18 research outputs found
Numerical continuation for fractional PDEs: sharp teeth and bloated snakes
Partial differential equations (PDEs) involving fractional Laplace operators
have been increasingly used to model non-local diffusion processes and are
actively investigated using both analytical and numerical approaches. The
purpose of this work is to study the effects of the spectral fractional
Laplacian on the bifurcation structure of reaction-diffusion systems on bounded
domains. In order to do this we use advanced numerical continuation techniques
to compute the solution branches. Since current available continuation packages
only support systems involving the standard Laplacian, we first extend the
pde2path software to treat fractional PDEs. The new capabilities are then
applied to the study of the Allen-Cahn equation, the Swift-Hohenberg equation
and the Schnakenberg system (in which the standard Laplacian is each replaced
by the spectral fractional Laplacian). Our study reveals some common effects,
which contributes to a better understanding of fractional diffusion in generic
reaction-diffusion systems. In particular, we investigate the changes in
snaking bifurcation diagrams and also study the spatial structure of
non-trivial steady states upon variation of the order of the fractional
Laplacian. Our results show that the fractional order can induce very
significant qualitative and quantitative changes in global bifurcation
structures
pde2path - version 2.0: faster FEM, multi-parameter continuation, nonlinear boundary conditions, and periodic domains - a short manual
pdepath 2.0 is an upgrade of the continuation/bifurcation package pde2path
for elliptic systems of PDEs over bounded 2D domains, based on the Matlab
pdetoolbox. The new features include a more efficient use of FEM, easier
switching between different single parameter continuations, genuine
multi-parameter continuation (e.g., fold continuation), more efficient
implementation of nonlinear boundary conditions, cylinder and torus geometries
(i.e., periodic boundary conditions), and a general interface for adding
auxiliary equations like mass conservation or phase equations for continuation
of traveling waves. The package (library, demos, manuals) can be downloaded at
www.staff.uni-oldenburg.de/hannes.uecker/pde2pat
Pattern formation of a Schnakenberg-type plant root hair initiation model
This paper concentrates on the diversity of patterns in a quite general Schnakenberg-type model. We discuss existence and nonexistence of nonconstant positive steady state solutions as well as their bounds. By means of investigating Turing, steady state and Hopf bifurcations, pattern formation, including Turing patterns, nonconstant spatial patterns or time periodic orbits, is shown. Also, the global dynamics analysis is carried out
Continuation for thin film hydrodynamics and related scalar problems
This chapter illustrates how to apply continuation techniques in the analysis
of a particular class of nonlinear kinetic equations that describe the time
evolution through transport equations for a single scalar field like a
densities or interface profiles of various types. We first systematically
introduce these equations as gradient dynamics combining mass-conserving and
nonmass-conserving fluxes followed by a discussion of nonvariational amendmends
and a brief introduction to their analysis by numerical continuation. The
approach is first applied to a number of common examples of variational
equations, namely, Allen-Cahn- and Cahn-Hilliard-type equations including
certain thin-film equations for partially wetting liquids on homogeneous and
heterogeneous substrates as well as Swift-Hohenberg and Phase-Field-Crystal
equations. Second we consider nonvariational examples as the
Kuramoto-Sivashinsky equation, convective Allen-Cahn and Cahn-Hilliard
equations and thin-film equations describing stationary sliding drops and a
transversal front instability in a dip-coating. Through the different examples
we illustrate how to employ the numerical tools provided by the packages
auto07p and pde2path to determine steady, stationary and time-periodic
solutions in one and two dimensions and the resulting bifurcation diagrams. The
incorporation of boundary conditions and integral side conditions is also
discussed as well as problem-specific implementation issues