Posts

2017-08-22: A rebuttal on the beauty in applying math

2017-04-22: Free googles book library

2016-11-02: In search of Theodore von Karman

2016-09-25: Amath Timeline

2016-02-24: Math errors and risk reporting

2016-02-20: Apple VS FBI

2016-02-19: More Zika may be better than less

2016-02-17: Dependent Non-Commuting Random Variable Systems

2016-01-14: Life at the multifurcation

2015-09-28: AI ain't that smart

2015-06-24: MathEpi citation tree

2015-03-31: Too much STEM is bad

2015-03-24: Dawn of the CRISPR age

2015-02-12: A Comment on How Biased Dispersal can Preclude Competitive Exclusion

2015-02-09: Hamilton's selfish-herd paradox

2015-02-08: Risks and values of microparasite research

2014-11-10: Vaccine mandates and bioethics

2014-10-18: Ebola, travel, president

2014-10-17: Ebola comments

2014-10-12: Ebola numbers

2014-09-23: More stochastic than?

2014-08-17: Feynman's missing method for third-orders?

2014-07-31: CIA spies even on congress

2014-07-16: Rehm on vaccines

2014-06-21: Kurtosis, 4th order diffusion, and wave speed

2014-06-20: Random dispersal speeds invasions

2014-05-06: Preservation of information asymetry in Academia

2014-04-16: Dual numbers are really just calculus infinitessimals

2014-04-14: More on fairer markets

2014-03-18: It's a mad mad mad mad prisoner's dilemma

2014-03-05: Integration techniques: Fourier--Laplace Commutation

2014-02-25: Fiber-bundles for root-polishing in two dimensions

2014-02-17: Is life a simulation or a dream?

2014-01-30: PSU should be infosocialist

2014-01-12: The dark house of math

2014-01-11: Inconsistencies hinder pylab adoption

2013-12-24: Cuvier and the birth of extinction

2013-12-17: Risk Resonance

2013-12-15: The cult of the Levy flight

2013-12-09: 2013 Flu Shots at PSU

2013-12-02: Amazon sucker-punches 60 minutes

2013-11-26: Zombies are REAL, Dr. Tyson!

2013-11-22: Crying wolf over synthetic biology?

2013-11-21: Tilting Drake's Equation

2013-11-18: Why $1^\infty != 1$

2013-11-15: Adobe leaks of PSU data + NSA success accounting

2013-11-14: 60 Minutes misreport on Benghazi

2013-11-11: Making fairer trading markets

2013-11-10: L'Hopital's Rule for Multidimensional Systems

2013-11-09: Using infinitessimals in vector calculus

2013-11-08: Functional Calculus

2013-11-03: Elementary mathematical theory of the health poverty trap

2013-11-02: Proof of the area of a circle using elementary methods

Kurtosis, 4th order diffusion, and wave speed

Here's a very simple problem students can do on their own to investigate how higher-order effects in linear parabolic equations can alter the wave-speed properties.

Consider a linear partial differential equation exhibiting non-zero 4th derivative. This is applicable to local expansion of kernels in integral-differential equations. Murray talks about this some.

\[ \frac{\partial N}{\partial t} = RN + D\frac{\partial^{2}N}{\partial x^{2}} + K \frac{\partial^{4}N}{\partial x^{4}}. \] \[ \dot{u} = u + u'' + K u''''. \]

This is important in the limiting expansion of integro-differential equations. One can look for exponential traveling wave with intensity $\theta$ so that \[ u = \bar{C} e^{\theta(x-ct)}, \] and this gives \[ 0 = \bar{C} e^{\theta(x-ct)} \left( K \theta^{4} + \theta^{2} + c\theta + 1 \right). \] Applying elimination theory with criticality condition $\partial c / \partial \theta = 0$, $4K\theta^{3}+2\theta+c_{min}=0$, so the minimum wave speed $c_{min}$ must satisfy \[ (9Kc_{min}^{2}-8K+2)\theta +c_{min}(1+12K)=0. \] This leads to nasty 8th degree polynomial in $c_{min}$ and $K.$ However, using groebner basis theory, one finds a rotated hyperbola and an ellipse/hyperbola govern the solution: \begin{eqnarray} 2 \theta^{2} + 3 \theta c_{min} + 4 &=& 0 \label{eq:1}\\ 9 c_{min}^{2} - 4(12K+1) \theta^{2} &=& 32 \label{eq:2} \end{eqnarray} These allow the problem to be reduced to simple 1-d root-finding.

The question is whether there exists a minimum speed for these waves. Well, for given $c$, it is straight forward to show when $K>0$ there exist $0$ or $2$ invasion speeds for given wave intensities $\theta$ by inspecting $$ K \theta^{4} + \theta^{2} + 1 = - c \theta.$$ When $K<0$ there are will be $2$ or $4$ solutions for $\theta$, but inspecting \[ c = - \frac{ K \theta^{4} + \theta^{2} + 1 }{ \theta },\] we see that for every possible $c$, there exists at least one $\theta > 0$ -- there is no band-gap in allowed speeds. Thus, we can not define any wave speeds for $K < 0$ without changing model fundamentals.

Produced by GNUPLOT 4.4 patchlevel 3 -10 -5 0 5 10 -4 -3 -2 -1 0 1 2 3 4 Wave speed (c) Wave number ($\theta$) K=-0.05 K=0.00 K=0.20 K=0.40

In this case ($K\geq 0$), $\frac{\partial c_{min}}{\partial K}>0$, such that $c_{min}$ grows without bound. For $K>0, c_{min}>2$ there is the very nice asymptotic approximation that $$K \sim \frac{27}{256}c_{min}^{4} - \frac{9}{16}c_{min}^{2} + \frac{1}{2}$$ which isn't too bad even when $K\sim 0$. For small $K$, $c_{min} = 2 + K + O(K^2)$, which recovers the standard diffusion behavior.

Produced by GNUPLOT 4.4 patchlevel 3 -4 -3 -2 -1 0 1 2 3 4 -1 0 1 2 3 4 Minimum wave speed (c min ) Kurtosis (K) Exact Asymptotic approximation