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I have a friend working in Earth Sciences who appears to be doing something with DEs. If I understand his data correctly, it appears as though he's solving a differential equation of the form

[tex] \frac{dy}{dx} = f(x) [/tex]

Let [0,L] be the interval over which this is to be solved, [itex] y(0) = y_0 [/itex] and take a partition [itex] 0 =x_0 < x_1 < \cdots < x_{n-1} < x_n = L [/itex]. Then it appears that his differencing method is giving the approximation of the [itex] (i+1)^{st} [/itex] value as

[tex] y(x_{i+1}) = \frac{ f(x_{i+1}) (x_{i+1} - x_i) - \left(\sum_{j=0}^i y(x_j)\right) (x_i - x_{i-1}) }{x_{i+1}} [/tex]

I don't recognize the formula. The summation term would be theoretically reminiscent of an integral yes? Does anyone recognize this?

Edit: Sorry, that summation might actually only be [itex] y(x_i) [/itex]. I'm not quite sure yet since the data is a little fuzzy. If that's the case, this is almost an Euler method right? But it still doesn't quite seem there.

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# Odd Differencing Method for Solving DE

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