Difference equation --> differential equation

Jens

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Nov 27, 2019
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Hi folks,

I have a bit of a hard time finding the right math theory behind the following transformation of the difference equation into a differential one ('delta' means the difference):

Difference equation: 'delta'xi(k) = pij(xj(k)-xi(k))
Differential equation according the scientific article: xi'(t) = pij(xj(t)-xi(t)) -- they say they assume infinitesimal changes and use taylor expansion to derive this

For the record, xi and xj mean the strategy of player i and j, respectively. pij is a transition probability.

Does someone have an idea?
 

HallsofIvy

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The "difference", \(\displaystyle \Delta(x_i(k))\) is, by definition, \(\displaystyle x_i(k+1)- x_i(k)\). We can think of that as \(\displaystyle \frac{x_i(k+1)- x_i(k)}{1}\) where the denominator, 1, is "(k+1)- k". If we allow "k" to be non-integer and write \(\displaystyle \Delta k\) to be the difference between successive "k"s (no longer 1, may be less than 1) then we have \(\displaystyle \frac{x_i(k+\Delta k)- x_i(k)}{\Delta k}\). That is the "difference quotient" and the derivative is the limit as \(\displaystyle \Delta k\) goes to 0 (I would nor use the word "infinitesimal").
 
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Jens

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Thanks for your reply. The part that is not yet clear to me is that when the derivate w.r.t. time is obtained, one can replace the k's for t's. Is this because we allow k to be non-integer so that for each point in time (e.g. 0.01 seconds) the equation makes sense?
 

Subhotosh Khan

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Hi folks,

I have a bit of a hard time finding the right math theory behind the following transformation of the difference equation into a differential one ('delta' means the difference):

Difference equation: 'delta'xi(k) = pij(xj(k)-xi(k))
Differential equation according the scientific article: xi'(t) = pij(xj(t)-xi(t)) -- they say they assume infinitesimal changes and use taylor expansion to derive this

For the record, xi and xj mean the strategy of player i and j, respectively. pij is a transition probability.

Does someone have an idea?
It will help if you provide reference to the article mentioned.
 

HallsofIvy

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All they have done is change the integer step, k, to the continuous variable, t, and change the difference, \(\displaystyle \Delta x_i(k)\) to the derivative, \(\displaystyle \dot{x_i}(t)\).
 

Jens

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Nov 27, 2019
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If I rewrite that in a few more lines, I do not get the same result. I added a picture of my 'thinking' and the explanation of the article. Where does the interval in the numerator come from? It appears out of the blue to me
 

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