Some recent advances in validated methods for IVPs for ODEs

Compared to standard numerical methods for initial value problems (IVPs) for ordinary differential equations (ODEs), validated methods (often called interval methods) for IVPs for ODEs have two important advantages: if they return a solution to a problem, then (1) the problem is guaranteed to have a unique solution, and (2) an enclosure of the true solution is produced.We present a brief overview of interval Taylor series (ITS) methods for IVPs for ODEs and discuss some recent advances in the theory of validated methods for IVPs for ODEs. In particular, we discuss an interval Hermite-Obreschkoff (IHO) scheme for computing rigorous bounds on the solution of an IVP for an ODE, the stability of ITS and IHO methods, and a new perspective on the wrapping effect, where we interpret the problem of reducing the wrapping effect as one of finding a more stable scheme for advancing the solution.

[1]  Eldon Hansen,et al.  Topics in Interval Analysis , 1969 .

[2]  Nedialko S. Nedialkov,et al.  A New Perspective on the Wrapping Effect in Interval Methods for Initial Value Problems for Ordinary Differential Equations , 2001, Perspectives on Enclosure Methods.

[3]  L. Pastur,et al.  On the spectral and wave propagation properties of the surface Maryland model , 2003 .

[4]  P. Hartman.  Ordinary Differential Equations , 1965 .

[5]  Gerhard Wanner,et al.  On the integration of stiff differential equations , 1977 .

[6]  G. Alefeld,et al.  Introduction to Interval Computation , 1983 .

[7]  Nedialko S. Nedialkov,et al.  Validated solutions of initial value problems for ordinary differential equations , 1999, Appl. Math. Comput..

[8]  David S. Watkins,et al.  Understanding the $QR$ Algorithm , 1982 .

[9]  Richard S. Varga,et al.  DISCRETIZATION ERRORS FOR WELL-SET CAUCHY PROBLEMS.I., , 1965 .

[10]  R. Milner.  Mathematical Centre Tracts , 1976 .

[11]  N. Nedialkov,et al.  Computing rigorous bounds on the solution of an initial value problem for an ordinary differential equation , 1999 .

[12]  R. Lohner.  Einschliessung der Lösung gewöhnlicher Anfangs- und Randwertaufgaben und Anwendungen , 1988 .

[13]  N. F. Stewart.  A heuristic to reduce the wrapping effect in the numerical solution ofx′=f(t,x) , 1971 .

[14]  Martin Berz,et al.  Verified Integration of ODEs and Flows Using Differential Algebraic Methods on High-Order Taylor Models , 1998, Reliab. Comput..

[15]  B. L. Ehle.  A-Stable Methods and Padé Approximations to the Exponential , 1973 .

[16]  John D. Pryce,et al.  An Effective High-Order Interval Method for Validating Existence and Uniqueness of the Solution of an IVP for an ODE , 2001, Reliab. Comput..

[17]  J. H. Wilkinson.  The algebraic eigenvalue problem , 1966 .

[18]  Nedialko S. Nedialkov,et al.  An Interval Hermite-Obreschkoff Method for Computing Rigorous Bounds on the Solution of an Initial Value Problem for an Ordinary Differential Equation , 1998, SCAN.

[19]  J. Lambert.  Computational Methods in Ordinary Differential Equations , 1973 .

[20]  A. Ralston.  A first course in numerical analysis , 1965 .

[21]  H. Spreuer,et al.  On the Existence and the Verified Determination of Homoclinic and Heteroclinic Orbits of the Origin for the Lorenz Equations , 1993 .

[22]  J. G. F. Francis,et al.  The QR Transformation A Unitary Analogue to the LR Transformation - Part 1 , 1961, Comput. J..