2016/04/19 by Laura Murphy, Chandrasekhar Venkataraman, Murphy, Laura +3
Computer Science · Materials Science · #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #FOS: Mathematics #Nonlinear Dynamics and Pattern Formation #Numerical Analysis (math.NA) #Solidification and crystal growth phenomena
paper · pdf · doi:10.48550/arxiv.1604.05653
openalex publication_date 2016/04/19 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
In this article we present a computational framework for isolating spatial\npatterns arising in the steady states of reaction-diffusion systems. Such\nsystems have been used to model many different phenomena in areas such as\ndevelopmental and cancer biology, cell motility and material science. Often one\nis interested in identifying parameters which will lead to a particular\npattern. To attempt to answer this, we compute eigenpairs of the Laplacian on a\nvariety of domains and use linear stability analysis to determine parameter\nvalues for the system that will lead to spatially inhomogeneous steady states\nwhose patterns correspond to particular eigenfunctions. This method has\npreviously been used on domains and surfaces where the eigenvalues and\neigenfunctions are found analytically in closed form. Our contribution to this\nmethodology is that we numerically compute eigenpairs on arbitrary domains and\nsurfaces. Here we present various examples and demonstrate that mode isolation\nis straightforward especially for low eigenvalues. Additionally we see that if\ntwo or more eigenvalues are in a permissible range then the inhomogeneous\nsteady state can be a linear combination of the respective eigenfunctions.\nFinally we show an example which suggests that pattern formation is robust on\nsimilar surfaces in cases that the surface either has or does not have a\nboundary.\n