Research of Computer aided Nonlinear Analysis with Flexibility

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We had the purpose to establish fundamental theories and to constitute the elements of computer aided mathematical analyzing software. We developed our research as we planned as the following :(1) We proposed the fixed point theorem for fuzzy map which is obtained by modeling the system with uncertain property.(2) We make the theory to prove numerically the existence of solution for nonlinear operator equations.(3) Based on C++ and an object oriented language in which rational number arithmetic is implemented, we constituted 3 prototypes of object oriented software which can deal with various objects corresponding to interval analysis, automatic differentiation, function representation and so on.(4) By extending the methods to prove numerically the existence of bifurcation point, we developed the theory to cancel singular points. We also applied our theory to various types of bifurcation phenomena and indicated that we can prove the existence of actual bifurcation phenomena.(5) We prop … More osed the theory to prove the existence of homoclinic orbits or heteroclinic orbits and prove their existence for actual examples.(6) We proposed an algorithm to prove the existence of all solutions in a bounded region for finite dimensional nonlinear equations and proved that this algorithm stops within finite steps under the certain conditions.(7) We proposed a method to prove the existence of all solutions with high speed in a bounded region for finite dimensional nonlinear equations with separability, whose example is VLSI circuit. (8) We could change the speed of calculation by the accuracy. Concretely, We proposed the method in which we can obtain the calculated results with super high speed when we demand its low accuracy and in which we can obtain the results with high speed even when we demand its high accuracy.(9) We realized the obtained techniques of numerical method with guaranteed accuracy on our prototypes of the software. We applied our software to various nonlinear functional equations and indicated its usefulness.(10) We combined the numerical method in the case that we demand its low accuracy and the numerical one in the case that we demand its high accuracy, by which we proposed the numerical method with high speed at our request of accuracy. We realized this method on our software and indicated its usefulness by the example of nonlinear circuit systems.(11) We integrated the above organized investigations and remade a prototype of the software for computer aided nonlinear analysis which can correspond to the changes of the problem or the accuracy. We also indicated its usefulness by applying it to nonlinear circuit problems. Less