial value problems for nonlinear differential equations of fractional order

We discuss the solvability of integral equations associated with initial value problems for a nonlinear differential equation of fractional order. The differential operator is the Caputo fractional derivative and the inhomogeneous term depends on the fractional derivative of lower orders. We obtain the existence of at least one solution for integral equations using the Leray–Schauder Nonlinear Alternative for several types of initial value problems. In addition, using the Banach contraction principle, we establish sufficient conditions for unique solutions. Our approach in obtaining integral equations is the “reduction” of the fractional order of the integro-differential equations based on certain semigroup properties of the Caputo operator.     

Stability of some boundary value methods for the solution of initial value problems

The stability properties of three particular boundary value methods (BVMs) for the solution of initial value problems are considered. Our attention is focused on the BVMs based on the midpoint rule, on the Simpson method and on an Adams method of order 3. We investigate their BV-stability regions by considering the scalar test problem and constant stepsize. The study of the conditioning of the coefficient matrix of the discrete problem is extended to the case of variable stepsize and block ODE problems. We also analyse an appropriate choice for the stepsize for stiff problems. Numerical tests are reported to evidentiate the effectiveness of the BVMs and the differences among the BVMs considered.Work supported by the Ministero della Ricerca Scientifica, 40% project, and C.N.R. (contract of research # 92.00535.01).     

On dynamic iteration for delay differential equations

In this paper we study dynamic iteration techniques for systems of nonlinear delay differential equations. After pointing out a close connection to the truncated infinite embedding, as proposed by Feldstein, Iserles, and Levin, we give a proof of the superlinear convergence of the simple dynamic iteration scheme. Then we propose a more general scheme that in addition allows for a decoupling of the equations into disjoint subsystems, just like what we are used to from dynamic iteration schemes for ODEs. This scheme is also shown to converge superlinearly.     

Limit behavior of solutions of ordinary linear differential equations

A classification of classes of equivalent linear differential equations with respect to -limit sets of their canonical representations is introduced. Some consequences of this classification with respect to the oscillatory behavior of solution spaces are presented.     

Diagonally implicit general linear methods for ordinary differential equations

We investigate some classes of general linear methods withs internal andr external approximations, with stage orderq and orderp, adjacent to the class withs=r=q=p considered by Butcher. We demonstrate that interesting methods exist also ifs+1=r=q, p=q orq+1,s=r+1=q, p=q orq+1, ands=r=q, p=q+1. Examples of such methods are constructed with stability function matching theA-acceptable generalized Padé approximations to the exponential function.The work of Z. Jackiewicz was partially supported by the National Science Foundation under grant NSF DMS-9208048.     

Point initial value problem for linear functional differential equations in Banach spaces

Summary The problem of existence and uniqueness of solutions defined on the whole real line and satisfying given initial point data for general abstract linear functional differential equations is considered. The equation is not assumed to be of the delay type. The essence of the method presented here consists in the representation of a solution in the form analogous to the variation of constants formula known for linear ordinary differential equations. It is shown that such an approach can be effectively applied to the problem of existence and uniqueness of solutions satisfying an exponential growth estimate, provided that the deviation of the argument is sufficiently small. The proofs are based on the Banach fixed point principle. Detailed comparison and discussion of the hypotheses ensuring the existence and uniqueness of solutions are presented.     

On then-th order iterative ordinary linear differential equations

Summary It is shown that a change of the basis in the solution space of a second order linear differential equation induces a covariant change in the solution space of the corresponding iterative equation. Also studied is the problem to what extent a solution of an iterative equation determines the equation.Dedicated to the memory of Alexander M. Ostrowski on the occasion of the 100th anniversary of his birth.     

On the stability of solutions of linear boundary value problems for a system of ordinary differential equations

Linear boundary value problems for a system of ordinary differential equations are considered. The stability of the solution with respect to small perturbations of coefficients and boundary values is investigated.     

Solving linear ordinary differential equations by exponentials of iterated commutators

Summary A sequence of transformations of a linear system of ordinary differential equations is investigated. It is shown that these transformations produce new systems which represent progressively smaller perturbations of the original set of equations.The transformations are implemented as a basis of a numerical method. Order, stability and error control of this method are analyzed. Numerical examples demonstrate the potential of this approach.     

A Lax equivalence theorem for stochastic differential equations

In this paper, a stochastic mean square version of Lax’s equivalence theorem for Hilbert space valued stochastic differential equations with additive and multiplicative noise is proved. Definitions for consistency, stability, and convergence in mean square of an approximation of a stochastic differential equation are given and it is shown that these notions imply similar results as those known for approximations of deterministic partial differential equations. Examples show that the assumptions made are met by standard approximations.     

Iterative solutions of nonlinear initial value differential equations in Chebyshev series using Lie series

Summary In this paper Lie series are presented in Chebyshev form and applied to the iterative solution of initial value problems in differential equations. The resulting method, though algebraically complicated, is of theoretical interest as a generalisation of Taylor series methods and iterative Chebyshev methods. The theory of the method is discussed and the solutions of some simple scalar equations are analysed to illustrate the behaviour of the process.     

Perturbation of the Drazin inverse for closed linear operators

We investigate the perturbation of the Drazin inverse of a closed linear operator recently introduced by second author and Tran, and derive explicit bounds for the perturbations under certain restrictions on the perturbing operators. We give applications to the solution of perturbed linear equations, to the asymptotic behaviour ofC ₀-semigroups of linear operators, and to perturbed differential equations. As a special case of our results we recover recent perturbation theorems of Wei and Wang.     

Convergence of linear multistep and one-leg methods for stiff nonlinear initial value problems

To prove convergence of numerical methods for stiff initial value problems, stability is needed but also estimates for the local errors which are not affected by stiffness. In this paper global error bounds are derived for one-leg and linear multistep methods applied to classes of arbitrarily stiff, nonlinear initial value problems. It will be shown that under suitable stability assumptions the multistep methods are convergent for stiff problems with the same order of convergence as for nonstiff problems, provided that the stepsize variation is sufficiently regular.     

Average value problems in ordinary differential equations

Using Schauder's fixed point theorem, with the help of an integral representation in ‘Sharp conditions for weighted 1-dimensional Poincaré inequalities’, Indiana Univ. Math. J., 49 (2000) 143-175, by Chua and Wheeden, we obtain existence and uniqueness theorems and ‘continuous dependence of average condition’ for average value problem:

y^′=F(x,y),     

Symmetries of distributions and quadrature of ordinary differential equations

We present a geometric exposition of S. Lie's and E. Cartan's theory of explicit integration of finite-type (in particular, ordinary) differential equations. Numerous examples of how this theory works are given. In one of these, we propose a method of hunting for particular solutions of partial differential equations via symmetry preserving overdetermination.     

Least squares completions for nonlinear differential algebraic equations

Summary A method has been proposed for numerically solving lower dimensional, nonlinear, higher index differential algebraic equations for which more classical methods such as backward differentiation or implicit Runge-Kutta may not be appropriate. This method is based on solving nonlinear DAE derivative arrays using nonlinear singular least squares methods. The theoretical foundations, generality, and limitations of this approach remain to be determined. This paper carefully examines several key aspects of this approach. The emphasis is on general results rather than specific results based on the structure of various applications.Research supported in part by the U.S. Army Research Office under DAALO3-89-D-0003 and the National Science Foundation under ECS-9012909 and DMS-9122745