Um ambiente computacional de aprendizagem para métodos de resolução de equações diferenciais parciais

Abstract

The analysis and solution of physical problems associated with partial differential equations of a subject in which students traditionally find many difficulties. Thus, to assist in the learning process of analytical and numerical methods for solving these equations, developed a computational learning environment by adopting an interdisciplinary approach involving symbolic computation, foundations of the education and advanced topics of numerical calculation . They used Myers (1971) as the basis of educational content and theory of meaningful learning of Ausubel to organize the structure of this content. Emphasizing aspects of theoretical description and mathematical detail, are presented in three solution methods learning environment for partial differential equations: the analytical method of separation of variables and numerical methods of finite differences in EULER variants and Crank- Nicolson. The environment was implemented in a symbolic computation system and therefore has a symbolic manipulation and graphical display capabilities. This makes it possible for students to develop and use the analytical solution obtained by variable separation method to interpret and analyze, through graphics and animations, the physical phenomenon. The environment also provides the student with the means to calculate the numerical solution interactively, using the finite difference methods of Euler and Crank- Nicolson, visualizing step-by-step function values ​​in the loop and comparing them with the numerical results calculated from the analytical solution. The learning environment also offers resources for student generate charts and graphs to analyze and compare the numerical results obtained by the three methods, adopting as benchmark the solution obtained by the variable separation method. Thus, the learner can access the information in interactive and dynamic character, providing the autonomous learning. As an example of application and in order to illustrate the study of physical-mathematical problems, adopted is the conduction heat transfer problem, which is used to study the process of cooling of electronic circuits. Thus, it is assumed that the heat diffusion in a bar epoxy, initially at a known temperature and cooled suddenly at their ends. The graphs and tables provided by the environment also allow the learner observe critically the bar cooling process, either by viewing the evolution of temperature profiles, or the cooling curve of a point in the spatial domain.