By a system we mean a set of related entities sometimes called components of elements. For instance, a hospital can be considered a system with doctor, nurses and patients are elements/components.
The elements have certain characteristics or attribute that have logical or numerical values. In the above example an attribute can be for instance the number of bags the number of X-ray machine skill quantity and so on. We say that, there exists internal and external relationships in the system. The internal relationship connects the elements with the system, while the external relationship connects the elements with the environment.
We can classify a system into a variety of ways-
Suppose that over a period of time the number of patient increases. If the hospital adds more staff to handle the increased workload??
The first step in studying a system is building a model:
Types of Models:
- Symbolic – Mathematical/Logical Operators (Abstract)
Advantages of Using Mathematical Models
- Enable investigators to organize their theoretical believes and empirical observation about a system and to deduce the logical implication.
- Improve system understanding
- Bring into perspective the need for detail and relevance
- Expedite the analysis
- Provide a framework for testing the desirability of system modification
- Easy manupilation
- Less costly than the system
Mathematical Models can be classified into:
- Static – Ohm’s Law
- Dynamic – Newton’s Laws of Motion
Mathematical model can be classified into:
- Deterministic- (All mathematical and logical relationship bet elements are fixed)
- Stochastic – Random
In order to be useful a scientific model should have two attributes:
- Realism – Resemble close approximation to the real system
After constructing a mathematical problem on the consideration, the next step is to derive a solution for this model. There are analytic and numerical solution methods.
An analytic solution is usually obtained directly from its mathematical representation in the form of a formula.
A numerical solution is generally an approximate solution obtained as a result of substitution of numerical values for the variables and parameters of the model. Many numerical methods are iterative i.e., each successive step in the solution uses the result from the previous step.
Newtons Method of approximating the roots of a non-linear equation can serve as an example.
Two special types of numerical methods are-
- Simulation and
- Monte Carlo Methods