User:Jaroslaw Karcz/Modelling Sandbox

THE KEY
$$\nabla \times \nabla \times v = ...$$ $$\nabla \times v = \epsilon_{spq}v_{q,p}$$ $$\nabla \times \nabla \times v = \epsilon_{irs}(\epsilon_{spq}v_{q,p})_{,r}$$ $$\Rightarrow \epsilon_{irs} \epsilon_{spq} (v_{q,p})_{r}$$ $$\Rightarrow \epsilon_{sir}\epsilon_{spq}v_{q,pr}$$....cyclic permutation $$\Rightarrow (\delta_{ip}\delta_{rq} - \delta_{iq}\delta_{rp})v_{q,pr}$$ $$\Rightarrow \delta_{ip}\delta_{rq}v_{q,pr} - \delta_{iq}\delta_{rp}v_{q,pr}$$ $$\Rightarrow \delta_{ip}v_{r,pr} - \delta_{iq}v_{q,rr}$$ $$\Rightarrow v_{r,ir} - v_{i,rr}$$ $$\Rightarrow v_{r,ri} - v_{i,rr} ......since..... v_{r,ri} = v_{r,ir}$$ ...continuous function

Convert to vector notation
$$\Rightarrow\nabla (\nabla \cdot v) - \nabla^{2}v$$

Model Development
The process of modelling consists of a number of layers; the following is a description of the modelling workflow:
 * 1) Definition of the problem
 * 2) Verification of information available
 * 3) Selection of model structure
 * 4) Establishing a simple model
 * 5) Sensitivity analysis
 * 6) Experimental tests of the model predictions
 * 7) Stating the agreements and divergences between experimental and modelling results, including any emergent behaviour
 * 8) Iterative refinement of model

$$f_{obj}(k) = \sum_{i=1}^q (f_{obs}(i) - f_{per}(i,k))^2$$

 Part Main Page Transfer Function Specificity Response time Stability Add Data

Introduction
The real world is dominated by complexity, especially biological systems Mathematical modelling and computer simulations provide a means of understanding the innate funtioning of system - dynamics, and to arrive at well-founded predictions about their future development and the effect of interactions with the environment. So what is a model? A model is an abstract representation of objects and processes that explain the features/nature of these objects or processes. We present the model of our construct, as a system of differential equations to describe the dynamics of that network.