Numerical Verification of Solutions of Nekrasov’s Integral Equation
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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.
AMS Subject Classifications:45G05 65G20 65R20
KeywordsNekrasov’s integral equation singular integral equation numerical verification
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