Definite Article with Prepositional Phrases

I am trying to learn how to correctly use articles in English. It is a bit hard for me since I do not have this concept in my native language.

I am confused by two rules: the first rule is to use "a" whenever you introduce things, while the second one is to use "the" with prepositional phrases. I am wondering how I can identify when a prepositional phrase makes a noun definite.

For instance, which article shall I use in the next sentence?

I am working on a(the) Web application to assess (the) current performance of traffic companies.

I came up with a(the) solution that was based on my previous experience to solve our current problem.

On one hand, I mention the Web application for the first time. On the other hand, you know what Web application I am talking about (to assess the current performance of traffic companies).

Thanks in advance!

Answer

In contexts like these, use "a". Even if the hearer is familiar with the particular app, they don't know that that is the subject of the conversation until you introduce it.

The only time you might use "the" here is if a collection of which the app is a part has already been introduced in the conversation; eg

My boss let me choose which parts of our new software suite to contribute to. I'm working on the Web application to assess the current performance of traffic companies.

In that context "a" and "the" would both be possible, with "the" suggesting that the hearer already knows something about what is in the software suite, and "a" suggesting that they don't.

I'm not sure what your rule is about 'to use "the" with prepositional phrases': I don't recognise that as a rule of English at all.