Partial differential equation
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When one writes a down a differential equation, inevitably it must pass through a stage of incompleteness before it is finished being written down (assuming that relativity is negligible). This is known as a partial differential equation.
This then immediately tells us that every differential equation must go through a stage of "partial-ness" before becoming a proper, fully-fledged differential equation. The most obvious way, then, to solve a partial differential equation is (as described in Cauchy's textbook "Moi, je suis magnifique avec les equations partials differentiels, et tres bon dans le lit" -- written in his native dialect of Badly Translated French) to finish writing it down.
However, this is not always possible. There are numerous things which can interrupt the writing of a differential equation: an attack of the hungries, absent-mindedness, the time on your car parking running out, terrorism, being seduced into bed by Cauchy, or the bar closing. In this case, the differential equation remains partial, and another mathemagician may find the partial differential equation and not know what you had in mind for it. (This is known as an elliptic partial differential equation, or in the jargon of mathematics, a Bendy Urkel.) In this case it is impossible to solve the partial differential equation by the Cauchy method, described above.
In these cases, a variety of alternatives exist. If a corresponding partial differential equation can be located, it is possible to sew the edges together to create a complete differential equation. Great strides are being made in this field thanks to the newly developed Sellotape Theory.
Alternatively, the partial differential equation can be solved by Green's Method, a development now credited to Cauchy. This method involves finding a so-called 'Delta Function' and corresponding 'Green's function'.
A Delta Function is defined to be zero everywhere, except for one bit where it is infinity. However, it still has an area underneath it, as proved in Cauchy's text "Je suis toujours vrai, parce que j'ai un pistolet." Because of these useful and practical properties, Delta Functions are considered by mathematicians to be of use in solving applied, real-world equations.
A Green's Function, on the other hand, consists of the original partial differential equation but with all the bits that don't work scribbled out and replaced with pictures of kittens. (This process is often referred to as a Fourier Transform, despite the fact that it's clearly something else entirely.) Some simple Applied Analysis now shows that the integral of the Green's Function multiplied by the Delta Function produces, give or take some Hand waving, an answer.
