Geodesic
Numerical solution of vector Sturm–Liouville problems with Dirichlet conditions and nonlinear dependence on the spectral parameter
Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki, Tome 57 (2017) no. 9, pp. 1503-1516

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A numerical-analytical iterative method is proposed for solving generalized self-adjoint regular vector Sturm-Liouville problems with Dirichlet boundary conditions. The method is based on eigenvalue (spectral) correction. The matrix coefficients of the equations are assumed to be nonlinear functions of the spectral parameter. For a relatively close initial approximation, the method is shown to have second-order convergence with respect to a small parameter. Test examples are considered, and the model problem of transverse vibrations of a hinged rod with a variable cross section is solved taking into account its rotational inertia. 
@article{ZVMMF_2017_57_9_a7,
     author = {L. D. Akulenko and A. A. Gavrikov and S. V. Nesterov},
     title = {Numerical solution of vector {Sturm{\textendash}Liouville} problems with {Dirichlet} conditions and nonlinear dependence on the spectral parameter},
     journal = {\v{Z}urnal vy\v{c}islitelʹnoj matematiki i matemati\v{c}eskoj fiziki},
     pages = {1503--1516},
     publisher = {mathdoc},
     volume = {57},
     number = {9},
     year = {2017},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZVMMF_2017_57_9_a7/}
}
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L. D. Akulenko; A. A. Gavrikov; S. V. Nesterov. Numerical solution of vector Sturm–Liouville problems with Dirichlet conditions and nonlinear dependence on the spectral parameter. Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki, Tome 57 (2017) no. 9, pp. 1503-1516. http://geodesic.mathdoc.fr/item/ZVMMF_2017_57_9_a7/