"A first Course in mathematical Modeling" Modeling of-chaper1-difference equation

From today on, I will be using this column to learn "mathematical modeling". In fact, for the "mathematical modeling" of its own connotation or meaning does not need too much explanation, concise and clear interpretation of the meaning of mathematical modeling.

In fact, the mathematical modeling itself can be regarded as a different angle to interpret mathematics, we are familiar with the mathematical model to the real life of the specific problems.

To model changes:

As shown, a great function of the mathematical model is to speculate on the future, which is a conjecture of "change".

First, the most basic definition of "change" is given:

Change = future value-present value.

So if we set up a time function for this "change", then based on the current value, we can calculate the future value well.

For discrete-time variations, we use differential equations for modeling, and for continuous-time changes, we often use differential equations for modeling, where the modeling of differential equations is discussed separately in Chaper10, and the modeling of variations in discrete time is mainly discussed in Chaper1.

About the concept of difference equations:

Set sequence a={a0,a1,a2......an}, the first order difference is as follows;

A0=a1-a0

A1=a2-a1

A2=a3-a2

Nth first-order difference is an=a (n+1)-an

We use an example to simply apply it.

EX1: A savings fund with an initial value of $1000, with a monthly interest rate of 1%, set its value to a, and a value of 1010,1020.10,10030.30 after n months respectively.

Establish first-order differential:

a0 = A1–A0 = 10

A1 = A2–A1 = 10.10

a2 = A3–A2 = 10.20

Not difficult to establish nth first order differential: an = A (n+1) –an = 0.01an

So we can write the general formula of the difference equation:

A (n+1) = 1.01an

At this point we add the initial conditions of the a0=1000 and we get the dynamical system model.

The so-called dynamical system, is a can express infinite number of algebra, it can characterize the adjacent discrete time point (that is, within a period) of the change situation. It is very easy to see that, with the difference equation, we know an item, we can calculate its next item, but we cannot directly calculate the value of a particular item.

"A first Course in mathematical Modeling" Modeling of-chaper1-difference equation