5,695 research outputs found
Lie symmetries and exact solutions for a fourth-order nonlinear diffusion equation
In this paper, we consider a fourth-order nonlinear diffusion partial differential
equation, depending on two arbitrary functions. First, we perform an analysis
of the symmetry reductions for this parabolic partial differential equation by
applying the Lie symmetry method. The invariance property of a partial differential
equation under a Lie group of transformations yields the infinitesimal
generators. By using this invariance condition, we present a complete classification
of the Lie point symmetries for the different forms of the functions that
the partial differential equation involves. Afterwards, the optimal systems of
one-dimensional subalgebras for each maximal Lie algebra are determined, by
computing previously the commutation relations, with the Lie bracket operator,
and the adjoint representation. Next, the reductions to ordinary differential
equations are derived from the optimal systems of one-dimensional subalgebras.
Furthermore, we study travelling wave reductions depending on the form of the
two arbitrary functions of the original equation. Some travelling wave solutions
are obtained, such as solitons, kinks and periodic waves
Evaluating the Evans function: Order reduction in numerical methods
We consider the numerical evaluation of the Evans function, a Wronskian-like
determinant that arises in the study of the stability of travelling waves.
Constructing the Evans function involves matching the solutions of a linear
ordinary differential equation depending on the spectral parameter. The problem
becomes stiff as the spectral parameter grows. Consequently, the
Gauss--Legendre method has previously been used for such problems; however more
recently, methods based on the Magnus expansion have been proposed. Here we
extensively examine the stiff regime for a general scalar Schr\"odinger
operator. We show that although the fourth-order Magnus method suffers from
order reduction, a fortunate cancellation when computing the Evans matching
function means that fourth-order convergence in the end result is preserved.
The Gauss--Legendre method does not suffer from order reduction, but it does
not experience the cancellation either, and thus it has the same order of
convergence in the end result. Finally we discuss the relative merits of both
methods as spectral tools.Comment: 21 pages, 3 figures; removed superfluous material (+/- 1 page), added
paragraph to conclusion and two reference
Takens-Bogdanov bifurcation of travelling wave solutions in pipe flow
The appearance of travelling-wave-type solutions in pipe Poiseuille flow that
are disconnected from the basic parabolic profile is numerically studied in
detail. We focus on solutions in the 2-fold azimuthally-periodic subspace
because of their special stability properties, but relate our findings to other
solutions as well. Using time-stepping, an adapted Krylov-Newton method and
Arnoldi iteration for the computation and stability analysis of relative
equilibria, and a robust pseudo-arclength continuation scheme we unfold a
double-zero (Takens-Bogdanov) bifurcating scenario as a function of Reynolds
number (Re) and wavenumber (k). This scenario is extended, by the inclusion of
higher order terms in the normal form, to account for the appearance of
supercritical modulated waves emanating from the upper branch of solutions at a
degenerate Hopf bifurcation. These waves are expected to disappear in
saddle-loop bifurcations upon collision with lower-branch solutions, thereby
leaving stable upper-branch solutions whose subsequent secondary bifurcations
could contribute to the formation of the phase space structures that are
required for turbulent dynamics at higher Re.Comment: 26 pages, 15 figures (pdf and png). Submitted to J. Fluid Mec
Thermoelasticity and generalized thermoelasticity viewed as wave hierarchies
It is seen how to write the standard\^E form of the four partial differential
equations in four unknowns of anisotropic thermoelasticity as a single equation
in one variable, in terms of isothermal and isentropic wave operators. This
equation, of diffusive type, is of the eighth order in the space derivatives
and seventh order in the time derivatives and so is parabolic in character.
After having seen how to cast the 1D diffusion equation into Whitham's wave
hierarchy form it is seen how to recast the full equation, for uni-directional
motion, in wave hierarchy form. The higher order waves are isothermal and the
lower order waves are isentropic or purely diffusive. The wave hierarchy form
is then used to derive the main features of the solution of the initial value
problem, thereby bypassing the need for an asymptotic analysis of the integral
form of the exact solution. The results are specialized to the isotropic case.
The theory of generalized thermoelasticity associates a relaxation time with
the heat flux vector and the resulting system of equations is hyperbolic in
character. It is seen also how to write this system in wave hierarchy form
which is again used to derive the main features of the solution of the initial
value problem. Simpler results are obtained in the isotropic case.Comment: 16 page
A non-linear degenerate equation for direct aggregation and traveling wave dynamics
The gregarious behavior of individuals of populations is an important factor in avoiding predators or for reproduction. Here, by using a random biased walk approach, we build a model which, after a transformation, takes the general form [u_{t}=[D(u)u_{x}]_{x}+g(u)] . The model involves a density-dependent non-linear diffusion coefficient [D] whose sign changes as the population density [u] increases. For negative values of [D] aggregation occurs, while dispersion occurs for positive values of [D] . We deal with a family of degenerate negative diffusion equations with logistic-like growth rate [g] . We study the one-dimensional traveling wave dynamics for these equations and illustrate our results with a couple of examples. A discussion of the ill-posedness of the partial differential equation problem is included
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