Thesis (M.S.)--Rochester Institute of Technology, 1994. / Typescript. Includes bibliographical references.

We consider a dynamical system that undergoes frequent random switches according to Markovian laws between different states and where the associated transition rates change with the position of the system. These systems are called random evolutions; in engineering they are also known as stochastic switching systems. Since these kinds of dynamical systems combine deterministic and stochastic features, they are used for modelling in a variety of fields including biology, economics and communication networks. However, to gather information on future states, it is useful to search for alternative descriptions of this system. In this thesis, we present and study a partial differential equation of Fokker-Planck type and a stochastic differential equation that both serve as approximations of a random evolution. Furthermore, we establish a link between the two differential equations and conclude our analysis on the approximations of the random evolution with a numerical case study. / Graduate / 0405 / henrikek@uvic.ca

Thesis (Ph. D.)--University of Washington, 2007. / Vita. Includes bibliographical references (p. 127-131).

Diss. Uppsala : Univ., 2004.

Thesis (Ph. D.)--University of Chicago, Committee of Neurobiology, December 1999. / Includes bibliographical references. Also available on the Internet.

Approved for public release, distribution is unlimited. / High-resolution combat simulations that model urban combat currently use computationally expensive algorithms to represent urban target acquisition at the entity level. While this may be suitable for small-scale urban combat scenarios, simulation run time can become unacceptably long for larger scenarios. Consequently, there is a need for models that can lend insight into target acquisition in urban terrain for largescale scenarios in an acceptable length of time. This research develops urban target acquisition models that can be substituted for existing physicsbased or computationally expensive combat simulation algorithms and result in faster simulation run time with an acceptable loss of aggregate simulation accuracy. Specifically, this research explores (1) the adaptability of probability of line of sight estimates to urban terrain; (2) how cumulative distribution functions can be used to model the outcomes when a set of sensors is employed against a set of targets; (3) the uses for Markov Chains and Event Graphs to model the transition of a target among acquisition states; and (4) how a system of differential equations may be used to model the aggregate flow of targets from one acquisition state to another. / Captain, United States Marine Corps

The present dissertation aims to discuss diagonalization of linear operators, so that we can explore these concepts in the solution of systems of ordinary differential equations and in the identification of conics. A linear operator in a vector space of finite dimension can be represented by a matrix. Since diagonal arrays are the simplest from the point of view of matrix operations, we will show under what conditions, given a linear operator it is possible to represent it by a diagonal matrix. Thus, this paper presents the process of operator diagonalization, introduces basic concepts about systems of ordinary differential equations and applications. / A presente dissertação tem como objetivo discorrer sobre diagonalização de operadores lineares, de modo que possamos explorar esses conceitos na solução de sistemas de equações diferenciais ordinárias e na identificação de cônicas. Um operador linear em um espaço vetorial de dimensão finita, pode ser representado por uma matriz. Sendo as matrizes diagonais as mais simples do ponto de vista das operações matriciais, mostraremos sob que condições, dado um operador linear é possível representá-lo por uma matriz diagonal. Dessa forma, este trabalho apresenta o processo de diagonalização de operadores, introduz conceitos básicos sobre sistemas de equações diferenciais ordinárias e aplicações. / São Cristóvão, SE

The solving of stiff systems is still a contemporary sophisticated problem. The basic problem is the absence of precise definition of stiff systems. A question is also how to detect the stiffness in a given system of differential equations. Implicit numerical methods are commonly used for solving stiff systems. The stability domains of these methods are relatively large but the order of them is low.   The thesis deals with numerical solution of ordinary differential equations, especially numerical calculations using Taylor series methods. The source of stiffness is analyzed and the possibility how to reduce stiffness             in systems of ordinary differential equations (ODEs) is introduced. The possibility of detection stiff systems using explicit Taylor series terms is analyzed. The stability domains of explicit and implicit Taylor series are presented. The solutions of stiff systems using implicit Taylor series method are presented in many examples. The multiple arithmetic must be used in many cases. The new suitable parallel algorithm based on implicit Taylor series method with recurrent calculation of Taylor series terms and Newton iteration method (ITMRN) is proposed.

Nous présentons dans cette thèse des théorèmes d’existence pour des systèmes d’équations différentielles non-linéaires d’ordre trois, pour des systèmes d’équa- tions et d’inclusions aux échelles de temps non-linéaires d’ordre un et pour des systèmes d’équations aux échelles de temps non-linéaires d’ordre deux sous cer- taines conditions aux limites. Dans le chapitre trois, nous introduirons une notion de tube-solution pour obtenir des théorèmes d’existence pour des systèmes d’équations différentielles du troisième ordre. Cette nouvelle notion généralise aux systèmes les notions de sous- et sur-solutions pour le problème aux limites de l’équation différentielle du troisième ordre étudiée dans [34]. Dans la dernière section de ce chapitre, nous traitons les systèmes d’ordre trois lorsque f est soumise à une condition de crois- sance de type Wintner-Nagumo. Pour admettre l’existence de solutions d’un tel système, nous aurons recours à la théorie des inclusions différentielles. Ce résultat d’existence généralise de diverses façons un théorème de Grossinho et Minhós [34]. Le chapitre suivant porte sur l’existence de solutions pour deux types de sys- tèmes d’équations aux échelles de temps du premier ordre. Les résultats d’exis- tence pour ces deux problèmes ont été obtenus grâce à des notions de tube-solution adaptées à ces systèmes. Le premier théorème généralise entre autre aux systèmes et à une échelle de temps quelconque, un résultat obtenu pour des équations aux différences finies par Mawhin et Bereanu [9]. Ce résultat permet également d’obte- nir l’existence de solutions pour de nouveaux systèmes dont on ne pouvait obtenir l’existence en utilisant le résultat de Dai et Tisdell [17]. Le deuxième théorème de ce chapitre généralise quant à lui, sous certaines conditions, des résultats de [60]. Le chapitre cinq aborde un nouveau théorème d’existence pour un système d’in- clusions aux échelles de temps du premier ordre. Selon nos recherches, aucun résultat avant celui-ci ne traitait de l’existence de solutions pour des systèmes d’inclusions de ce type. Ainsi, ce chapitre ouvre de nouvelles possibilités dans le domaine des inclusions aux échelles de temps. Notre résultat a été obtenu encore une fois à l’aide d’une hypothèse de tube-solution adaptée au problème. Au chapitre six, nous traitons l’existence de solutions pour des systèmes d’équations aux échelles de temps d’ordre deux. Le premier théorème d’existence que nous obtenons généralise les résultats de [36] étant donné que l’hypothèse que ces auteurs utilisent pour faire la majoration a priori est un cas particulier de notre hypothèse de tube-solution pour ce type de systèmes. Notons également que notre définition de tube-solution généralise aux systèmes les notions de sous- et sur-solutions introduites pour les équations d’ordre deux par [4] et [55]. Ainsi, nous généralisons également des résultats obtenus pour des équations aux échelles de temps d’ordre deux. Finalement, nous proposons un nouveau résultat d’exis- tence pour un système dont le membre droit des équations dépend de la ∆-dérivée de la fonction. / In this thesis, we present existence theorems for systems of third order nonli- near differential equations, for systems of first order nonlinear time scales equa- tions and inclusions and for systems of second order nonlinear time scales equa- tions under some boundary conditions. In chapter three, we introduce a concept of solution-tube to get existence theorems for systems of third order differential equations. This new definition generalizes to systems the notions of lower- and upper-solution to third order differential equations introduced in [34]. In the last part of this chapter, we study third order systems when the right member f sa- tisfies a Wintner-Nagumo growth condition. To obtain an existence result in this case, we use the theory of differential inclusions. This result generalizes in many ways a theorem due to Grossinho and Minhós [34]. The next chapter concerns the existence of solutions for two kind of systems of first order time scales equations. Existence results for these problems are obtained with new notions of solution-tube adapted to these systems. Our first theorem ge- neralizes to systems and to an arbitrary time scale a result for difference equations due to Mawhin and Bereanu [9]. Our result permits to deduce the existence of so- lutions for systems which could not be treated in a result of Dai and Tisdell [17]. The second theorem of this chapter generalizes under few conditions some results of [60]. The fifth chapter presents a new existence theorem for a system of first order time scales inclusions. As far as we know, there is no result in the littera- ture for this kind of system of inclusions. Therefore, this chapter opens new doors in the branch of time scales inclusions. Again, our new result is obtained with the introduction of an hypothesis of solution-tube adapted to the problem studied. In the last chapter, existence of solutions for systems of second order time scales equations are obtained. The first result of this chapter generalizes theo- rems of [36] since the hypothesis used by these authors to get a priori bounds for solutions is a particular case of our definition of solution-tube for this type of problems. Let us mention also that our notion of solution-tube generalizes to systems the definitions of lower- and upper-solution used for second order time scales equations by [4] and [55]. We also generalize to systems, results obtained for second order time scales equations. Finally, we conclude this chapter with a new existence result for systems of second order time scales equations with a right member depending on the ∆-derivative.

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