Open Access

ARTICLE

A Meshless Method for Nonlinear, Singular and Generalized Sturm-Liouville Problems

by ,

1 Science and Technology Center of Magnetism of Technical Objects. The National Academy of Science of Ukraine, Industrialnaya St.,19, 61106, Kharkov, Ukraine.

Computer Modeling in Engineering & Sciences 2008, 34(3), 227-252. https://doi.org/10.3970/cmes.2008.034.227

Abstract

A new numerical technique for solving generalized Sturm--Liouville problem d²w/dx² + q(x, λ )w = 0, b_l[ λ ,w(a)] = b_r[ λ ,w(b)] = 0 is presented. In is presented. In particular, we consider the problems when the coefficient q(x, λ) or the boundary conditions depend on the spectral parameter λ in an arbitrary nonlinear manner. The method presented is based on mathematically modelling of physical response of a system to excitation over a range of frequencies. The response amplitudes are then used to determine the eigenvalues. The same technique can be applied to a very wide class of the eigenproblems: the Sturm--Liouville problems, the Schrodinger equation, the non-classical non-linear Sturm--Liouville problems, periodic problems. The results of the numerical experiments justifying the method are presented.

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