Validated solutions of initial value problems for ordinary differential equations

Nedialkov, Nedialko S.; Jackson, Kenneth R.; Corliss, George F.

doi:10.1016/s0096-3003(98)10083-8

Cited by 325 publications

(331 citation statements)

References 20 publications

Supporting

Mentioning

325

Contrasting

Unclassified

Order By: Relevance

“…Complete details of the computation of T yj+1 are given by Lin and Stadtherr (2007). An implementation of this approach, called VSPODE (Verifying Solver for Parametric ODEs), has been developed and tested by Lin and Stadtherr (2007), who compared its performance with results obtained using the popular VNODE package (Nedialkov et al, 1999;Nedialkov et al, 2001). For the test problems used, VSPODE provided tighter enclosures on the state variables than VNODE, and required significantly less computation time.…”

Section: Solution Proceduresmentioning

confidence: 99%

“…Excellent reviews of interval methods for IVPs are available in the literature (Nedialkov, Jackson and Corliss, 1999;Neher, Jackson and Nedialkov, 2007). Much work has been done for the case in which the initial values are given by intervals, and there are several available software packages that deal with this case, including AWA (Lohner, 1992), VNODE (Nedialkov, Jackson and Pryce, 2001), and COSY VI (Berz and Makino, 1998).…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Verified Solution of Nonlinear Dynamic Models in Epidemiology

Enszer

Stadtherr

2010

Progress in Industrial Mathematics at ECMI 2008

View full text Add to dashboard Cite

Epidemiological models can be used to study the impact of an infection within a population. These models often involve parameters that are not known with certainty. Using a method for the verified solution of nonlinear dynamic models we can bound the disease trajectories that are possible for given bounds on the uncertain parameters. The method used is based on the use of an interval Taylor series to represent dependence on time and the use of Taylor models to represent dependence on uncertain parameters and/or initial conditions. The use of this method in epidemiology is demonstrated using the SIRS model, and other variations of Kermack-McKendrick models, including the case of time-dependent transmission.

show abstract

Section: Solution Proceduresmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Verified Solution of Nonlinear Dynamic Models in Epidemiology

Enszer

Stadtherr

2010

Progress in Industrial Mathematics at ECMI 2008

View full text Add to dashboard Cite

show abstract

“…In this paper, we use interval arithmetic (Section 2.1) for rigorous over-approximation. As a theory solver, we adopt a technique for hybrid systems proposed in our previous work [11] that integrates an interval-based method for nonlinear ODEs [13] and an interval-based constraint programming framework [8]. These interval-based methods guarantee that computed intervals or boxes enclose the solutions of a given problem.…”

Section: Introductionmentioning

confidence: 99%

An interval-based SAT modulo ODE solver for model checking nonlinear hybrid systems

Ishii

Ueda

Hosobe

2011

Int J Softw Tools Technol Transfer

View full text Add to dashboard Cite

Abstract. This paper presents a bounded model checking (BMC) tool called hydlogic for hybrid systems. It translates a reachability problem of a nonlinear hybrid system into a predicate logic formula involving arithmetic constraints, and checks the satisfiability of the formula based on a satisfiability modulo theories (SMT) method. We tightly integrate (i) an incremental SAT solver to enumerate the possible sets of constraints and (ii) an interval-based solver for hybrid constraint systems (HCSs) to solve the constraints described in the formulas. The HCS solver verifies the occurrence of a discrete change by enclosing continuous states that may cause the discrete change by a set of boxes. We adopt the existence property of a unique solution in the boxes computed by the HCS solver as (i) a proof of the reachability of a model, and (ii) a guide in the over-approximation refinement procedure. Our hydlogic implementation successfully handled several examples including those with nonlinear constraints.

show abstract

“…By construction, HBT(12)9 uses lower order derivatives than the traditional Taylor method of order 12, denoted by T12 [16]. Taylor methods have been an excellent choice in astronomical calculations [3] and sensitivity analysis of ODEs/DAEs [2], and in solving general problems [7] and validating solutions of ODEs/DAEs by means of interval analysis [14,17]. Deprit and Zahar [9] proved that recurrent power series in Taylor methods achieve high accuracy, with less computing time and larger stepsize than other methods.…”

Section: Introductionmentioning

confidence: 99%

Nine-stage multi-derivative Runge-Kutta method of order 12

Nguyen‐Ba

Bozic

Kengne

et al. 2009

Publ Inst Math (Belgr)

View full text Add to dashboard Cite

Abstract. A nine-stage multi-derivative Runge-Kutta method of order 12, called HBT(12)9, is constructed for solving nonstiff systems of first-order differential equations of the form y = f (x, y), y(x 0 ) = y 0 . The method uses y and higher derivatives y (2) to y (6) as in Taylor methods and is combined with a 9-stage Runge-Kutta method. Forcing an expansion of the numerical solution to agree with a Taylor expansion of the true solution leads to order conditions which are reorganized into Vandermonde-type linear systems whose solutions are the coefficients of the method. The stepsize is controlled by means of the derivatives y (3) to y (6) . The new method has a larger interval of absolute stability than Dormand-Prince's DP(8,7)13M and is superior to DP(8,7)13M and Taylor method of order 12 in solving several problems often used to test high-order ODE solvers on the basis of the number of steps, CPU time, maximum global error of position and energy. Numerical results show the benefits of adding high-order derivatives to Runge-Kutta methods.

show abstract

Validated solutions of initial value problems for ordinary differential equations

Cited by 325 publications

References 20 publications

Verified Solution of Nonlinear Dynamic Models in Epidemiology

Verified Solution of Nonlinear Dynamic Models in Epidemiology

An interval-based SAT modulo ODE solver for model checking nonlinear hybrid systems

Nine-stage multi-derivative Runge-Kutta method of order 12

Contact Info

Product

Resources

About