Geodesic
The Immersion Method for the Solution of the Sturm --- Liouville Problem in the Matrix Statement
Dalʹnevostočnyj matematičeskij žurnal, Tome 12 (2012) no. 2, pp. 136-145

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In this paper we propose a method for solving boundary-value wave problems described by the matrix system of Helmholtz equations. It has been shown that the Sturm — Liouville problems with singular matrices in the boundary conditions are reduced to the forms with nondegenerate matrices by using algebraic ransformations. This allows to obtain matrix equations using the invariant imbedding method. The solution of the Sturm — Liouville problem is reduced to the solution of the Cauchy problem for the matrix Riccati equation. It has been shown that the solution of the matrix Riccati equation can be constructed for arbitrary boundary conditions chosen for reasons of convenience, and the solutions for given boundary conditions are expressed in terms of the reference solution of the matrix Riccati equation with algebraic transformations. An equation for the eigenvalues of the Sturm – Liouville problem has also been formulated. It is expressed through the solution of the matrix Riccati equation, and the evolution equation for the spectral parameter of Sturm — Liouville problem has been obtained. 
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     author = {O. V. Alexandrova and O. S. Gromasheva and G. Yu. Kosolapkin},
     title = {The {Immersion} {Method} for the {Solution} of the {Sturm} --- {Liouville} {Problem} in the {Matrix} {Statement}},
     journal = {Dalʹnevosto\v{c}nyj matemati\v{c}eskij \v{z}urnal},
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O. V. Alexandrova; O. S. Gromasheva; G. Yu. Kosolapkin. The Immersion Method for the Solution of the Sturm --- Liouville Problem in the Matrix Statement. Dalʹnevostočnyj matematičeskij žurnal, Tome 12 (2012) no. 2, pp. 136-145. http://geodesic.mathdoc.fr/item/DVMG_2012_12_2_a1/