Far Eastern Mathematical Journal

To content of the issue


The Immersion Method for the Solution of the Sturm — Liouville Problem in the Matrix Statement


O. V. Alexandrova, O. S. Gromasheva, G. Yu. Kosolapkin

2012, issue 2, Ñ. 136–145


Abstract
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.

Keywords:
Sturm — Liouville problem, matrix Riccati equation, the immersion method

Download the article (PDF-file)

References

[1] S. Bramley , L. Dieci , R. D. Russell, “Numerical Solution of Eigenvalue Problems for Linear Boundary Value ODES”, Journal of Computational Physics, 94 (1991), 382–402.
[2] V. I. Kliatskin, Metod pogruzheniia v teorii rasprostranenii voln, Nauka, M., 1986, 284 s.
[3] V. I. Kliatskin, Stokhasticheskie uravneniia: teoriia i ee prilozheniia k akustike, gidrodinamike i radiofizike, t. 1, Osnovnye polozheniia, tochnye rezul'taty i asimptoticheskie priblizheniia, FIZMATLIT, M., 2008, 320 s.
[4] R. F. Pannatoni, “Coupled mode theory for irregular acoustic waveguides with loss”, Akust. Zhurn., 51:1 (2011), 41–55.
[5] V.N. Zyrianov, “Vtorichnye toroidal'nye vikhri Teilora nad vozmushcheniiami dna vo vrashchaiushcheisia zhidkosti”, DAN, 427:2 (2009), 192–198.
[6] V. A. Sadovnichii , Ia. T. Sultanaev , A. M. Akhtiamov, Obratnye zadachi Shturma — Liuvillia s neraspadaiushchimisia kraevymi usloviiami, Izd-vo Moskovskogo universiteta, M., 2009, 183 s.
[7] K. V. Koshel', “Chislennoe reshenie zadachi troposfernogo rasprostraneniia korotkikh radiovoln dlia konechnogo poverkhnostnogo impedansa”, Radiotekh. i Elektr., 35:3 (1990), 647–649.
[8] Dzh. Kasti , R. Kalaba, Metod pogruzheniia v prikladnoi matematike, Mir, M., 1973
[9] V. I. Goland, K. V. Koshel', “Chislennoe reshenie v zadache zagorizontnogo rasprostraneniia ul'tra-korotkikh radiovoln dlia konechnogo poverkhnostnogo impedansa”, Radiotekh. i Elektr., 35:9 (1990), 1805–1809.
[10] V. I. Goland, V. I. Kliatskin, “Asimptoticheskii metod analiza stokhasticheskoi zadachi Shturma — Liuvillia. Metod pogruzheniia v teorii rasprostranenii voln”, Akust. Zhurn., 35:5 (1989), 942–944.
[11] R. Bellman, G. M.P. Wing, “An Introduction to Invariant Imbedding”, Classics in Applied Mathematics., 1992, ¹ 8, SIAM, Philadelphia.
[12] A. I. Egorov, Uravnenie Rikkati, FIZMATLIT, M., 2001, 328 s.[13] M. I. Zelikin, “K teorii matrichnogo uravneniia Rikkati”, Matem. sb., 182:7 (1991), 970–984.
[14] M. Kh. Zakhar-Itkin, “Matrichnoe differentsial'noe uravnenie Rikkati i polugruppa drobno-lineinykh preobrazovanii”, UMN, 28:3(171) (1973), 83–120.
[15] I. O. Iaroshchuk, “O chislennom modelirovanii odnomernykh stokhasticheskikh volnovykh zadach”, Zh. vychisl. matem. i matem. fiz., 24:11 (1984), 1748–-1751.
[16] F. R. Gantmakher, Teoriia matrits, FIZMATLIT, M., 2004, 560 s.

To content of the issue