EQUATIONS IN DIFFERENTIALS

Author: ESTELA CARBONELL M. ROSA.
Year: 2005.
University: POLITÉCNICA DE CATALUÑA [www.upc.edu].
Place of defense: ETSECCPB, Edifici C2, Campus Nord.
Place of preparation: U FACULTAT DE MATEMATIQUES I ESTADISTICA SUD.

Summary: This thesis presents applications Analysis Intervalar Modal Study of differential problems basically applied to solving problems on the field, preceded by a brief review of the basic theory of the system of manners and an exhaustive study of the partial optimality the roles sound, incorporating the concepts of optimality equivalent and optimality conditions, which represent an extension to the theory of Analysis Intervalar Modal existing one. Define intervals identification with the set of predicates to accept or reject predicates on the real line, a fact of course, which allows correct structural weaknesses and semantic Analysis Intervalar Classic, but mostly based on the theory intervalar the basic function of intervals as references to the system of real numbers consistent with the inevitable need for any truncation accompanying numerical value pilot. Revised the basic theory of intervalar modal analysis, we intend to apply it to solving problems on the field. Thus, to consider the resolution of even elementary problems, such as the propagation of heat in one dimension, we are faced with problems in raising the application of the theory intervalar owing to the restrictions imposed by the possibility of optimum calculations. This situation leads to the study of optimality subject that has been presented in the third chapter of the thesis. Admitting restrictions on the modalities of the arguments of rational functions are obtained new concepts such as the mode of departure or optimality side, which was finally allowed to introduce the concept of rational function syntactically c-conmutativa, giving a broader set of functions to which we can associate an optimal calculation. About all the intervals can define different operating systems such as getting the system modal intervals gifted their fundamental arithmetic or with a linear or parallel arithmetic. This last arithmetic is introduced in the fourth chapter of the thesis. From the standpoint of modal analysis intervalar we studied differences in equations defined as numerical solution to differential equations. The model intervalar and numerical calculation methods are objectively different: while the estimated trajectories unique numerical calculation approximate calculation intervalar estimated beam trajectories associated with a strategy determined by the patterns of intervals. In addition, the calculation is based on intervalar inclusiveness of the solutions intervalares and therefore gives rise to essentially accurate models from the point of view of the semantics associated with the inclusion, compared to the case of numerical calculation is based essentially on the concept of approximation. A property structurally Basic Analysis intervalar is that it is not appropriate algorithms take advantage of the numerical methods classics like algorithms intervalares, since the structure intervalar is essentially âmayor "that of the real numbers and we must therefore ask every problem intervalar always ab initio, within the context intervalar. Essentially this is determined by the fact that it makes no sense to raise the relations of inclusiveness in all the real numbers, reduced to the identity, and it makes no sense to dispense with them in the 8 context 3e8 or intervalar. Chapters 5, 6 and 7 are studying various problems posed by equations in differential intervalares, distinguishing situations that require a linear context and thus support interval arithmetic of brands. were also studied problems contour that arise in the numerical calculation classic.