Team:ETH Zurich/Modeling/Analytical Approximation
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We derived the gradient formation dynamics analytically already in the [[Team:ETH_Zurich/Modeling/Microfluidics#Model|reaction-diffusion model]]. However, there we only solved the resulting partial differential equation numerically. '''Here, we are going to venture forth and give a suitable analytical approximation!''' | We derived the gradient formation dynamics analytically already in the [[Team:ETH_Zurich/Modeling/Microfluidics#Model|reaction-diffusion model]]. However, there we only solved the resulting partial differential equation numerically. '''Here, we are going to venture forth and give a suitable analytical approximation!''' | ||
== Kolmogorov-Petrovsky-Piskounov Equation == | == Kolmogorov-Petrovsky-Piskounov Equation == | ||
+ | [[File:ETH-AcAl-Reaction-Diffusion.png|317px|center|thumb|General partial differential equation for an acetaldehyde reaction-diffusion system. '''D(AcAl(x,z),z)''' is the diffusive term, '''R(AcAl(x,z))''' is the uniform (independent of the spatial z coordinate) reaction term.]] | ||
+ | In mathematics, the problem posed by general reaction-diffusion systems of this form is known as the '''Kolmogorov-Petrovsky-Piskounov Equation'''. | ||
+ | |||
== Dead Zone Concept == | == Dead Zone Concept == | ||
== Analytical Solution == | == Analytical Solution == |
Revision as of 18:11, 27 October 2011
Analytical Approximation
GFP Band: Dimensionless Model
1. Dimensionless Species
We nondimensionalized all species occurring in the equations for the GFP band circuit.
2. Dimensionless Equations
The dimensionless equations for the dynamics of the band-generating system then read
3. Steady State
4. Dimensionless Groupings
The equation system can be simplified by introducing the following dimensionless groupings:
This yields the final equations for steady state:
Gradient Approximation
We derived the gradient formation dynamics analytically already in the reaction-diffusion model. However, there we only solved the resulting partial differential equation numerically. Here, we are going to venture forth and give a suitable analytical approximation!
Kolmogorov-Petrovsky-Piskounov Equation
In mathematics, the problem posed by general reaction-diffusion systems of this form is known as the Kolmogorov-Petrovsky-Piskounov Equation.
Dead Zone Concept
Analytical Solution
References