COLLOCATION METHOD FOR NUMERICAL SOLUTION OF NONLINEAR SPECTRAL PROBLEMS FOR MULLER BOUNDARY INTEGRAL EQUATIONS

Spiridonov Aleksandr Olegovich, Postgraduate student, Kazan (Volga region) Federal University (18 Kremlevskaya street, Kazan, Russia), sasha_ens@mail.ru
Karchevskiy Evgeniy Mikhaylovich, Doctor of physical and mathematical sciences, professor, sub department of applied mathematics, Kazan (Volga region) Federal University (18 Kremlevskaya street, Kazan, Russia), sasha_ens@mail.ru
Nosich Aleksandr Iosifovich, Doctor of physical and mathematical sciences, professor, laboratory of micro and nanooptics, Institute of Radio Physics and Electronics of the National Academy of Sciences of Ukraine (12 Akademika Proskury street, Kharkov, Ukraine), sasha_ens@mail.ru

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Background. A number of spectral problems of the theory of optical waveguides and resonators is reduced to nonlinear problem of searching eigenvalues of Muller boundary integral equations. One of effective numerical methods of solution of such problems is the collocation method. The present work has the following aims: to realize the collocation method for searching surface and leaky own waves of a weakly guiding waveguide with piecewise constant dielectric permittivity; to theoretically prove the convergence of the given method. Numerical solution of the problem under investigation was previously carried out by the collocation method on the basis of integral equations, built by the method of simple fiber potential. Therefore, one of the aims of the work is to find out which method of integral equations construction is the most efficient from the practical point of view at numerical solution of the set problem: the method of Muller boundary integral equations or the method of simple fiber potential.
Materials and methods. The proving of the collocation method convergence relies on general results of the theory of discrete convergence of projection methods of nonlinear spectral problem solution and the theory of approximation of weakly singular integral equations. The comparative analysis of practical efficiency of the mehtod of Muller boundary integral equations and the method of simple fiber potential was carried out on the basis of numerical experiments of model problems solution.
Results. It is proved that if there exists a solution to the set problems, there exists a sequence of eigenvalues of the collocation method matrix, coverging as a number of points of collocation to the exact solution increases. On the other hand, if there exists a converging sequence of the above mentioned eigenvalues, it converges to the exact problem solution. Numerical experiments displayed the collocation method convergence and stability.
Conclusions. The collocation method is a theoretically substantiated method of the set problem solution with guaranteed convergence. However, discretization of integral equations with logarithmic peculiarity of kernels by the suggested variant of the collocation method (method of spline-collocations of zeroth order) is unefficient minor grid pitches. Besides, the method of simple fiber offers no advantage in counting time in comparison with the Muller boundary integral equations. Therefore, due to absolute equivalence of the system of Muller boundary integral equations to the initial differential problem it is more preferable to use the given equations for numerical solution of the latter.

dielectric waveguide, eigenvalue problem, integral equations, collocation method.

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