Introduction to the Modeling and Analysis of Complex Systems

276 CHAPTER 14. CONTINUOUS FIELD MODELS II: ANALYSISIn fact, we saw a similar situation before. When we discussed how to obtain analyticalsolutions for linear dynamical systems, the nuisances were the matrices that existed inthe equations. We “destroyed” those matrices by using their eigenvectors, i.e., vectorsthat can turn the matrix into a scalar eigenvalue when applied to it. Can we do somethingsimilar to destroy those annoying spatial derivatives?The answer is, yes we can. While the obstacle we want to remove is no longer a simplematrix but a linear differential operator, we can still use the same approach. Instead ofusing eigenvectors, we will use so-called eigenfunctions. An eigenfunction of a linearoperator L is a function that satisfiesLf = λf, (14.40)where λ is (again) called an eigenvalue that corresponds to the eigenfunction f. Look atthe similarity between the definition above and the definition of eigenvectors (Eq. (5.37))!This similarity is no surprise, because, after all, the linear operators and eigenfunctionsare straightforward generalizations of matrices and eigenvectors. They can be obtainedby increasing the dimensions of matrices/vectors (= number of rows/columns) to infinity.Figure 14.2 gives a visual illustration of how these mathematical concepts are related toeach other, where the second-order spatial derivative of a spatial function f(x) in a [0, 1]1-D space is shown as an example.When the space is discretized into n compartments, the function f(x) is also definedon a discrete spatial grid made of n cells, and the calculation of its second-order spatialderivative (or, to be more precise, its discrete equivalent) can be achieved by adding itstwo nearest neighbors’ values and then subtracting the cell’s own value twice, as shownin Eq. (13.34). This operation can be represented as a matrix with three diagonal linesof non-zero elements (Fig. 14.2, left and center). Because it is a square matrix, we cancalculate its eigenvalues and eigenvectors. And if the number of compartments n goes toinfinity, we eventually move into the realm of continuous field models, where what used tobe a matrix is now represented by a linear operator (∂ 2 /∂x 2 , i.e., ∇ 2 in 1-D space), whilethe eigenvector is now made of infinitely many numbers, which entitles it to a new name,“eigenfunction.”In fact, this is just one instance of how mathematical concepts for matrices and vectorscan be generalized to linear operators and functions in a continuous domain. Nearly allthe concepts developed in linear algebra can be seamlessly extended continuous linearoperators and functions, but we won’t go into details about this in this textbook.Most linear operators we see in PDE-based continuous field models are the secondorder(and sometimes first-order) differential operators. So here are their eigenfunctions(which are nothing more than general solutions of simple linear differential equations):