Numerical verification for nonlinear equations including a principal value integral

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It is needless to say that numerical calculation is required to solve mathematical models in the field of engineering and science. We try to obtain approximate solutions using numerical calculation. Then error estimate is very important, in particular for nonlinear problems. The main subject of this project is development of the numerical verification method which is rigorous estimation of errors included in numerical solutions using computers.This work considered the numerical verification method of solutions of nonlinear, periodic and singular integral equation which can be often found in the two-dimensional potential problems in elastic body theory, fluid mechanics, acoustics, electromagnetism, and so on. The basic idea is to transform an original equation into a fixed point form, to set a suitable function space, and to apply Schauder's fixed point theorem. The neighbourhood of approximate solutions is given by product of the finite dimensional part and the truncated part. The fixed point theorem guarantees that exact solutions exist in the neighbourhood under some conditions. As an example, Nekrasov's integral equation for water waves is investigated. It was found that solutions of this equation can be verified using the proposed method for the case of moderate wave height. When the wave height is large, this method did not work due to computational complexity. This problem remains as a future work.Also this work considered numerical verification of bifurcation points, in particular double turning points. The necessary and sufficient conditions for existence of the bifurcation points and the verification method using them are shown.