A (1 + 2)-Dimensional Simplified Keller–Segel Model: Lie Symmetry and Exact Solutions. II
A simplified Keller–Segel model is studied by means of Lie symmetry based approaches. It is shown that a (1 + 2)-dimensional Keller–Segel type system, together with the correctly-specified boundary and/or initial conditions, is invariant with respect to infinite-dimensional Lie algebras. A Lie symme...
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| Format: | Article |
| Language: | English |
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MDPI AG
2017-01-01
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| Series: | Symmetry |
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| Online Access: | http://www.mdpi.com/2073-8994/9/1/13 |
| _version_ | 1811300304803594240 |
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| author | Roman Cherniha Maksym Didovych |
| author_facet | Roman Cherniha Maksym Didovych |
| author_sort | Roman Cherniha |
| collection | DOAJ |
| description | A simplified Keller–Segel model is studied by means of Lie symmetry based approaches. It is shown that a (1 + 2)-dimensional Keller–Segel type system, together with the correctly-specified boundary and/or initial conditions, is invariant with respect to infinite-dimensional Lie algebras. A Lie symmetry classification of the Cauchy problem depending on the initial profile form is presented. The Lie symmetries obtained are used for reduction of the Cauchy problem to that of (1 + 1)-dimensional. Exact solutions of some (1 + 1)-dimensional problems are constructed. In particular, we have proved that the Cauchy problem for the (1 + 1)-dimensional simplified Keller–Segel system can be linearized and solved in an explicit form. Moreover, additional biologically motivated restrictions were established in order to obtain a unique solution. The Lie symmetry classification of the (1 + 2)-dimensional Neumann problem for the simplified Keller–Segel system is derived. Because Lie symmetry of boundary-value problems depends essentially on geometry of the domain, which the problem is formulated for, all realistic (from applicability point of view) domains were examined. Reduction of the the Neumann problem on a strip is derived using the symmetries obtained. As a result, an exact solution of a nonlinear two-dimensional Neumann problem on a finite interval was found. |
| first_indexed | 2024-04-13T06:49:07Z |
| format | Article |
| id | doaj.art-1677a6f8b7b843009ce3ee82f2bdfdc9 |
| institution | Directory Open Access Journal |
| issn | 2073-8994 |
| language | English |
| last_indexed | 2024-04-13T06:49:07Z |
| publishDate | 2017-01-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Symmetry |
| spelling | doaj.art-1677a6f8b7b843009ce3ee82f2bdfdc92022-12-22T02:57:28ZengMDPI AGSymmetry2073-89942017-01-01911310.3390/sym9010013sym9010013A (1 + 2)-Dimensional Simplified Keller–Segel Model: Lie Symmetry and Exact Solutions. IIRoman Cherniha0Maksym Didovych1Institute of Mathematics, National Academy of Sciences of Ukraine, 3 Tereshchenkivs’ka Street, Kyiv 01004, UkraineInstitute of Mathematics, National Academy of Sciences of Ukraine, 3 Tereshchenkivs’ka Street, Kyiv 01004, UkraineA simplified Keller–Segel model is studied by means of Lie symmetry based approaches. It is shown that a (1 + 2)-dimensional Keller–Segel type system, together with the correctly-specified boundary and/or initial conditions, is invariant with respect to infinite-dimensional Lie algebras. A Lie symmetry classification of the Cauchy problem depending on the initial profile form is presented. The Lie symmetries obtained are used for reduction of the Cauchy problem to that of (1 + 1)-dimensional. Exact solutions of some (1 + 1)-dimensional problems are constructed. In particular, we have proved that the Cauchy problem for the (1 + 1)-dimensional simplified Keller–Segel system can be linearized and solved in an explicit form. Moreover, additional biologically motivated restrictions were established in order to obtain a unique solution. The Lie symmetry classification of the (1 + 2)-dimensional Neumann problem for the simplified Keller–Segel system is derived. Because Lie symmetry of boundary-value problems depends essentially on geometry of the domain, which the problem is formulated for, all realistic (from applicability point of view) domains were examined. Reduction of the the Neumann problem on a strip is derived using the symmetries obtained. As a result, an exact solution of a nonlinear two-dimensional Neumann problem on a finite interval was found.http://www.mdpi.com/2073-8994/9/1/13KeywordsLie symmetryalgebra of invariancenonlinear boundary-value problemKeller–Segel modelCauchy problemexact solution |
| spellingShingle | Roman Cherniha Maksym Didovych A (1 + 2)-Dimensional Simplified Keller–Segel Model: Lie Symmetry and Exact Solutions. II Symmetry Keywords Lie symmetry algebra of invariance nonlinear boundary-value problem Keller–Segel model Cauchy problem exact solution |
| title | A (1 + 2)-Dimensional Simplified Keller–Segel Model: Lie Symmetry and Exact Solutions. II |
| title_full | A (1 + 2)-Dimensional Simplified Keller–Segel Model: Lie Symmetry and Exact Solutions. II |
| title_fullStr | A (1 + 2)-Dimensional Simplified Keller–Segel Model: Lie Symmetry and Exact Solutions. II |
| title_full_unstemmed | A (1 + 2)-Dimensional Simplified Keller–Segel Model: Lie Symmetry and Exact Solutions. II |
| title_short | A (1 + 2)-Dimensional Simplified Keller–Segel Model: Lie Symmetry and Exact Solutions. II |
| title_sort | 1 2 dimensional simplified keller segel model lie symmetry and exact solutions ii |
| topic | Keywords Lie symmetry algebra of invariance nonlinear boundary-value problem Keller–Segel model Cauchy problem exact solution |
| url | http://www.mdpi.com/2073-8994/9/1/13 |
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