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Mathematics LibreTexts

13.1: Continuous Field Models with Partial Differential Equations


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Spatio-temporal dynamics of complex systems can also be modeled and analyzed using partial differential equations (PDEs), i.e., differential equations whose independent variables include not just time, but also space. As a modeling framework, PDEs are much older than CA. But interestingly, the applications of PDEs to describe self-organizing dynamics of spatially extended systems began about the same time as the studies of CA. As discussed in Section 11.5, Turing’s monumental work on the chemical basis of morphogenesis [44] played an important role in igniting researchers’ attention to the PDE-based continuous field models as a mathematical framework to study self-organization of complex systems. 

There are many different ways to formulate a PDE-based model, but here we stick to the following simple first-order mathematical formulation:

\[\frac{\partial{f}}{\partial{t}}=F(f, \frac{\partial{f}}{\partial{x}}, \frac{\partial^{2}f}{\partial{x^{2}}}, ....,x, t)\label{13.1}\]

Now the partial derivatives (e.g., \(∂f/∂t\)) have begun to show up in the equations, but don’t be afraid; they are nothing different from ordinary derivatives (e.g., \(df/dt\)). Partial derivatives simply mean that the function being differentiated has more than one independent variable (e.g., \(x\), \(t\)) and that the differentiation is being done while other independent variables are kept as constants. The above formula is still about instantaneous change of something over time (as seen on the left hand side), which is consistent with what we have done so far, so you will find this formulation relatively easy to understand and simulate.

Note that the variable \(x\) is no longer a state variable of the system, but instead, it represents a position in a continuous space. In the meantime, the state of the system is now represented by a function, or a field \(f(x,t)\), which is defined over a continuous space as well as over time (e.g., Fig. \(\PageIndex{1}\)). The value of function \(f\) could be scalar or vector. Examples of such continuous fields include population density (scalar field) and flows of ocean currents (vector field).


Figure \(\PageIndex{1}\): Example of a spatial function (field) defined over a two-dimensional space. The function \(f(x,y) = e^{-(x^2+y^2)}\)  is shown here.

In Equation \ref{13.1}, the right hand side may still contain space \(x\) and time \(t\)), which means that this may be a non-autonomous equation. But as we already know, non-autonomous equations can be converted into autonomous equations using the trick discussed in Section 6.2. Therefore, we will focus on autonomous models in this chapter. 

As you can see on the right hand side of Equation \ref{13.1}, the model can now include spatial derivatives of the field, which gives information about how the system’s state is shaped spatially. The interaction between spatial derivatives (shape, structure) and temporal derivatives (behavior, dynamics) makes continuous field models particularly interesting and useful. 

In a sense, using a continuous function as the system state means that the number of variables (i.e., the degrees of freedom) in the system is now infinite. You must be proud to see this milestone; we have come a long way to reach this point, starting from a single variable dynamical equation, going through CA with thousands of variables, to finally facing systems with infinitely many variables!

But of course, we are not actually capable of modeling or analyzing systems made of infinitely many variables. In order to bring these otherwise infinitely complex mathematical  models down to something manageable for us who have finite intelligence and lifespan, we usually assume the smoothness of function f. This is why we can describe the shape and behavior of the function using well-defined derivatives, which may still allow us to study their properties using analytical means1.


1But we should also beaware that not all physically important, interesting systems can be represented by  smooth spatial functions. For example, electromagnetic and gravitational fields can have singularities where smoothness is broken and state values and/or their derivatives diverge to infinity. While such singularities do play an important role in nature, here we limit ourselves to continuous, smooth spatial functions only.