Should I write PDE or partial differential equation in the body of a mathematics paper? In a similar way, should I expand Dirichlet-to-Neumann map or write DN map or DtN map?
A somewhat related question Can I use abbreviations in the title of a research paper? considers only the title; I am more interested in the rest of the paper.
I use four factors to decide when an abbreviation should be used:
Is the "thing" known principally or exclusively by its abbreviation? DNA, for example, is almost always called "DNA"; no one mentally expands it to deoxyribonucleic acid every time they encounter it. In cases like these, I would find it off-putting if you insisted on spelling it out; in fact, it even seems a little weird to define it as in "We extracted deoxyribonucleic acid, or DNA, by...." I suspect that PDEs are also treated similarly, at least for some audiences.
Do you want to draw attention to one component of the abbreviation? If, for some reason, I were writing about the acidity of DNA or the partial-ness of PDEs, I would spell the words out. Otherwise, they can be abbreviated.
Do I want the reader to treat the term as a single entity? Abbreviations tend to reify a concept. Mentioning rodents of unusual size makes one think about mice and rats that happen to be at the far ends of a bell curve: they're rodents, but rather large (or small) ones. If we call them ROUSes instead, then one begins to think about them as a separate thing. What are the properties of an ROUS? Why are they here? Why aren't they "normal" rats?
How often is the term used? Each definition (and reference) of an abbreviation incurs a little bit of mental overhead. If you're only using the term occasionally, just write the words out (unless #1 applies). This is particularly true if the mentions are far apart. Pay attention to this when editing, since you may add or delete a lot of the mentions.
If the term is sufficiently long and sufficiently frequently used, it is reasonable to use abbrevations, as it may be very helpful for reading. I once reviewed a paper that was almost unreadable due to using some very long and complicated term all over the place (and I recommended introducing an abbrevation). For example, compare the following:
Using a solver for partial differential equations to solve a problem after applying the Dirichlet-to-Neumann map was first suggested in Ref. 42.
Using a PDE solver for a DtN-mapped problem was first suggested in Ref. 42.
However, you should explain the abbrevation the first time you are using it, e.g.:
partial differential equation (PDE)
The only exception from this are abbrevations that are extremely common like JPEG or Laser, i.e., abbrevations you could also use in the title of a paper. Depending on your field, PDE might be such an abbrevation.
As a sidenote: I recommend to decide about the usage of abbrevations and explaining them in a piecewise manner. For example if you use a term as follows:
I recommend not to use or introduce the abbrevation in section 1, but only use it in sections 3, 5 and 6, introducing it when it’s first used in section 3 and 5, respectively. Depending on how likely it is that somebody reads section 6 without reading section 5, it may also be wise to introduce it again at the beginning of section 6.
As a fellow mathematician, you can certainly use PDE without further explanation anywhere in the paper (title, abstract, body) but I would write "the DtN map (Dirichlet-to-Neumann)" in the first use and then DtN as suggested by the other answer. This is presuming you're submitting to a math journal.
My personal rule is always explain the abbreviation first, even for stuff that is supposed to be "known". As mentioned in the comments, the same abbreviation can have several different meanings, even in the same general field.
Another issue is not to use several abbreviations in the same sentence, which is even worse if some of them are similar... the objective is to make the reader understand, not get him dizzy with so many letters :)
