Synthetical Researches on Applied Analysis and Computational Mathematics

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New topics of applied analysis and computational mathematics in the wide sense are investigated with the collaboration of investigators. Research achievements are presented, discussed and evaluated in the two joint symposium on applied mathematics. Also several workshops and international conferences on selected topics are organized and partly supported by this project. These activated research communications and stimulated young mathematicians.1. The research group of head investigator obtained the following results:(1)Consolidation of the method of validated numerical computation. It gives a computer aided proof of the existence of periodic solution for the autonomous system of ordinary differential equations which depends delicately on the value of parameters. In the proof a software of interval analysis by the computer is necessary to evaluate the round-off errors at the each step of numerial computations. One software for the big computer Fujitsu M-1800 and one for Sun Workstations are completed and used in the researches.(2)Waves on the free surface of viscous fluid flowing sown on an inclined plane. A gloval in time existence theorem is obtained under appropriate conditions. Spectral analysis for the linearized system will be our next project which will require an improvement of the method of the above computer aided proof.2. Co-investigators obtained the following results:(1)Accurate error estimates for the solution of finite element method of evolution systems.(H.Fujita)(2)Analysis of structure change of the equilibrium solution over the wide range of parameter for reaction diffusion sytems by the reduction method of dimension.(M.Mimura)(3)Solution of boundary value problems by the double exponential transformation using Sine approximation.(M.Mori)(4)Approximation of a function by a generating function of a chaotic mapping.(M.Yamaguti)(5)Improvement of convergent radins of simplified Newton method.(T.Yamamoto)