Differential Equations (Schaum’s Outlines) 4th Edition
Book Preface
MATHEMAATICAL MODELS
Mathemaatical models can be thought of as equations. In this chapter, and in other parts of the book (see Chapter 7, Chahapter 14 and Chapter 31, for example), we will consider equations which model certain realworld situations.
For exampple, when considering a simple direct current (DC) electrical circuit, the equation V = RI models the voltage droop (measured in volts) across a resistor (measured in ohms), where I is the current (measured in amperes). This equation is called Ohm’s Law, named in honor of G. S. Ohm (1787–1854), a German physicist.
Once conststructed, some models can be used to predict many physical situations. For example, weather forecasting, the growth of a tumor, or the outcome of a roulette wheel, can all be connected with some form of mathematical mmodeling.
In this chappter, we consider variables that are continuous and how differential equations can be used in modeling. Chaptter 34 introduces the idea of difference equations. These are equations in which we consider discrete variableses; that is, variables which can take on only certain values, such as whole numbers. With few modifications, evverything presented about modeling with differential equations also holds true with regard to modeling with dififference equations.
THE “MODELINNG CYCLE”
Suppose we haave a reallife situation (we want to find the amount of radioactive material in some element). Research may be aable to model this situation (in the form of a “very difficult” differential equation). Technology may be used to heelp us solve the equation (computer programs give us an answer). The technological answers are then interpreteted or communicated in light of the reallife situation (the amount of radioactive material). Figure 21 illustrarates this cycle.
QUALITATIVE METHODS
To build a model can be a long and arduous process; it may take many years of research. Once they are formulated, models may be virtually impossible to solve analytically. Then the researcher has two options:
• Simplify, or “tweak”, the model so that it can be dealt with in a more manageable way. This is a valid approach, provided the simplification does not overly compromise the “realworld” connection, and therefore, its usefulness.
• Retain the model as is and use other techniques, such as numerical or graphical methods (see Chapter 18, Chapter 19, and Chapter 20). This represents a qualitative approach. While we do nott possess an exact, analytical solution, we do obtain some information which can shed some light on the model and its application. Technological tools can be extremely helpful with this approach (see Appendix B).
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