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Abstract:
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The analysis of the two-dimensional wave equation in a membrane with fixed boundary is a field which was studied by mathematicians for decades. Solving problems such as Mark Kac´s can one hear the shape of a drum? were open for as long as 30 years. With the use of powerful mathematical techniques, it was possible to answer this intriguing question. The purpose of this work was to introduce the problem posed Mark Kac and to explain how mathematicians have reached to the solution. The thesis was divided into five sections. A brief introduction explained the use of partial differential equations in biological, medical or engineering activities. The second part of the thesis focused on the two-dimensional wave equation, the theory behind the discrete theory of pure tones and the problem can one hear the shape of a drum? The third section is dedicated entirely to isospectrality, stating some practical problems with the use of MATLAB. One pair of isospectral drums is presented and a comparison of the results of different geometries is established. The geometries studied are: circular, curved, pentagonal and octagonal drums. |