Once the model has been developed and applied to the problem, your resulting model solution must be analyzed and interpreted with respect to the problem. The interpretations and conclusions should be checked for accuracy by answering the following questions:
1. Is the information produced reasonable?
2. Are the assumptions made while developing the model reasonable?
3. Are there any factors that were not considered that could affect the outcome?
4. How do the results compare with real data, if available?
In answering these questions, you may need to modify your model. This refining process should continue until you obtain a model that agrees as closely as possible with the real world observations of the phenomenon that you have set out to model.
Variables and Parameters
Mathematical models typically contain three distinct types of quantities: output variables, input variables, and parameters (constants). Output variables give the model solution. The choice of what to specify as input variables and what to specify as parameters is somewhat arbitrary and often model dependent. Input variables characterize a single physical problem while parameters determine the context or setting of the physical problem. For example, in modeling the decay of a single radioactive material, the initial amount of material and the time interval allowed for decay could be input variables, while the decay constant for the material could be a parameter. The output variable for this model is the amount of material remaining after the specified time interval.
Continuous-in-Time vs Discrete-in-Time Models
Mathematical models of time dependent processes can be split into two categories depending on how the time variable is to be treated. A continuous-in-time mathematical model is based on a set of equations that are valid for any value of the time variable.
A discrete-in-time mathematical model is designed to provide information about the state of the physical system only at a selected set of distinct times.
The solution of a continuous-in-time mathematical model provides information about the physical phenomenon at every time value. The solution of a discrete-in-time mathematical model provides information about the physical system at only a finite number of time values. Continuous-in-time models have two advantages over discrete-in-time models: (1) they provide
information at all times and (2) they more clearly show the qualitative effects that can be expected when a parameter or an input variable is changed. On the other hand, discrete in time models have two advantages over continuous in time models: (1) they are less demanding with respect to skill level in algebra, trigonometry, calculus, differential equations, etc. and (2) they are better suited for implementation on a computer.