What are Research and developement methodologies? or how to deal with the programming challenges?

My team is a set of web developers and it totally deals with the projects and issues the had already been doing. But when it comes to the projects "to-grow" where no one had any experience (new technology, new framework, or unknown type of problem) it results that know one is cabable give any estimations or any warranties for result (within the given budget, it is always limited).

So, there are R'n'D departments in many companies than perform such kind of work. They experiment and find proved way for solutions but they does not do actual product developement. The quation is, which are methodologies of Research and Developement? Is the there a better way of programming researching rather than do some random experiments and not to be sure if it gives the result?