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One must be careful to understand the exact question being asked, which is not made easy by the book, which is not very clear, despite the author's claim that it is easy.
The key question is "polynomial in what variable?". I have to assume that the intention is to write it as a polynomial in terms of [latex]x_r [/latex]. If that is the case, we can take your example and make it clear.
Hopefully, now it is clear because if one expands the [latex](x_3+\Delta x)^3 [/latex] using the Binomial theorem, it's obvious that the highest order term will subtract out and only powers of n-1 are left.
The value "r" is used as an integer index, hence the adjacent value is always a difference of 1 unit. The variable x, or t might be used for the actual distance or time values, and then you can talk about x+h or t+h, and x-h and t-h. Basically, it is customary to discretize continuous time variables and convert them into an integer domain, as typical in discrete-domain mathematics.
Think back to the discrete time notation x[n]=x(nT), where T is the sample time. This is sometimes notated at [latex] x_n[/latex]
I'm sorry for asking this now. I remember when I first read your reply a couple of days ago it made sense to me but now when I was going through the stuff again I'm stuck and confused. I'm sorry.
You could have a look on the first post again and I have quoted your reply below.
One must be careful to understand the exact question being asked, which is not made easy by the book, which is not very clear, despite the author's claim that it is easy.
The key question is "polynomial in what variable?". I have to assume that the intention is to write it as a polynomial in terms of [latex]x_r [/latex]. If that is the case, we can take your example and make it clear.
Hopefully, now it is clear because if one expands the [latex](x_3+\Delta x)^3 [/latex] using the Binomial theorem, it's obvious that the highest order term will subtract out and only powers of n-1 are left.
First, you asked how I knew x4=x3+Δx. That is simply a result of the fact that we define the grid spacing as Δx, so all adjacent points on the grid differ by Δx. If you are asking, how I know to use this known fact, remember I said before that it is just my guess that the problem is asking for a polynomial in xr, which is x3 in this case. Hence, it makes sense to express x4 as a function of x3.
Your main question is if you are doing the math correctly. I think that you are, but there seems to be a couple of typos there. I've attached a picture of the result using the symbolic tool "mupad" in Matlab. If you aren't familiar with mupad yet, then try this example to get a start on learning this useful tool.
In short, delta x is a constant so this:
3*dx*x^2+3*dx^2*x+dx^3
turns into this for example:
3*x^2+3*x+1
which is clearly second degree.
It works like derivatives. When you take the derivative of a third degree poly you get a second degree poly, take the derivative of a second degree you get a first degree, and finally taking the derivative of a first degree you get a constant. In fact, you'll note that the differences are what you would start with if you were going to take simple numerical derivatives with step size delta x:
(f(x+deltaX)-f(x))/deltaX which is the same as (y2-y1)/deltaX, the differences being y2-y1.
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