Exact solutions of the simplified Keller–Segel model

Cherniha, Roman; Didovych, Maksym

doi:10.1016/j.cnsns.2013.04.020

Cited by 11 publications

(9 citation statements)

References 13 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Using these algebras we find exact solutions for (1+1) and (1+2)-dimensional BVPs. This research is a natural continuation of the recent paper [9].…”

Section: Introductionmentioning

confidence: 98%

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

Didovych

2015

Symmetry

Self Cite

View full text Add to dashboard Cite

This research is a natural continuation of the recent paper "Exact solutions of the simplified Keller-Segel model" (Commun Nonlinear Sci Numer Simulat 2013, 18, 2960-2971. It is shown that a (1+2)-dimensional Keller-Segel type system is invariant with respect infinite-dimensional Lie algebra. All possible maximal algebras of invariance of the Neumann boundary value problems based on the Keller-Segel system in question were found. Lie symmetry operators are used for constructing exact solutions of some boundary value problems. Moreover, it is proved that the boundary value problem for the (1+1)-dimensional Keller-Segel system with specific boundary conditions can be linearized and solved in an explicit form.

show abstract

“…Using these algebras we find exact solutions for (1+1) and (1+2)-dimensional BVPs. This research is a natural continuation of the recent paper [9].…”

Section: Introductionmentioning

confidence: 98%

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

Didovych

2015

Symmetry

Self Cite

View full text Add to dashboard Cite

show abstract

“…whose properties have been studied in several papers. [8][9][10][11][12] In particular, in Cherniha and Didovych 9 and Didovych, 10 Lie symmetries have been found and applied for construction of exact solutions.…”

Section: Introductionmentioning

confidence: 99%

“…By assuming

V (u, v) \nabla v : = c u \nabla v,

we can get the following simplified Keller–Segel system:

\begin{matrix} eqnarrayleft center right \\ eqnarray-1 u_{t} & eqnarray-2 = u_{x x} + c u v_{x x} - c u_{x} v_{x}, \\ eqnarray-1 0 & eqnarray-2 = v_{x x} - β v + u, \end{matrix}

whose properties have been studied in several papers 8‐12 …”

Section: Introductionmentioning

confidence: 99%

“…In particular, in Cherniha and Didovych 9 and Didovych, 10 Lie symmetries have been found and applied for construction of exact solutions.…”

Section: Introductionmentioning

confidence: 99%

“…Here, taking into account other studies, 5,9,10 we restrict ourselves to the following set of the constitutive relations:

D_{u} = D_{v} = 1, V false(u, v false) = c false(N - u false), g false(u, v false) = g false(u false), h false(u, v false) = - f false(v false) + u,

where f = f ( v ) and g = g ( u ) are non‐negative smooth functions. So we are able to rewrite () as

\begin{matrix} eqnarrayleft center right \\ eqnarray-1 u_{t} & eqnarray-2 = \nabla^{2} u - c (\nabla \cdot ((N - u) \nabla v)) + g (u) \\ eqnarray-1 v_{t} & eqnarray-2 = \nabla^{2} v - f (v) + u, \end{matrix}

that in one space dimension reads:

\begin{matrix} right & left u_{t} = u_{x x} - c (N - u) v_{x x} + c u_{x} v_{x} + g (u) & right \\ right & left v_{t} = v_{x x} - f (v) + u . \end{matrix}

…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Symmetries and special solutions of a parabolic chemotaxis system

Bruzón

Gandarias

Torrisi

et al. 2020

Math Methods in App Sciences

View full text Add to dashboard Cite

show abstract

Similarity Solutions for Keller‐Segel model with fractional diffusion of cells

Ray

2020

Math Methods in App Sciences

View full text Add to dashboard Cite

In this paper, the similarity method has been used to solve fractional Keller-Segel model where the diffusion is represented by a nonlocal fractional Laplace operator. In order to verify the results, fractional centred difference method and weighted shifted Grüwald-Letnikov difference method have been employed for reduction equations. The comparison of results establishes the accuracy and efficiency of the proposed two numerical schemes.

show abstract

Exact solutions of the simplified Keller–Segel model

Cited by 11 publications

References 13 publications

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

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

Symmetries and special solutions of a parabolic chemotaxis system

Similarity Solutions for Keller‐Segel model with fractional diffusion of cells

Contact Info

Product

Resources

About