In initial value problems for ODEs with interval-valued parameters and/or initial values, it is desirable in many applications to be able to determine a validated enclosure of all possible solutions to the ODE system. Much work has been done for the case in which initial values are given by intervals, and there are available software packages that deal with this case. However, less work has been done on the case in which parameters are given by intervals. We describe here a new method for obtaining validated solutions of initial value problems for ODEs with interval-valued parameters. The method also accounts for interval-valued initial values. The effectiveness of the method is demonstrated using several numerical examples involving parametric uncertainties.

Characterization of liquid-liquid equilibrium (LLE) in system containing ionic liquids (ILs) is important in evaluating ILs as candidates for replacing traditional extraction and separation solvents. Though an increasing amount of experimental LLE data is becoming available, comprehensive coverage of ternary liquid-phase behavior via experimental observation is impossible. Therefore, it is important to model the LLE of mixtures containing ILs.Experimental binary and ternary LLE data involving ILs can be correlated using standard excess Gibbs energy models. However, the predictive capability of these models in this context has not been widely studied. In this paper, we study the effectiveness with which excess Gibbs energy models can be used to predict ternary LLE solely from binary measurements. This is a stringent test of the suitability of various models for describing LLE in systems containing ILs. Three different excess Gibbs free energy models are evaluated: the NRTL, UNIQUAC and electrolyte-NRTL (eNRTL) models. In the case of eNRTL, a new formulation of the model is used, based on a symmetric reference state. To our knowledge, this is the first time that an electrolyte excess Gibbs energy model has been formulated for and applied to the modeling of multicomponent LLE for mixtures involving ILs. Ternary systems (IL, solvent, cosolvent) exhibiting experimental phase diagrams of various types have been chosen from the literature for comparison with the predictions. Comparisons of experimental and predicted octanol-water partition coefficients are also used to evaluate the models studied.

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A method based on interval analysis is presented for determining parameter values from mutual solubility data in two-parameter activity coefficient models for liquid-liquid equilibrium. The method is mathematically and computationally guaranteed to locate all sets of parameter values corresponding to stable phase equilibria. The technique is demonstrated with examples using the NRTL and electrolyte-NRTL (eNRTL) models. In two of the NRTL examples, results are found that contradict previous work. In the eNRTL examples, binary systems of an ionic liquid and an alcohol are considered. This appears to be the first time that a method for parameter estimation in the eNRTL model from binary LLE data (mutual solubility) has been presented.

A method is presented for deterministic global optimization in the estimation of parameters in models of dynamic systems. The method can be implemented as an -global algorithm, or, by use of the interval-Newton method, as an exact algorithm. In the latter case, the method provides a mathematically guaranteed and computationally validated global optimum in the goodness of fit function. A key feature of the method is the use of a new validated solver for parametric ODEs, which is used to produce guaranteed bounds on the solutions of dynamic systems with intervalvalued parameters, as well as on the first-and second-order sensitivities of the state variables with respect to the parameters. The computational efficiency of the method is demonstrated using several benchmark problems.