Application of the Method of Summation Identities in Solving a Boundary-Value Problem for the Lame Equations
A boundary-value problem on an interval for a one-dimensional system of the Lame equations corresponding to a physical problem of propagation of an elastic wave through the gradient layer is considered. In this case, the coefficients of the equations are complex-valued continuous functions. The boun...
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Kazan Federal University
2016-03-01
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| Series: | Учёные записки Казанского университета: Серия Физико-математические науки |
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| Online Access: | http://kpfu.ru/portal/docs/F1317621846/158_1_phys_mat_2.pdf |
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| author | A.V. Anufrieva E.V. Rung D.N. Tumakov |
| author_facet | A.V. Anufrieva E.V. Rung D.N. Tumakov |
| author_sort | A.V. Anufrieva |
| collection | DOAJ |
| description | A boundary-value problem on an interval for a one-dimensional system of the Lame equations corresponding to a physical problem of propagation of an elastic wave through the gradient layer is considered. In this case, the coefficients of the equations are complex-valued continuous functions. The boundary conditions of the most general type corresponding to an additional condition, which, from the physical point of view, means absence of any surface waves at the working frequency, are considered. The concept of the generalized solution in the Sobolev space is formulated. Equivalence of the generalized and classical solutions is proven. A finite-difference scheme is constructed by the method of summation identities. For the case in which the equation coefficients and the desired functions are sufficiently smooth, it is shown that the error of approximation is of the order O(h2). |
| format | Article |
| id | doaj-art-c1b56d65e2184505bf31400ff243e45e |
| institution | Kabale University |
| issn | 2541-7746 2500-2198 |
| language | English |
| publishDate | 2016-03-01 |
| publisher | Kazan Federal University |
| record_format | Article |
| series | Учёные записки Казанского университета: Серия Физико-математические науки |
| spelling | doaj-art-c1b56d65e2184505bf31400ff243e45e2024-12-02T08:47:03ZengKazan Federal UniversityУчёные записки Казанского университета: Серия Физико-математические науки2541-77462500-21982016-03-0115812639Application of the Method of Summation Identities in Solving a Boundary-Value Problem for the Lame EquationsA.V. Anufrieva0E.V. Rung1D.N. Tumakov2Kazan Federal University, Kazan, 420008 RussiaKazan Federal University, Kazan, 420008 RussiaKazan Federal University, Kazan, 420008 RussiaA boundary-value problem on an interval for a one-dimensional system of the Lame equations corresponding to a physical problem of propagation of an elastic wave through the gradient layer is considered. In this case, the coefficients of the equations are complex-valued continuous functions. The boundary conditions of the most general type corresponding to an additional condition, which, from the physical point of view, means absence of any surface waves at the working frequency, are considered. The concept of the generalized solution in the Sobolev space is formulated. Equivalence of the generalized and classical solutions is proven. A finite-difference scheme is constructed by the method of summation identities. For the case in which the equation coefficients and the desired functions are sufficiently smooth, it is shown that the error of approximation is of the order O(h2).http://kpfu.ru/portal/docs/F1317621846/158_1_phys_mat_2.pdfboundary-value problemLame equationsgeneralized solutionmethod of summation identitiesdifference scheme |
| spellingShingle | A.V. Anufrieva E.V. Rung D.N. Tumakov Application of the Method of Summation Identities in Solving a Boundary-Value Problem for the Lame Equations Учёные записки Казанского университета: Серия Физико-математические науки boundary-value problem Lame equations generalized solution method of summation identities difference scheme |
| title | Application of the Method of Summation Identities in Solving a Boundary-Value Problem for the Lame Equations |
| title_full | Application of the Method of Summation Identities in Solving a Boundary-Value Problem for the Lame Equations |
| title_fullStr | Application of the Method of Summation Identities in Solving a Boundary-Value Problem for the Lame Equations |
| title_full_unstemmed | Application of the Method of Summation Identities in Solving a Boundary-Value Problem for the Lame Equations |
| title_short | Application of the Method of Summation Identities in Solving a Boundary-Value Problem for the Lame Equations |
| title_sort | application of the method of summation identities in solving a boundary value problem for the lame equations |
| topic | boundary-value problem Lame equations generalized solution method of summation identities difference scheme |
| url | http://kpfu.ru/portal/docs/F1317621846/158_1_phys_mat_2.pdf |
| work_keys_str_mv | AT avanufrieva applicationofthemethodofsummationidentitiesinsolvingaboundaryvalueproblemforthelameequations AT evrung applicationofthemethodofsummationidentitiesinsolvingaboundaryvalueproblemforthelameequations AT dntumakov applicationofthemethodofsummationidentitiesinsolvingaboundaryvalueproblemforthelameequations |