Validating an A Priori Enclosure Using High-Order Taylor Series

ValidatinganAPrioriEnclosureUsingHigh-OrderTaylorSeriesGeorgeF. CorlissandRob ert RihmAbstractWe use Taylor series plus an enclosure of the remainder term to validate theexistence of a uniquesolutionfor initialvalue problems in ordinarydi erentialequations and to compute a coarse enclosure of that solution.The signi cance ofthis result is its applicationto Lohner's AWA algorithm for validated solutions,not to the theory of ordinary di erential equations.By using high-order TaylorseriesinLohner's AlgorithmI, we are ableto validatethe solutionover muchlonger time steps than is done in the current AWA co de.For Lohner's enclosureby p olynomial s, the enclosures are exp ensive to compute, but it is easy to checkfor enclosure.For our enclosure by Taylor series, the enclosures are free b ecausethey are already b eing computed, but checking for enclosure requires 2np oly-nomialro ot ndings.Work iscontinuingon an implementationthat willallowdirect computational comparisons of the e ectiveness of the two metho ds.Keywords:ordinary di erential equations, Lohner's algorithm, validated computa-tion, Taylor series, enclosure metho ds.0Intro ductionThe AWA (Anfangswertaufgab e) program by Rudolf Lohner [4, 5, 6, 7, 8] computesan enclosure of the solution of an initial value problem (IVP) in ordinary di erentialequations (ODE)u0=f();t0) =(1)where we only know an interval enclosure [u0] of the vectorin general.Without lossof generality, Lohnerassumes thesystem is autonomous only to simplify the pro ofs.We assume thatfis at leastp1 times continuously di erentiablein a domainDwith [u0]DIRn,p2.Thenthereis auniqueatleasttimes continuouslydi erentiable solutionu(t) in a neighb orho o d of0.AWA is a single-step metho d.t each integration time step, AA applies two algo-rithms:Algorithm I:(Existenceandenclosure)Findastepsizehcoarseenclosureinterval [u0]Dsuch that fort2[0] := [; t+h], the solution() exists andsatis esu(t)2[0].