Discrete and Continuous Models and Applied Computational ScienceDiscrete and Continuous Models and Applied Computational Science2658-46702658-7149Peoples' Friendship University of Russia named after Patrice Lumumba (RUDN University)13388Research ArticleAlgorithms and Programs for Solving Boundary-Value Problems for Systems of Second-Order ODEs with Piecewise Constant Potentials: Multichannel Scattering and Eigenvalue ProblemsGusevA Agooseff@jinr.ruChuluunbaatarOchuka@jinr.ruVinitskyS IRUDN University, Moscow, Russiavinitsky@theor.jinr.ruHaiL LBelgorod State University, Belgorod, Russiauonglehai_tcl@yahoo.com.vnDerbovV Lderbov@sgu.ruGόźdźAandrzej.gozdz@umcs.plJoint Institute for Nuclear ResearchSaratov State UniversityInstitute of Physics, University of M. Curie-Sklodowska150320163385217092016Copyright © 2016,2016The new algorithms and programs, implemented in Maple, for solving waveguide-type multichannel scattering and eigenvalue problems for systems of the second-order ODEs with N х N matrix piecewise constant coefficients on the axis are proposed. New algorithm and program for solving the boundary-value problems by method of matching the fundamental solutions (MMFS) of the system of ODEs at the points of discontinuity of potentials are elaborated. In each of the subintervals of an axis the general solution of the system of ODEs are sought in the form of linear combination of 2N fundamental solutions with unknown coefficients. Each fundamental solution explicitly dependent on spectral parameter and eigenvalues and eigenvectors of algebraic eigenvalue problems with N х N matrix of constant potentials. From the condition of continuity for the solutions and their derivatives at the discontinuity points of the potentials, the system of algebraic equations is followed. In the case of bound or metastable state problem the obtained system of algebraic equations contains nonlinear dependence of unknown spectral parameter. For solving such nonlinear problem symbolic-numerical algorithm is formulated. The benchmark calculations of bound, metastable and scattering states of BVPs for systems of the second-order ODEs obtained using program of the MMFS are compared with those obtained using program of the finite element method.multichannel scattering problemeigenvalue problemsystem of second order ordinary differential equationsmethod of matching the fundamental solutionsмногоканальная задача рассеяниязадача на собственные значениясистема ОДУ второго порядкаметодом сшивки фундаментальных решений[C.-C. Huang, C.-C. Huang, J.-Y. Yang. An Eﬃcient Method for Computing Optical Waveguides With Discontinuous Refractive Index Proﬁles Using Spectral Collocation Method With Domain Decomposition, Journal of Lightwave Technology 21 (10) (2003) 2284.][A.L. Sevastyanov, L.A. Sevastianov, A.A. Tyutyunnik. Analytical Calculations of Derivation Partial Diﬀerential Equations for Coeﬃcient Kantorovich Functions, Matem. Mod. 27 (2) (2015) 103.][M.N. Gevorkyan, D.S. Kulyabov, K.P. Lovetskiy, A.L. Sevastyanov, L.A. Sevastyanov. Waveguide Modes of a Planar Optical Waveguide, Mathematical Modelling and Geometry 3 (1) (2015) 43.][A.A. Gusev, L.L. Hai, O. Chuluunbaatar, S.I. Vinitsky. Program KANTBP 4M for Solving Boundary-Value Problems for Systems of Ordinary Diﬀerential Equations of the Second Order. URL http://wwwinfo.jinr.ru/programs/jinrlib/kantbp4m][A.A. Gusev, L.L. Hai, O. Chuluunbaatar, V. Ulziibayar, S.I. Vinitsky, V.L. Derbov, A. Gόźdź, V.A. Rostovtsev. Symbolic-Numeric Solution of Boundary-Value Problems for the Schr¨odinger Equation using the Finite Element Method: Scattering Problem and Resonance States, Lecture Notes in Computer Science 9301 (2015) 182.][Huang C.-C., Huang C.-C., Yang J.-Y. An Eﬃcient Method for Computing Optical Waveguides With Discontinuous Refractive Index Proﬁles Using Spectral Collocation Method With Domain Decomposition // Journal of Lightwave Technology. - 2003. - Vol. 21, No 10. - P. 2284.][Sevastyanov A.L., Sevastianov L.A., Tyutyunnik A.A. Analytical Calculations of Derivation Partial Diﬀerential Equations for Coeﬃcient Kantorovich Functions // Matem. Mod. - 2015. - Vol. 27, No 2. - P. 103.][Waveguide Modes of a Planar Optical Waveguide / M.N. Gevorkyan, D.S. Kulyabov, K.P. Lovetskiy, A.L. Sevastyanov, L.A. Sevastyanov // Mathematical Modelling and Geometry. - 2015. - Vol. 3, No 1. - P. 43.][Gusev A.A., Hai L.L., Chuluunbaatar O., Vinitsky S.I. Program KANTBP 4M for Solving Boundary-Value Problems for Systems of Ordinary Diﬀerential Equations of the Second Order. - http://wwwinfo.jinr.ru/programs/jinrlib/kantbp4m.][Symbolic-Numeric Solution of Boundary-Value Problems for the Schr¨odinger Equation using the Finite Element Method: Scattering Problem and Resonance States / A.A. Gusev, L.L. Hai, O. Chuluunbaatar, V. Ulziibayar, S. I. Vinitsky, V.L. Derbov, A. Gόźdź, V.A. Rostovtsev // Lecture Notes in Computer Science. - 2015. - Vol. 9301. - P. 182.]