Publication Date:
2015-04-02
Description:
In this paper we use trigonometric approximation of a nonlinear Riemann–Hilbert problem, with bounded and closed restriction curves, on the annulus. We reformulate the boundary conditions for the holomorphic function as a system of nonlinear singular integral equations A ( w )= 0 , where A : H 1+ μ ( )colone H 1+ μ ( 1 ) x H 1+ μ ( 2 ) -〉 H 1+ μ ( ) is defined via a Nemytski operator and is a Fredholm quasiruled mapping. The operator A is approximated by operators A n : H 1+ μ ( ) -〉 H 1+ μ ( ), obtained by applying trigonometric collocation. Following the results of Šnirel’man (1972, Math. USSR-Sb. , 18 , 373–396), Efendiev (2009, AIMS Series on Differential Equations and Dynamical Systems, vol. 3. Springfield) and Efendiev & Wendland (1996, Nonlinear Anal. , 27 , 37–58; 2001, Topol. Methods Nonlinear Anal. , 17 , 111–124; 2003, Proc. Roy. Soc. London Ser. A , 459 , 945–955; 2007a, J. Math. Anal. Appl. , 329 , 383–391; 2007b, J. Math. Anal. Appl. , 329 , 425–444), we use their degree of mapping and show under additional assumptions the existence of trigonometric polynomial solutions of the semidiscrete equations A n ( w )= 0 , for n large enough. We discuss the solvability of the nonlinear collocation method and the convergence of the approximate solutions. We conclude the paper by giving a procedure for the construction of the solution, using Newton's method, and show a numerical example which converges exponentially.
Print ISSN:
0272-4979
Electronic ISSN:
1464-3642
Topics:
Mathematics
