MATHEMATICAL MODEL

PRESENTED BY: KATHERINE SILVA

NUMERICAL MOTHODS IN PETROLEUM ENGINEERING

image 1Source: www.emi s.de/.../NNJ/images_number1/MGC-05.jpg

A mathematical model is the mathematical description of a real situation.

In developing the model, some assumptions are made and we consider some simplifications of reality.

A model can represent it using:

• Relations

• functions

DEFINITION

TYPES OF MODELSThere are three types of models:

•Linear model: We call linear models to situations can be represented by a linear function. can be determined graphically or by means of an equation.

images 2 and 3Source: http://entren.dgsca.unam.mx/ModMat/mm05.html

• quadratic model: We say that the model is quadratic if we can express by means of a quadratic function.

A quadratic model can be determined through an equation or by means of a graph that made the best approximates of the data.

image 4Source: http://entren.dgsca.unam.mx/ModMat/mm05.html

• exponential model: We call exponential models to situations that are represented by an exponential function.

The exponential models are very common in the study of population increases, the calculation of bank interest, so as various physical phenomena.

image 5Source: http://entren.dgsca.unam.mx/ModMat/mm05.html

HOW DO YOU DEVELOP A MATHEMATICAL MODEL?

The process for develop a mathematical model is as following: 1. Find a real world problem

2. Formulate a mathematical model of the problem, identifying variables and establishing hypotheses simple to be treated mathematically.

3. Apply the mathematical knowledge that you have to reach mathematical conclusions.

4.Compare the data obtained as predictions with real data. If the data are different, the process is restarted.

DIFFERENTIAL EQUATIONS AS MATHEMATICAL MODEL

A single differential equation mathematical model can be of many different phenomena.

a mathematical model is formed by an initial value problem, or also value problem at the border.

image 6Source: http://entren.dgsca.unam.mx/ModMat/mm05.html

EXAMPLE OF MATHEMATICAL MODEL• An analysis of the spread of a contagious flu, for

example, is reasonable to assume that the rate or reason that spreads not only is proportional to the number of people, x (t), which have contracted at time t, but also the number of subjects, and (t), which have not yet been exposed to infection. If the rate is dx / dt, then

• Where k is the usual constant of proportionality.

• If, for example, introduces an infected person in a constant population of n people, then x and y are related by x + y = n + 1. We use this equation to eliminate and in equation (1) and obtain the model

 

• An obvious initial condition accompanying equation (2) is x (0) = 1.

BIBLIOGRAFHY

• MONOGRAFIAS• http://

entren.dgsca.unam.mx/ModMat/mm05.html• http://

mazinger.sisib.uchile.cl/repositorio/ap/ciencias_quimicas_y_farmaceuticas/apmat4a/01b.html