Methods of Modelling 1

Forward problem/evaluation question

Given all information about the state of a system and the relation between its variables, can we quantitatively predict its other properties and how it will function? Examples: What is the maximum attainable speed of this car? How quickly will this disease spread through the population of this city? (Assuming we know everything about whatever ’this’ is.)

Inverse problems/detection questions

If some information about a phenomenon is not directly available, can it be ’reverse engineered’ from observations? Examples: How can we use data from CT scans to estimate the location of a tumour? Can we determine the damping of an oscillating pendulum from the decay of its time series height data?

Control problems/design questions

Here, the objective is to create or generate a solution that best meets a specified goal. We may also say that we want to steer the system into a specific state. Examples: What shape is best for a paper airplane to make it fly furthest? How should a pill be coated to release its drug at a constant rate over an entire day?

linear ODE

no products of the dependent variable

autonomous ODE

equation does not depend explicitly on the independent variable

homogeneous ODE

s(t) = 0

Principle of dimensional homogeneity

All terms that are summed to contribute to a mathematical statement have the same dimension.

Principle of dimensional completeness

It is possible to combine the independent variables (in the sense of a product of suitable powers) to produce a variable of the same dimension as the dependent variable.