Numerical Verification of Solutions of Nekrasov’s Integral Equation |
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Authors: | S Murashige S Oishi |
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Affiliation: | (1) Department of Complexity Science and Engineering, The University of Tokyo, Tokyo 113-0033, Japan;(2) School of Science and Engineering, Waseda University, Tokyo 169-8555, Japan |
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Abstract: | This paper describes numerical verification of solutions of Nekrasov’s integral equation which is a mathematical model of two-dimensional water waves. This nonlinear and periodic integral equation includes a logarithmic singular kernel which is typically found in some two-dimensional potential problems. We propose the verification method using some properties of the singular integral for trigonometric polynomials and Schauder’s fixed point theorem in the periodic Sobolev space. A numerical example shows effectiveness of the present method. |
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Keywords: | 45G05 65G20 65R20 |
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