The rank of projection-algebraic representations of some differential operators
The Lie-algebraic method approximates differential operators that are formal polynomials of ${1,x,frac{d}{dx}}$ by linear operators acting on a finite dimensional space of polynomials. In this paper we prove that the rank of the $n$-dimensional representation of the operator $$K=a_k frac{d^k}{dx^k}+...
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| Format: | Article |
| Language: | deu |
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Ivan Franko National University of Lviv
2011-03-01
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| Series: | Математичні Студії |
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| Online Access: | http://matstud.org.ua/texts/2011/35_1/9-21.pdf |
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| author | O. Bihun M. Prytula |
| author_facet | O. Bihun M. Prytula |
| author_sort | O. Bihun |
| collection | DOAJ |
| description | The Lie-algebraic method approximates differential operators that are formal polynomials of ${1,x,frac{d}{dx}}$ by linear operators acting on a finite dimensional space of polynomials. In this paper we prove that the rank of the $n$-dimensional representation of the operator $$K=a_k frac{d^k}{dx^k}+a_{k+1}frac{d^{k+1}}{dx^{k+1}}+ldots +a_{k+p}frac{d^{k+p}}{dx^{k+p}}$$ is $n-k$ and conclude that the Lie-algebraic reductions of differential equations allow to approximate only {it some} of solutions of the differential equation $K[u]=f$. We show how to circumvent this obstacle when solving boundary value problems by making an appropriate change of variables. We generalize our results to the case of several dimensions and illustrate them with numerical tests. |
| format | Article |
| id | doaj-art-bd59d89fd2b045a498e44a3836d86c92 |
| institution | Kabale University |
| issn | 1027-4634 |
| language | deu |
| publishDate | 2011-03-01 |
| publisher | Ivan Franko National University of Lviv |
| record_format | Article |
| series | Математичні Студії |
| spelling | doaj-art-bd59d89fd2b045a498e44a3836d86c922025-08-20T03:39:19ZdeuIvan Franko National University of LvivМатематичні Студії1027-46342011-03-01351921The rank of projection-algebraic representations of some differential operatorsO. BihunM. PrytulaThe Lie-algebraic method approximates differential operators that are formal polynomials of ${1,x,frac{d}{dx}}$ by linear operators acting on a finite dimensional space of polynomials. In this paper we prove that the rank of the $n$-dimensional representation of the operator $$K=a_k frac{d^k}{dx^k}+a_{k+1}frac{d^{k+1}}{dx^{k+1}}+ldots +a_{k+p}frac{d^{k+p}}{dx^{k+p}}$$ is $n-k$ and conclude that the Lie-algebraic reductions of differential equations allow to approximate only {it some} of solutions of the differential equation $K[u]=f$. We show how to circumvent this obstacle when solving boundary value problems by making an appropriate change of variables. We generalize our results to the case of several dimensions and illustrate them with numerical tests.http://matstud.org.ua/texts/2011/35_1/9-21.pdfLie-algebraic methodboundary value problem |
| spellingShingle | O. Bihun M. Prytula The rank of projection-algebraic representations of some differential operators Математичні Студії Lie-algebraic method boundary value problem |
| title | The rank of projection-algebraic representations of some differential operators |
| title_full | The rank of projection-algebraic representations of some differential operators |
| title_fullStr | The rank of projection-algebraic representations of some differential operators |
| title_full_unstemmed | The rank of projection-algebraic representations of some differential operators |
| title_short | The rank of projection-algebraic representations of some differential operators |
| title_sort | rank of projection algebraic representations of some differential operators |
| topic | Lie-algebraic method boundary value problem |
| url | http://matstud.org.ua/texts/2011/35_1/9-21.pdf |
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