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Concept# Ordinary differential equation

Summary

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.
Differential equations
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
:a_0(x)y +a_1(x)y' + a_2(x)y'' +\cdots +a_n(x)y^{(n)}+b(x)=0,
where a_0(x), ..., a_n(x) and b(x) are arbitrary differentiable functions that do not need to be linear, and y', \ldots, y^{(n)} are the successive derivatives of the unknown function y of the variable x.
Among ordinary differential equations, linear differential equations play a prominent rol

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Differential equation

In mathematics, a differential equation is an equation that relates one or more unknown functions and their derivatives. In applications, the functions generally represent physical quantities, the

Linear differential equation

In mathematics, 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

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 mathema

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Identification of kinetic models and estimation of reaction and mass-transfer parameters can be performed using the extent-based identification method, whereby each chemical/physical process is treated individually. This method is used here to analyze gas-liquid systems under unsteady-state mass transfer. Such a situation is common in the case of diffusion-controlled reactions and can be modeled by the film theory. In both the gas and liquid bulks, mass-balance relations describe the species dynamics as ordinary differential equations (ODE) and serve as boundary conditions for the film. On the other hand, the dynamic accumulation in the film is described by Fick's second lay. The resulting partial differential equation (PDE) system is solved by discretization and rearrangement into ODEs. Kinetic models are assessed and the corresponding parameters are estimated using extents of reaction. The estimation of diffusion coefficients follows a tow-steps procedure. First, the extents of mass transfer are computed from measurements in the two bulks. Diffusion coefficients are then estimated individually by fitting each extent of mass transfer to the extent obtained by solving the corresponding PDE. Comparison of the estimated diffusion coefficients with their literature values serves to validate the models identified in the two bulks. The estimation of both kinetic parameters and diffusion coefficients is investigated for gas-liquid reaction systems with unsteady-state diffusion. The approach is illustrated with simulated examples.

2013The modeling of a system composed by a gas phase and organic aerosol particles, and its numerical resolution are studied. The gas-aerosol system is modeled by ordinary differential equations coupled with a mixed-constrained optimization problem. This coupling induces discontinuities when inequality constraints are activated or deactivated. Two approaches for the solution of the optimization-constrained differential equations are presented. The first approach is a time splitting scheme together with a fixed-point method that alternates between the differential and optimization parts. The ordinary differential equations are approximated by the Crank-Nicolson scheme and a primal-dual interior-point method combined with a warm-start strategy is used to solve the minimization problem. The second approach considers the set of equations as a system of differential algebraic equations after replacing the minimization problem by its first order optimality conditions. An implicit 5th-order Runge-Kutta method (RADAU5) is then used. Both approaches are completed by numerical techniques for the detection and computation of the events (activation and deactivation of inequality constraints) when the system evolves in time. The computation of the events is based on continuation techniques and geometric arguments. Moreover the first approach completes the computation with extrapolation polynomials and sensitivity analysis, whereas the second approach uses dense output formulas. Numerical results for gas-aerosol system made of several chemical species are proposed for both approaches. These examples show the efficiency and accuracy of each method. They also indicate that the second approach is more efficient than the first one. Furthermore theoretical examples show that the method for the computation of the activation is of second order for the first approach and exact for the second one.

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The topic of this thesis is the study of several stochastic control problems motivated by sailing races. The goal is to minimize the travel time between two locations, by selecting the fastest route in face of randomly changing weather conditions, such as wind direction. When a sailboat is travelling upwind, the key is to decide when to tack. Since this maneuver slows down the yacht, it is natural to model this time lost by a "tacking penalty" which places the problem in the context of optimal stochastic control problems with switching costs. An objective of this work is to propose and to study mathematical models that capture some of the features of a sailing race, but which remain amenable to an explicit rigorous solution that can be proved to be optimal. We consider three different models in which the wind direction is described by a stochastic process. In the first model, we consider a wind that changes randomly only once. In the second model, the wind oscillates between two possible directions according to a continuous-time Markov chain. We exhibit a free boundary problem for the value function involving hyperbolic partial differential equations of Klein-Gordon type. The last model considers the wind direction as a Brownian motion. We prove the existence of a finite value function and exhibit a free boundary problem involving parabolic partial differential equations with non-constant coefficients. In these three models, the optimal solution consists of a partition of the state space into a region where it is optimal to tack immediately and a region where it is optimal to continue on the current tack. The boundaries between these regions are given by one or more "switching curves" and in the cases where we have been able to exhibit them, the optimality of the solution is established by a verification theorem based on the martingale method. We also solve two other control problems in which a player tries to minimize or maximize the exit time from an interval of a Brownian particle by controlling its drift and subject to a switching penalty. In each problem, the value function is written as the solution of a second order ordinary differential equations problem whose unknown boundaries are found by applying the principle of smooth fit. For both problems, we exhibit a candidate strategy as a function of the switching cost and we prove its optimality as well as its generic uniqueness.