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Open AccessFeature PaperArticle

A (1 + 2)-Dimensional Simplified Keller–Segel Model: Lie Symmetry and Exact Solutions. II

Institute of Mathematics, National Academy of Sciences of Ukraine, 3 Tereshchenkivs’ka Street, Kyiv 01004, Ukraine
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Academic Editor: Hari M. Srivastava
Symmetry 2017, 9(1), 13; https://doi.org/10.3390/sym9010013
Received: 31 October 2016 / Revised: 27 December 2016 / Accepted: 10 January 2017 / Published: 20 January 2017
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. View Full-Text
Keywords: Keywords; Lie symmetry; algebra of invariance; nonlinear boundary-value problem; Keller–Segel model; Cauchy problem; exact solution Keywords; Lie symmetry; algebra of invariance; nonlinear boundary-value problem; Keller–Segel model; Cauchy problem; exact solution
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MDPI and ACS Style

Cherniha, R.; Didovych, M. A (1 + 2)-Dimensional Simplified Keller–Segel Model: Lie Symmetry and Exact Solutions. II. Symmetry 2017, 9, 13. https://doi.org/10.3390/sym9010013

AMA Style

Cherniha R, Didovych M. A (1 + 2)-Dimensional Simplified Keller–Segel Model: Lie Symmetry and Exact Solutions. II. Symmetry. 2017; 9(1):13. https://doi.org/10.3390/sym9010013

Chicago/Turabian Style

Cherniha, Roman; Didovych, Maksym. 2017. "A (1 + 2)-Dimensional Simplified Keller–Segel Model: Lie Symmetry and Exact Solutions. II" Symmetry 9, no. 1: 13. https://doi.org/10.3390/sym9010013

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