chapter 1: modeling

BY: MARIA FERNANDA VERGARA M. UNIVERSIDAD INDUSTRIAL DE SANTANDER

A mathematical model is a description of a physical phenomenon, process, even an economic system, using a mathematical formulation or equation.It can be seen as a functional relationship, taking into account it’s parameters:

Dependent variable f Independent

variablesparameters

Forcing functions, ,

Dependent variable f Independent

variablesparameters

Forcing functions, ,

Reflects the system behavior

These ones are dimensions, for example: time

These ones tell us about system

properties

These ones are external influences that affect the

system

To formulate a mathematical model you will need to follow the next steps:

Hypothesis

TestingGetting solutions

Mathematical formulation or

equation

Express the hypotesis in terms of differential

equations

Solving the D.E.

Showing model predictions

If required, raise the complexity of the model or change the hypothesis

W

Fr Newton’s Second Law:

Where:a is the dependent parameterF is the forcing functionm is the parameter

¿Which is the terminal velocity of a free-falling body near the earth’s surface?

W

Fr

Net Force: Fr + W

Fr = -cv W= mgWhere and

Drag Coefficient

Solving, and taking into account that initial velocity is 0:

Analytical or exact solution

An analytical solution satisfies the differential equation, but there are many mathematical models that cannot be solved exactly, here is when we need numerical methods to solve the equation and get an aproximated solution.This way we can solve the problem of the parachute getting a numerical solution:

Source: CHAPRA,Steven C., Numerical Methods for Engineers. Mc Graw Hill

Using Newtons law, but realizing that the time rate of change of velocity can be aproximated by:

We can get a numerical solution for the same problem of the parachute:

Now we got a numerical solution for the problem of the parachute, so if you have an initial time and velocity for some time ti , you can easily get the velocity at a time ti+1. This velocity at the time ti+1 can be used to extend the computation to the velocity at ti+2 and so on.

CHAPRA, Steven C. Numerical Methods for engineers. Mc Graw Hill.

ZILL, Dennis. Differential Equations with modeling applications. International Thomson Publishing Company.