*Mathematical Model*

A mathematical model can be broadly defined as a formulation or equation that expresses the essential features of a physical system or process in mathematical terms. In a very general sense, it can be represented as a functional relationship of the form

Dependent variable = f (independent variables, parameters, forcing functions)

(Eq (1))

Fig. The engineering problem: solving process.)

where the dependent variable is a characteristic that usually reflects the behavior or state of the system; the independent variable are usually dimensions, such as time and space, along wich the system's behavior is being determined; the parameters are reflective of the system's properties or composition; and the forcing functions are external influences acting upon the system.

The actual mathematical expression of Eq. (1) can range from a simple algebraic relationship to large complicated sets of differential equations. For example, on the basis of his observations, Newton formulated his second law of motion, which states that the time rate of change of momentum of a body is equal to the resultant force acting on it. The mathematical expression, or model,

of the second law is the well-known equation

F = m.a (Eq (2))

where F = net force acting on the body, m = mass of the object, and a = its acceleration.

The second law can be recast in the format of Eq (1) by merely dividing both sides by m to give

Characteristics that are typical of mathematical models of the physical world:

1. It describes a natural process or system in mathematical terms.

2. It represents an idealization and simplification of reality. That is, the model ignores negligible details of the natural process and focuses on its essential manifestations. Thus, the second law does not include the effects of relativity that are minimal importance when applied to objects and forces that interact on or about the earth's surface at velocities and on scales visible to humans.

3. Finally, it yields reproducible results and, consequently, can be used for predictive purposes.

This equation, is called an analytical, or exact, solution because it exactly satisfies the original differential equation. Unfortunately, there are many mathematical models that cannot be solved exactly. In many of these cases, the only alternative is to develop a numerical solution that approximates the exact solution.

As conclution, numerical methods are those in wich the matematical problem is reformulated so it can be solved by arithmetic operations.


Source:
Numerical methods for Engineers
Fifth Edition
Steven C. Chapra
Raymond P. Canale
Mc Graw Hill