Numerical analysis of the chaotic bahavior of the free boundary

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In the term of the project (three years) many results are obtained. Important results are shown as follows.1. Development of numerical methods for analysis of chaotic phenomena.The group of Tokushima Univ. developed a method for analysis of chaotic phenomena in free boundary problems.It is based on numerical methods, so it is general. Moreover, it has a surprising feature that attractors in the infinite-dimensional solution space can be approximated arbitrarily. This means the method connects theoretical analysis and numerical analysis, therefore it enhances development of theoretical and numerical researches.2. A proposal of a free boundary problem with attractors.The group of Tokushima Univ. proposed a one-dimensional free boundary problem with some parameters. It has exact solutions for special values of the parameters, so it is covenient for the check of the method and programs. Various attractors are found numerically.3. Analysis of the pattern.Nishiura solved self-replicating dynamics of the dissipative system. He also revealed the minimizer of the system which describes the pattern formation of the diblock copolymer has the fine structure with the meso-scale. Sakaguchi proposed a model to formation of colony patterns by a bacterial cell population. From numerical simulation various patterns are observed in spite of the simplicity of the model.4. Development of fast numerical computation and its application.The group of Tokushima Univ., Ikeda and Okamoto developed methods for fast numerical computation in the environment of parallel computing offered by PVM. These methods enabled large-scaled numerical simulation of free boundary problems or fluid mechanics. Some new phenomena were found.