AMP1514 mathematical modeling difference equation ,discrete, continuous and autonomous

Mathematical modeling is the process of using mathematical equations to describe real-world phenomena. This can be done for a wide variety of systems, including those that are discrete, continuous, or autonomous. In this script, we will walk through the process of mathematical modeling for each of these types of systems.

SECTION 1: DIFFERENCE EQUATION
A difference equation is a type of equation that describes how a system changes over time based on its current state. To create a mathematical model using difference equations, we need to follow these steps:

Step 1: Define the system
The first step in creating a difference equation model is to define the system we want to model. For example, let's say we want to model the population growth of a certain species of animal.

Step 2: Determine the variables
Next, we need to determine the variables that will be used in the difference equation. In the case of population growth, we might use the variable "P" to represent the population size.

Step 3: Define the difference equation
The difference equation is an equation that describes how the system changes over time. In the case of population growth, the equation might look like this:

P(t+1) = P(t) + rP(t)(1 - P(t)/K)

where:
P(t) = population size at time t
P(t+1) = population size at time t+1
r = growth rate
K = carrying capacity (maximum population size)

Step 4: Interpret the model
Finally, we need to interpret the model. In this case, the model tells us how the population size will change over time based on the growth rate and carrying capacity. We can use this model to make predictions about future population sizes.

SECTION 2: DISCRETE SYSTEM
A discrete system is a system in which the variables change only at certain intervals. To create a mathematical model using discrete systems, we need to follow these steps:

Step 1: Define the system
As with difference equations, the first step in creating a mathematical model using discrete systems is to define the system we want to model.

Step 2: Determine the variables
Next, we need to determine the variables that will be used in the model.

Step 3: Define the discrete system
The discrete system is an equation that describes how the system changes at each interval. For example, if we were modeling the temperature in a room that was being heated, the discrete system might look like this:

T(n+1) = T(n) + k(H - T(n))

where:
T(n) = temperature at time n
T(n+1) = temperature at time n+1
k = constant
H = temperature of the heater

Step 4: Interpret the model
Finally, we need to interpret the model. In this case, the model tells us how the temperature will change over time as the heater is turned on and off.

SECTION 3: CONTINUOUS SYSTEM
A continuous system is a system in which the variables change continuously over time. To create a mathematical model using continuous systems, we need to follow these steps:

Step 1: Define the system
As with the other types of systems, the first step in creating a mathematical model using continuous systems is to define the system we want to model.

Step 2: Determine the variables
Next, we need to determine the variables that will be used in the model.

Step 3: Define the continuous system
The continuous system is a differential equation that describes how the system changes continuously over time. For example, if we were modeling the growth of a tumor

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