What is the difference between perturbation theory and numerical analysis?
Both subjects are trying to obtain the approximate answer.
What are they study specifically?
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First, the broad concept of numerical analysis is that we are studying algorithms to obtain some kind of approximation (numerical in nature) to problems in pure maths (particularly analysis). It has applications in huge swathes of pure and applied mathematics.
A couple of examples of topics in numerical analysis are:
Numerical linear algebra: This is the study of algorithms that will solve problems in linear algebra that we cannot do by hand (such as extensive matrix calculations)
Finite element methods: This topic is used to find an approximation to PDE boundary value problems (BVPs)
On the other hand, perturbation theory is also trying to find an approximate solution to a given problem, but looking at a related problem with an exact solution. What if there was another parameter (say $\epsilon$)/small term that would mean the solution is available by setting $\epsilon =0$? Perturbation theory tells us how the solution will change for arbitrarily small $\epsilon$.
Perturbation theory is closely related to numerical analysis, and can in fact be considered a sub-topic of numerical analysis.
I am not sure exactly what kind of answer you are looking for. Are there any topics you wanted to know about in depth?