WHAT IS MATHEMATICS AND ENGINEERING PROBLEM SOLVING ?

Mathematical modeling  and scintific problem Solving :

Knowledge and understanding are prerequisites for the effective implementation of any

tool. No matter how impressive your tool chest, you will be hard-pressed to repair a car if

you do not understand how it works.

This is particularly true when using computers to solve engineering problems. Although

they have great potential utility, computers are practically useless without a fundamental

understanding of how engineering systems work.

This understanding is initially gained by empirical means—that is, by observation and

experiment. However, while such empirically derived information is essential, it is only

half the story. Over years and years of observation and experiment, engineers and scientists

have noticed that certain aspects of their empirical studies occur repeatedly. Such general

behavior can then be expressed as fundamental laws that essentially embody the cumulative

wisdom of past experience. Thus, most engineering problem solving employs the twopronged

approach of empiricism and theoretical analysis (Fig. 1.1).

It must be stressed that the two prongs are closely coupled. As new measurements are

taken, the generalizations may be modified or new ones developed. Similarly, the generalizations

can have a strong influence on the experiments and observations. In particular,

generalizations can serve as organizing principles that can be employed to synthesize observations

and experimental results into a coherent and comprehensive framework from

which conclusions can be drawn. From an engineering problem-solving perspective, such

a framework is most useful when it is expressed in the form of a mathematical model.

The primary objective of this chapter is to introduce you to mathematical modeling

and its role in engineering problem solving. We will also illustrate how numerical method.