Mathematical Models
Mathematical Models
Contents
Definition of Mathematical Model Types of Variables The Mathematical Modeling Cycle Classification of Models
2
Definitions of Mathematical Model
Mathematical modeling is the process of creating a mathematical representation of some phenomenon in order to gain a better understanding of that phenomenon. It is a process that attempts to match observation with symbolic statement. A mathematical model uses mathematical language to describe a system. Building a model involves a trade-off between simplicity and accuracy. The success of a model depends on how easily it can be used and how accurate are its predictions.
3
Types of Variables
A mathematical model usually describes a system by a set of variables and a set of equations that establish relationships between the variables. The variables represent some properties of the system. There are four basic groups of variables:
– – – – Input variables Parameters Random variables Decision variables (output variables)
4
The Mathematical Modeling Cycle
Simplify Real World Problem
Interpret
Mathematical Model
Program
Conclusions Simulate
Computer Software
5
The Mathematical Modeling Cycle
1. Identify and understand the problem, draw diagrams 2. Define the terms in your problem 3. Identify important variables and constants and determine how they relate to each other. 4. State the assumptions as you focus on particular aspects of the phenomenon. 5. Develop the equation(s) that express the relationships between the variables and constants.
6
The Mathematical Modeling Cycle
The Real World
A non-mathematical setting
The Mathematical World
An abstract representation
Representation Pose a question Create a model
Identify relevant variables, determine the form of the answer and the objects that best capture the relationships between the variables
Visualize a problem or phenomenon, make questions
Manipulation
Analysis
Derive new
Contents
Definition of Mathematical Model Types of Variables The Mathematical Modeling Cycle Classification of Models
2
Definitions of Mathematical Model
Mathematical modeling is the process of creating a mathematical representation of some phenomenon in order to gain a better understanding of that phenomenon. It is a process that attempts to match observation with symbolic statement. A mathematical model uses mathematical language to describe a system. Building a model involves a trade-off between simplicity and accuracy. The success of a model depends on how easily it can be used and how accurate are its predictions.
3
Types of Variables
A mathematical model usually describes a system by a set of variables and a set of equations that establish relationships between the variables. The variables represent some properties of the system. There are four basic groups of variables:
– – – – Input variables Parameters Random variables Decision variables (output variables)
4
The Mathematical Modeling Cycle
Simplify Real World Problem
Interpret
Mathematical Model
Program
Conclusions Simulate
Computer Software
5
The Mathematical Modeling Cycle
1. Identify and understand the problem, draw diagrams 2. Define the terms in your problem 3. Identify important variables and constants and determine how they relate to each other. 4. State the assumptions as you focus on particular aspects of the phenomenon. 5. Develop the equation(s) that express the relationships between the variables and constants.
6
The Mathematical Modeling Cycle
The Real World
A non-mathematical setting
The Mathematical World
An abstract representation
Representation Pose a question Create a model
Identify relevant variables, determine the form of the answer and the objects that best capture the relationships between the variables
Visualize a problem or phenomenon, make questions
Manipulation
Analysis
Derive new