On Construction and Analysis of Finite Difference Schemes for Pseudoparabolic Problems with Nonlocal Boundary Conditions
In this paper the one- and two-dimensional pseudoparabolic equations with nonlocal boundary conditions are approximated by the Euler finite difference scheme. In the case of classical boundary conditions the stability of all schemes is investigated by the spectral method. Stability regions of finite...
| Main Authors: | , |
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| Format: | Article |
| Language: | English |
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Vilnius Gediminas Technical University
2014-04-01
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| Series: | Mathematical Modelling and Analysis |
| Subjects: | |
| Online Access: | https://journals.vgtu.lt/index.php/MMA/article/view/3290 |
| _version_ | 1818872457057009664 |
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| author | Raimondas Čiegis Natalija Tumanova |
| author_facet | Raimondas Čiegis Natalija Tumanova |
| author_sort | Raimondas Čiegis |
| collection | DOAJ |
| description | In this paper the one- and two-dimensional pseudoparabolic equations with nonlocal boundary conditions are approximated by the Euler finite difference scheme. In the case of classical boundary conditions the stability of all schemes is investigated by the spectral method. Stability regions of finite difference schemes approximating pseudoparabolic problem are compared with the stability regions of the classical discrete parabolic problem. These results are generalized for problems with nonlocal boundary conditions if a matrix of the finite difference scheme can be diagonalized. For the two-dimensional problem an efficient algorithm is constructed, which is based on the combination of the FFT method and the factorization algorithm. General stability results, known for the three level finite difference schemes, are applied to investigate the stability of some explicit approximations of the two-dimensional pseudoparabolic problem with classical boundary conditions. A connection between the energy method stability conditions and the spectrum Hurwitz stability criterion is shown. The obtained results can be applied for pseudoparabolic problems with nonlocal boundary conditions, if a matrix of the finite difference scheme can be diagonalized. |
| first_indexed | 2024-12-19T12:39:07Z |
| format | Article |
| id | doaj.art-3c7b40399ec841678c544972976ce8b5 |
| institution | Directory Open Access Journal |
| issn | 1392-6292 1648-3510 |
| language | English |
| last_indexed | 2024-12-19T12:39:07Z |
| publishDate | 2014-04-01 |
| publisher | Vilnius Gediminas Technical University |
| record_format | Article |
| series | Mathematical Modelling and Analysis |
| spelling | doaj.art-3c7b40399ec841678c544972976ce8b52022-12-21T20:21:02ZengVilnius Gediminas Technical UniversityMathematical Modelling and Analysis1392-62921648-35102014-04-0119210.3846/13926292.2014.910562On Construction and Analysis of Finite Difference Schemes for Pseudoparabolic Problems with Nonlocal Boundary ConditionsRaimondas Čiegis0Natalija Tumanova1Vilnius Gediminas Technical University Saulėtekio al. 11, LT-10223 Vilnius, LithuaniaVilnius Gediminas Technical University Saul_etekio al. 11, LT-10223 Vilnius, LithuaniaIn this paper the one- and two-dimensional pseudoparabolic equations with nonlocal boundary conditions are approximated by the Euler finite difference scheme. In the case of classical boundary conditions the stability of all schemes is investigated by the spectral method. Stability regions of finite difference schemes approximating pseudoparabolic problem are compared with the stability regions of the classical discrete parabolic problem. These results are generalized for problems with nonlocal boundary conditions if a matrix of the finite difference scheme can be diagonalized. For the two-dimensional problem an efficient algorithm is constructed, which is based on the combination of the FFT method and the factorization algorithm. General stability results, known for the three level finite difference schemes, are applied to investigate the stability of some explicit approximations of the two-dimensional pseudoparabolic problem with classical boundary conditions. A connection between the energy method stability conditions and the spectrum Hurwitz stability criterion is shown. The obtained results can be applied for pseudoparabolic problems with nonlocal boundary conditions, if a matrix of the finite difference scheme can be diagonalized.https://journals.vgtu.lt/index.php/MMA/article/view/3290pseudoparabolic problemsnonlocal boundary conditionsfinite difference methodstability |
| spellingShingle | Raimondas Čiegis Natalija Tumanova On Construction and Analysis of Finite Difference Schemes for Pseudoparabolic Problems with Nonlocal Boundary Conditions Mathematical Modelling and Analysis pseudoparabolic problems nonlocal boundary conditions finite difference method stability |
| title | On Construction and Analysis of Finite Difference Schemes for Pseudoparabolic Problems with Nonlocal Boundary Conditions |
| title_full | On Construction and Analysis of Finite Difference Schemes for Pseudoparabolic Problems with Nonlocal Boundary Conditions |
| title_fullStr | On Construction and Analysis of Finite Difference Schemes for Pseudoparabolic Problems with Nonlocal Boundary Conditions |
| title_full_unstemmed | On Construction and Analysis of Finite Difference Schemes for Pseudoparabolic Problems with Nonlocal Boundary Conditions |
| title_short | On Construction and Analysis of Finite Difference Schemes for Pseudoparabolic Problems with Nonlocal Boundary Conditions |
| title_sort | on construction and analysis of finite difference schemes for pseudoparabolic problems with nonlocal boundary conditions |
| topic | pseudoparabolic problems nonlocal boundary conditions finite difference method stability |
| url | https://journals.vgtu.lt/index.php/MMA/article/view/3290 |
| work_keys_str_mv | AT raimondasciegis onconstructionandanalysisoffinitedifferenceschemesforpseudoparabolicproblemswithnonlocalboundaryconditions AT natalijatumanova onconstructionandanalysisoffinitedifferenceschemesforpseudoparabolicproblemswithnonlocalboundaryconditions |