Mathematical modeling

ESCUELA DE INGENIERÍA DE PETROLEOS

RUBEN DARIO ARISMENDI RUEDA


CHAPTER 1: ‘MATHEMATICAL MODELING’


Mathematical Modeling, was made to formulate different problems or situations (essential physical systems) into mathematical language by equations that can be treated and solved by many ways. But the most important ways are: Numerical Methods, Graphics results and analytical solutions.


Steps of Mathematical Modeling

The next image taken from PhD. Eduardo Carrillo's presentation ''METODOS NUMERICOS EN INGENIERIA DE PETROLEOS''


COMPONENTS.

•Dependent Variable : Reflects the behavior of the system.

•Independent Variable : Are usually dimensions that determines system's behavior.

•Parameters: ''Are refelctive's of the systems properties or composition''

•Forcing Functions: External influences acting upon the system


Last components could be express like this:

Numerical Methods for Engineers . Steven C. Chapra, pag 11


When Mathematical Models are written in terms of differential rate of change (dv/dt), we can say that we have differential equations as a Model.

Example:

Every differential equation will have his own solution by algebraic manipulation or by other kind of techniques when it's not too easy to obtain the exact or analytical solution.

kxdt

sdm

2

2


Classifying Mathematical models.

•Linear or. Nonlinear: If all the operators in a mathematical model exhibit linerity, the resulting mathematical model is defined as linear. If one or more of the objective functions or constraints are represented with a nonlinear equation, then the model is known as a nonlinear model. Nonlinearity, even in fairly simple systems, is often associated with phenomena such as chaos and irreversibility. A common approach to nonlinear problems is linearization, but this can be problematic if one is trying to study aspects such as irreversibility, which are strongly tied to nonlinearity.

•Deterministic and probabilistic: A deterministic model is one in which every set of variable states is uniquely determined by parameters in the model and by sets of previous states of these variables. Therefore, deterministic models perform the same way for a given set of initial conditions. Conversely, in a probabilistic model, randomness is present, and variable states are not described by unique values, but rather by probability distributions.


•Static and dynamic: A static model does not account for the element of time, while a dynamic model does. Dynamic models typically are represented with difference equations or differential equations.

•Lumped and distributed parameters: If the model is heterogeneous (varying state within the system) the parameters are distributed. If the model is homogeneous (consistent state throughout the entire system), then the parameters are lumped. Distributed parameters are typically represented with partial differential equations.


EXAMPLEThe best and most easy example that we can treat in this topic, is the falling parachutist, who describes the second law of motion.

Hypothesis:El tiempo de caída depende de la altura inicial. Variables: M is the mass , g gravity, t time of the fall

THEN.


ANALYTICAL SOLUTION: M=68,1g=9,8C=12,5


Bibliography:

•Numerical Methods for Engineers . Steven C. Chapra•Prf. Eduardo Carrillo's presentation ''METODOS NUMERICOS EN INGENIERIA DE PETROLEOS''.•http://en.wikipedia.org/wiki/Mathematical_model

http://en.wikipedia.org/wiki/Mathematical_model

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Mathematical modeling