I am putting my effort into proposing a novel method to solve a set of new problems in my field which have never been defined and solved before. These problems are almost similar to existing problems that are solved by different methods proposed by researchers over a long period of time. However solutions for the new problems are different than those of existing problems and also these new set of problems will be evaluated by existing methods for verification of my novel proposed method. My question is that: Is this considered a contribution or not? Can reviewers point out that why am I not solving a set of famous existing problems with my novel method?
You can always define new problems and provide answers to them. It's your choice what you do with your time and intellectual ability.
The question you don't, but should, ask is whether that meaningfully advances science. In my field of computer simulations, for example, you can always come up with a new finite element and then ask the question of whether it converges. That is, in principle, a valid scientific question that can be answered. But I suspect that few will care because there are already a million finite elements and we have very good existing methods -- we don't really need any new ones; as a consequence, you would probably have a hard time publishing this work in a good journal.
In other words, when you ask whether you can define variations on previous problems and propose solutions to them, the answer is "yes" -- but you are not entitled to anyone caring about these variations, or entitled to get them published. That depends on whether these problems are of practical relevance or otherwise of interest to your community.
If you can successfully do that, and few can, it would be a good thing. Reviewer reaction can't be predicted in advance, however. But a good reviewer should, with suitable hints from you, say in an introduction to a paper, be able to make some judgement as to the possible impact and importance of new ideas.
In math, for example, variations on a theme can be welcome (as was my dissertation). And new proof methods can give insight beyond the specific application. A reviewer skilled in the general topic area should, in principle, be able to recognize this.
Note that both novelty and potential impact are important here. Some new but trivial variations might be interesting but only require a few moments of thought and a smile or two, but have no lasting impact on a field.
So, "can you do it" is subtle. You "can" if you are able, but it also requires that you meet some general, perhaps diffuse, criteria of impact. Neither of these can be predicted in advance.