Odd Differencing Method for Solving DE

• Kreizhn
In summary, my friend is solving a differential equation using data from Earth Sciences, and appears to be getting better results by dropping the term x_{i-1} from the equation. This equation looks exactly like an Euler method, but with some sort of weird scaling on the solution.
Kreizhn
Hey all,

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
$$\frac{dy}{dx} = f(x)$$
Let [0,L] be the interval over which this is to be solved, $y(0) = y_0$ and take a partition $0 =x_0 < x_1 < \cdots < x_{n-1} < x_n = L$. Then it appears that his differencing method is giving the approximation of the $(i+1)^{st}$ value as
$$y(x_{i+1}) = \frac{ f(x_{i+1}) (x_{i+1} - x_i) - \left(\sum_{j=0}^i y(x_j)\right) (x_i - x_{i-1}) }{x_{i+1}}$$
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 $y(x_i)$. 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.

Are you sure it's a y(x_j) in the sum and not f(x_j)? Given the DE, y(x_j) makes no sense at all.

EDIT: No, actually, it doesn't make sense either way.

Just checked the data. That summation is wrong, but it does represent the solution at the previous partition point. This would be $y(x_i)$.

It almost looks like an Euler method with some sort of bizarre scaling. We can re-arrange it to get

$$y(x_{i+1}) x_{i+1} = y(x_i)(x_i -x_{i-1}) + f(x_{i+1}) (x_{i+1} - x_i)$$

My friend now tells me that if we drop the $x_{i-1}$ term from the y(x_i) he gets better results. This would leave

$$y(x_{i+1}) x_{i+1} = y(x_i)x_i + f(x_{i+1}) (x_{i+1} - x_i)$$

Which looks exactly like an Euler method with some sort of weird scaling on the solution.

Looks more like Adams-Bashforth than Euler to me.

Isn't one-step AB just Euler?

1. What is the Odd Differencing Method for solving differential equations?

The Odd Differencing Method is a numerical method used to approximate the solution of a differential equation by taking the average of the left and right derivatives at each point. It is based on the idea that the derivative of an odd function is even, and vice versa.

2. How does the Odd Differencing Method work?

The Odd Differencing Method involves dividing the interval of the differential equation into smaller sub-intervals and computing the average of the left and right derivatives at each sub-interval. This average is then used to approximate the value of the function at the midpoint of the sub-interval. This process is repeated until the desired accuracy is achieved.

3. What are the advantages of using the Odd Differencing Method?

The Odd Differencing Method is easy to implement and does not require advanced mathematical knowledge. It is also suitable for solving a wide range of differential equations, including those with discontinuous or non-smooth solutions. Additionally, it can provide accurate approximations even when the step size is relatively large.

4. Are there any limitations to the Odd Differencing Method?

Like any numerical method, the accuracy of the Odd Differencing Method depends on the choice of step size. If the step size is too large, the approximation may not be accurate enough. Additionally, this method may not be suitable for differential equations with highly oscillatory solutions.

5. Can the Odd Differencing Method be used for systems of differential equations?

Yes, the Odd Differencing Method can be extended to solve systems of differential equations by applying it to each equation separately. However, this can be computationally intensive and may not be the most efficient method for solving systems of differential equations.

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