and here's the explanation post (or one of them, at least)!
welcome to... drumroll please... *drumroll*
fun math with sofia kovalevskaya !! (yayy!!)
Part 1: what are partial differential equations?
Partial differential equations are differential equations involving the following:
1 or more dependent variables
an unknown function dependent on the variables
partial derivatives of the function
Below is an example of a partial differential equation given the function f(x,y)!
Now, you might be asking: What is a partial derivative? Well, I'll get to that!
A partial derivative is a way of expressing the rate of change of a function depending on more than one variable.
A partial derivative is denoted by the ∂ symbol instead of a "d" like full derivatives. We use partial derivatives instead of full derivatives when a function partially depends on two or more variables (that are independent from each other). When getting the partial derivative of a function with respect to one of the variables the function depends on, we hold the other variables constant, then treat it like we would a full derivative. This only partially tells us about the rate of change of the function, since we assume the other variables are constant, hence the name "partial derivative."
The partial derivative of a function f(x,y) with respect to y (meaning that x is held constant) is shown below.
Now, what has my paper got to do with partial differential equations?
My paper built on Cauchy's findings on some partial differential equations having analytic solutions, and I was able to generalize this and prove that, with the right conditions, partial differential equations have unique solutions!!
Though this may not always work, especially for functions that are not analytic, this is still very cool, don't you think?
I hope you all enjoyed my lengthy ramble, and that's it for part 1 of fun math with sofia kovalevskaya!!! stay tuned for the others!
