Quasireversibility methods for non-well-posed problems
$$displaylines{ u_t+Au = 0, quad 0<t<T cr u(T) =f} $$ with positive self-adjoint unbounded $A$ is known to be ill-posed. One approach to dealing with this has been the method of quasireversibility, where the operator is perturbed to obtain a well-posed problem which approximates the original p...
| Main Authors: | , |
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| Format: | Article |
| Language: | English |
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Texas State University
1994-11-01
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| Series: | Electronic Journal of Differential Equations |
| Subjects: | |
| Online Access: | http://ejde.math.txstate.edu/Volumes/1994/08/abstr.html |
| _version_ | 1819030179164454912 |
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| author | G. W. Clark S. F. Oppenheimer |
| author_facet | G. W. Clark S. F. Oppenheimer |
| author_sort | G. W. Clark |
| collection | DOAJ |
| description | $$displaylines{ u_t+Au = 0, quad 0<t<T cr u(T) =f} $$ with positive self-adjoint unbounded $A$ is known to be ill-posed. One approach to dealing with this has been the method of quasireversibility, where the operator is perturbed to obtain a well-posed problem which approximates the original problem. In this work, we will use a quasi-boundary-value method, where we perturb the final condition to form an approximate non-local problem depending on a small parameter $alpha$. We show that the approximate problems are well posed and that their solutions $u_alpha$ converge on $[0,T]$ if and only if the original problem has a classical solution. We obtain several other results, including some explicit convergence rates. |
| first_indexed | 2024-12-21T06:26:02Z |
| format | Article |
| id | doaj.art-b0de3c55fbbc4c81ad1b38cd0cf58124 |
| institution | Directory Open Access Journal |
| issn | 1072-6691 |
| language | English |
| last_indexed | 2024-12-21T06:26:02Z |
| publishDate | 1994-11-01 |
| publisher | Texas State University |
| record_format | Article |
| series | Electronic Journal of Differential Equations |
| spelling | doaj.art-b0de3c55fbbc4c81ad1b38cd0cf581242022-12-21T19:13:08ZengTexas State UniversityElectronic Journal of Differential Equations1072-66911994-11-0119940819Quasireversibility methods for non-well-posed problemsG. W. ClarkS. F. Oppenheimer$$displaylines{ u_t+Au = 0, quad 0<t<T cr u(T) =f} $$ with positive self-adjoint unbounded $A$ is known to be ill-posed. One approach to dealing with this has been the method of quasireversibility, where the operator is perturbed to obtain a well-posed problem which approximates the original problem. In this work, we will use a quasi-boundary-value method, where we perturb the final condition to form an approximate non-local problem depending on a small parameter $alpha$. We show that the approximate problems are well posed and that their solutions $u_alpha$ converge on $[0,T]$ if and only if the original problem has a classical solution. We obtain several other results, including some explicit convergence rates.http://ejde.math.txstate.edu/Volumes/1994/08/abstr.htmlQuasireversibilityFinal Value ProblemsIll-Posed Problems. |
| spellingShingle | G. W. Clark S. F. Oppenheimer Quasireversibility methods for non-well-posed problems Electronic Journal of Differential Equations Quasireversibility Final Value Problems Ill-Posed Problems. |
| title | Quasireversibility methods for non-well-posed problems |
| title_full | Quasireversibility methods for non-well-posed problems |
| title_fullStr | Quasireversibility methods for non-well-posed problems |
| title_full_unstemmed | Quasireversibility methods for non-well-posed problems |
| title_short | Quasireversibility methods for non-well-posed problems |
| title_sort | quasireversibility methods for non well posed problems |
| topic | Quasireversibility Final Value Problems Ill-Posed Problems. |
| url | http://ejde.math.txstate.edu/Volumes/1994/08/abstr.html |
| work_keys_str_mv | AT gwclark quasireversibilitymethodsfornonwellposedproblems AT sfoppenheimer quasireversibilitymethodsfornonwellposedproblems |