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Publication# Solving Optimization-Constrained Differential Equations With Discontinuity Points, With Application To Atmospheric Chemistry

Abstract

Ordinary differential equations are coupled with mixed constrained optimization problems when modeling the thermodynamic equilibrium of a system evolving with time. A particular application arises in the modeling of atmospheric particles. Discontinuity points are created by the activation/deactivation of inequality constraints. A numerical method for the solution of optimization-constrained differential equations is proposed by coupling an implicit Runge-Kutta method (RADAU5), with numerical techniques for the detection of the events (activation and deactivation of constraints). The computation of the events is based on dense output formulas, continuation techniques, and geometric arguments. Numerical results are presented for the simulation of the time-dependent equilibrium of organic atmospheric aerosol particles, and show the efficiency and accuracy of the approach.

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Numerical analysis

Numerical analysis is the study of algorithms that use numerical approximation (as opposed to symbolic manipulations) for the problems of mathematical analysis (as distinguished from discrete mathematics). It is the study of numerical methods that attempt at finding approximate solutions of problems rather than the exact ones. Numerical analysis finds application in all fields of engineering and the physical sciences, and in the 21st century also the life and social sciences, medicine, business and even the arts.

Ordinary differential equation

In mathematics, an ordinary differential equation (ODE) is a differential equation (DE) dependent on only a single independent variable. As with other DE, its unknown(s) consists of one (or more) function(s) and involves the derivatives of those functions. The term "ordinary" is used in contrast with partial differential equations which may be with respect to one independent variable. A linear differential equation is a differential equation that is defined by a linear polynomial in the unknown function and its derivatives, that is an equation of the form where a_0(x), .

Time

Time is the continued sequence of existence and events that occurs in an apparently irreversible succession from the past, through the present, into the future. It is a component quantity of various measurements used to sequence events, to compare the duration of events or the intervals between them, and to quantify rates of change of quantities in material reality or in the conscious experience. Time is often referred to as a fourth dimension, along with three spatial dimensions.

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