A posteriori estimates of inverse operators for initial value problems in linear ordinary differential equations
We present constructive a posteriori estimates of inverse operators for initial value problems in linear ordinary differential equations (ODEs) on a bounded interval. Here, “constructive” indicates that we can obtain bounds of the operator norm in which all constants are explicitly given or are repr...
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| Vydáno v: | Journal of computational and applied mathematics Ročník 236; číslo 6; s. 1622 - 1636 |
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| Hlavní autoři: | , , |
| Médium: | Journal Article |
| Jazyk: | angličtina |
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Elsevier B.V
15.10.2011
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| ISSN: | 0377-0427, 1879-1778 |
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| Abstract | We present constructive a posteriori estimates of inverse operators for initial value problems in linear ordinary differential equations (ODEs) on a bounded interval. Here, “constructive” indicates that we can obtain bounds of the operator norm in which all constants are explicitly given or are represented in a numerically computable form. In general, it is difficult to estimate these inverse operators a priori. We, therefore, propose a technique for obtaining a posteriori estimates by using Galerkin approximation of inverse operators. This type of estimation will play an important role in the numerical verification of solutions for initial value problems in nonlinear ODEs as well as for parabolic initial boundary value problems. |
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| AbstractList | We present constructive a posteriori estimates of inverse operators for initial value problems in linear ordinary differential equations (ODEs) on a bounded interval. Here, “constructive” indicates that we can obtain bounds of the operator norm in which all constants are explicitly given or are represented in a numerically computable form. In general, it is difficult to estimate these inverse operators a priori. We, therefore, propose a technique for obtaining a posteriori estimates by using Galerkin approximation of inverse operators. This type of estimation will play an important role in the numerical verification of solutions for initial value problems in nonlinear ODEs as well as for parabolic initial boundary value problems. |
| Author | Kinoshita, T. Kimura, T. Nakao, M.T. |
| Author_xml | – sequence: 1 givenname: T. surname: Kinoshita fullname: Kinoshita, T. email: kinosita@kurims.kyoto-u.ac.jp organization: Research Institute for Mathematical Sciences, Kyoto University, Kyoto 606-8502, Japan – sequence: 2 givenname: T. surname: Kimura fullname: Kimura, T. email: tkimura@post.cc.sasebo.ac.jp organization: Sasebo National College of Technology, Nagasaki 857-1193, Japan – sequence: 3 givenname: M.T. surname: Nakao fullname: Nakao, M.T. email: mtnakao@post.cc.sasebo.ac.jp organization: Sasebo National College of Technology, Nagasaki 857-1193, Japan |
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| Cites_doi | 10.1006/jath.1998.3172 10.5109/13484 |
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| Keywords | 65L05 Finite element method Linear ODEs 65L70 65L60 34A30 65G99 Constructive a posteriori estimates 34L99 |
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| References | Nakao, Yamamoto, Kimura (br000010) 1998; 93 Rump (br000020) 1999 Kimura, Yamamoto (br000015) 1999; 31 Mitsuhiro T. Nakao, Takehiko Kinoshita, Takuma Kimura, On a posteriori estimates of inverse operators for linear parabolic initial-boundary value problems, RIMS Preprints, RIMS-1729, (2011). Nakao (10.1016/j.cam.2011.09.026_br000010) 1998; 93 Kimura (10.1016/j.cam.2011.09.026_br000015) 1999; 31 10.1016/j.cam.2011.09.026_br000005 Rump (10.1016/j.cam.2011.09.026_br000020) 1999 |
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| SubjectTerms | Construction Constructive a posteriori estimates Differential equations Estimates Finite element method Initial value problems Inverse Linear ODEs Mathematical analysis Mathematical models Operators |
| Title | A posteriori estimates of inverse operators for initial value problems in linear ordinary differential equations |
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