What's the solution to this difference equation?

In summary, the conversation discusses a nonlinear difference equation given by an+1=2an where n is a natural number and a0 is a fixed real number. The conversation also explores the possibility of letting n=1/2 and finding a function such that f(f(x))=2^x. The conversation mentions various shorthand notations for expressing the equation and discusses the order of operations in calculating the equation.
  • #1
phoenixthoth
1,605
2
an+1=2an where n is a natural number and a0 is some fixed real number. (in case it's not clear, what's on the right hand side is 2 to the an power.) thanks!

i'm wondering if it will be possible to let n=1/2 and have the half iterate to 2^x i was looking for earlier, where by that i mean a function such that f(f(x))=2^x. I'm guessing that solving one problem would be roughly as difficult as the other...

if you happen to know of a good article on nonlinear difference equations, i'd appreciate it. I'm also looking at the logistic.
 
Last edited:
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  • #2
Originally posted by phoenixthoth
an+1=2an where n is a natural number and a0 is some fixed real number.

Well,

[tex]
\begin{equation*}
\begin{split}
a_1 &= 2^{a_0}\\
a_2 &= 2^{a_1} = 2^{2^{a_0}}\\
a_3 &= 2^{a_2} = 2^{2^{2^{a_0}}}\\
&\cdots
\end{split}
\end{equation*}
[/tex]

That's about as simple as you can express it. You could write it with Knuth's or Conway's shorthand notation, or maybe with Ackermann's function.

http://www-users.cs.york.ac.uk/~susan/cyc/b/big.htm
 
  • #3
an+1 = 2an = 22an-1 = ...

And since: abc = abc

an = 22a0(n - 1)

I think. (And it seems to work too for a few calculations... the numbers do quickly get out of hand though.)
 
Last edited:
  • #4
You have the order of operations wrong.

Correct:

[tex]
a^{b^c} = a^{\left( b^c\right)}
[/tex]

wrong

[tex]
a^{b^c} = \left( a^b \right) ^c
[/tex]
 

FAQ: What's the solution to this difference equation?

What is a difference equation?

A difference equation is a mathematical equation that describes the relationship between the values of a sequence or function at different points in time or space. It is commonly used in various fields of science and engineering to model dynamic systems or processes.

How is a difference equation different from a differential equation?

A difference equation involves discrete variables, whereas a differential equation involves continuous variables. This means that a difference equation describes changes that occur at specific points in time or space, while a differential equation describes changes that occur continuously.

What is the solution to a difference equation?

The solution to a difference equation is a function that satisfies the equation for all values of the independent variable. It represents the relationship between the values of the sequence or function at different points in time or space.

How is the solution to a difference equation determined?

The solution to a difference equation can be determined through various methods, such as using algebraic techniques or numerical methods. The specific method used will depend on the complexity of the equation and the desired level of accuracy.

Can a difference equation accurately model real-world phenomena?

Yes, difference equations can accurately model real-world phenomena. They are commonly used in fields such as physics, biology, economics, and engineering to study and predict the behavior of dynamic systems or processes. However, the accuracy of the model depends on the assumptions and limitations of the equation and the accuracy of the input data.

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