This is an enthusiastic question: Currently I'm studying Akkadian for fun, but as it is a dead language, the vocabulary may have some missing words/concepts. For example, I didn`t found a word for "steel", or some mathematical terms (I'm a mathematician). How could I borrow some modern word to akkadian? Form which language should I look for? (hebrew? arabic?) How could the word be "transformed" to look akkadian?
There are several possible approaches here, and which you choose will depend somewhat on what your goals are for this project.
The simplest option would be to borrow the term as a loanword from a globally hegemonic language of today (likely English), making only the phonetic adaptations necessary to fit Akkadian phonology (e.g. you don't have any dental fricatives /θ ð/, so would need to replace these with some other consonant, e.g. the alveolar stops /t d/).
With this approach, you might have a word for steel as šaṭīlu- (š being the only coronal fricative, the a being inserted to break up the illegal onset cluster, and the unaspirated t being borrowed as emphatic).
You can instead choose to borrow the word from a language that was hegemonic at some point in the past, likely when the new technology was introduced. If you're doing this though, you need to be aware of the stage in the language's development you're borrowing into.
With this approach, you could do what Arabic and Hebrew did and borrow Middle Persian pwlʾpt' (pōlāwad), possibly as something like pūlâdu-, and then apply whatever sound changes would be expected to occur since the borrowing (if any).
You could also coin a word based on native roots, looking at the etymologies for the missing words in related or nearby languages (this technique was used a lot in the revival of Hebrew).
With this approach, your word for steel might be kûnu- (possibly phonemically kônu-, if there was an o phoneme) from the verb kânu "to be firm", following the same etymology as that of English "steel".
The other of your major choices is to look at the word in related languages, and construct what the Akkadian cognate would be, had the word existed at the Proto-Semitic stage (something that certainly would not have been the case here, seeing as Proto-Semitic predates the discovery of steel or most of mathematics by quite some time).
Unfortunately I can't use "steel" as an example here very well, because most of the nearby Semitic languages borrowed their word from Persian. Instead lets look at "equation". Here the Arabic word is مُعَادَلَة muʿādala, a verbal noun of a Form III verb. Form III verbs in Arabic (which promote the indirect object of a Form I verb to their direct object) do not have a direct cognate class in Akkadian, but the Gt-stem has the closest semantics in this case. The equivalent construction then might be something like tedāltu-.
As I say, which approach you should take depends on your goals.
- Borrowing from a modern hegemonic language is probably the easiest if you just want to make it usable today, and aren't interested in some alternate history for the language where it survived.
- Borrowing from a formerly hegemonic language on the other hand, is good if you are interested in that alternate history, and intend to apply various sound changes from the historical Akkadian to your present day form.
- Coining the word from native roots also works well if you're not interested in alternate history, but is likely to make it harder for other people to learn. This method has historically been popular only when there is already a community who uses the language (in at least some contexts), rather than those where someone is trying to build the community from the ground up.
- Constructing plausible cognates tends to make the most sense for accidental gaps in the lexicon where a word did presumably exist, in other instances it would be pretty unusual.
Languages adopt new words (in the sense you're talking about) four ways:
Immigration: simply borrow a word from another language, with some phonological changes if necessary to adapt it to the new language. English, as possibly the greatest vocabulary thief in history, does this all the time. Examples are too numerous to mention.
Adaptation/Evolution: you take an existing word from a related concept and either use it directly with a new meaning, or modify it a bit to differentiate it from the original word and meaning, or simply have the concept itself change over time. As an example, the word "toilet" made the transition "cloth cover for folded clothes" > "cloth cover for a dressing table" > "things related to dressing" > "washing (ie, thing you do prior to dressing" > "room where you wash" > "room for personal hygiene" > "the ceramic thing you sit on to defecate".
Metaphor: use a word or phrase that has some kind of metaphorical association with what you want to name. As an example, "steel" comes from a Proto-Germanic word meaning something like "to stand fast" (ie, be resilient). A computer mouse comes from the shape and cord resembling a real mouse and its tail, as a more modern example.
Deliberate coinage: more likely a modern phenomenon, but it could happen. See, for example, "kleenex"