Modification of initial value problems solution methods for systems of ordinary differential equations with interval parameters
Mathematica modeling, numerical technique and program complexes
Moscow Aviation Institute (National Research University), 4, Volokolamskoe shosse, Moscow, А-80, GSP-3, 125993, Russia
Studying applied problems, we frequently encounter systems of ordinary differential equations that contain uncertainty or ambiguity of initial conditions or parameters. A natural demand for such problems is obtaining an outer interval enclosure of the solution by the known values of initial conditions or parameters. In particular, such problems arise in applied mechanics, thermodynamics and chemical kinetics.
There exist a number of Taylor series-based numerical methods for solving an IVP for ODE systems with interval parameters, such as Moore’s direct method, the parallelepiped method and Lohner’s QR-factorization method. All these methods represent a certain extension of the classical Taylor series method that takes into account the interval set up of the problem. Applying rigorous interval methods, which include the class of Taylor series methods, to more real-life systems often gives unsatisfactory results due to excessive lengths of the obtained intervals. This effect (excessive overestimation of intervals’ lengths) is usually called the wrapping effect or Moore’s effect.
The methods at hand are based on representing each integration step as some transformation that deforms the solution region using an interval Jacobian matrix. In the majority of cases, the wrapping effect appears due to the usage of this matrix, which implies deforming every volume element by all real Jacobian matrices in the region of interest. The suggested approach consists in computing the interval Jacobian matrix only in some discrete points of the region, which allows us to significantly reduce the wrapping effect at the cost of the fact that the obtained enclosure is now slightly less guaranteed to contain the solution. In this paper, we present the test results of applying the suggested method to model problems, which show its efficiency and give evidence of the fact that this approach can be used for finding solution to the problems in the class of interest.
Keywords:Taylor series-based interval methods, Moore’s method, parallelepiped method, Lohner's QR-factorization method, wrapping effect, Lotka-Volterra model, interval ODE systems
Afanas’eva M.N., Kuznecov E.B. Trudy MAI, 2016, no.88: http://www.mai.ru/science/trudy/published.php?ID=70713
Markus Neher, Interval Methods and Taylor Model Methods for ODEs, TM VII, Key West, KIT.
Lee H.J., Schiesser W.E. Ordinary and partial differential equation routines in C, C++, Fortran, Java, Maple, and MATLAB. London: Chapman and Hall/CRC, 2004
Moore R.E. Interval Analysis. Prentice-Hall, Englewood Cliffs, NJ (1966).
Dobronets B.S. Interval’naya matematika (Interval Mathematics), Krasnoyarsk, Izdatel’stvo 2007, 287p.
Shary S.P. Sharyi S.P. Konechnomernyi interval’nyi analiz (Finite-dimensional Interval Analysis), Novosibirsk, Institut vychislitel’noi tekhniki SO RAN, 2015, 609 p.
Shary S.P. Interval’nye metody dlya resheniya zadach global’nogo poiska (Interval Methods for Solving Global Search Problems), Novosibirsk, Institut vychislitel’noi tekhniki SO RAN, 2010, 33p.
Shokin Yu.I. Interval’nyi analiz (Interval Analysis), Novosibirsk, Nauka, 1981, 112p.
Pozin A.V. Review of Methods and Tools for Solving the Cauchy Problem for Ordinary Differential Equations with Guaranteed Estimated Error [Digital resource], — http://conf.nsc.ru/niknik-90/reportview/37500
Eijgenraam P. The Solution of Initial Value Problems using Interval Arithmetic. Math. Centre Tracts 144, Amsterdam (1981).
Lohner R.J. Enclosing the solutions of ordinary initial and boundary value problems. Computer Arithmetic: Scientific Computation and Programming Languages. 1987. P. 255–286.