The notations are as follows: \begin{align*} & \operatorname{Diff}^+(\mathbb{D}^2):= \{ f:\mathbb{D}^2\to \mathbb{D}^2\ |\ f \text{ is an orientation preserving diffeomorphism} \},\\ & \operatorname{Diff}^+(\mathbb{S}^1):=\{ f:\mathbb{S}^1\to \mathbb{S}^1\ |\ f \text{ is an orientation preserving diffeomorphism} \},\\ & \operatorname{Diff}^+(\mathbb{D}^2_\partial):=\{ f \in \operatorname{Diff}^+(\mathbb{D}^2)\ |\ f |_{\partial \mathbb{D}^2}=id \}. \end{align*} Consider the following: $$ \operatorname{Diff}^+(\mathbb{D}^2_\partial)\xrightarrow{i} \operatorname{Diff}^+(\mathbb{D}^2) \xrightarrow{\pi} \operatorname{Diff}^+(\mathbb{S}^1). $$ I need to show that this is a fiber bundle with fiber $\operatorname{Diff}^+(\mathbb{D}^2_\partial)$. Since every diffeomorphism of a circle can be extended to a diffeomorphism of a disc and hence the map $\pi$ is surjective and also I have proved that the fiber will be $\operatorname{Diff}^+(\mathbb{D}^2_\partial).$ Now I am having problem in proving the local trivialization. I am unable to take the open sets that will be suitable for local trivialization.
Your help will be really helpful for me.
Thanks.
I will work with $C^1$-smooth diffeomorphisms and with the $C^0$ topology. In this setting, the restriction map $\pi$ admits a global section $$ \beta: \operatorname{Diff}^+(\mathbb{S}^1) \to \operatorname{Diff}^+(\mathbb{D}^2) $$ given by the Douady-Earle extension, see
A. Douady, C. Earle, Conformally natural extension of homeomorphisms of the circle. Acta Math. 157 (1986), no. 1-2, 23–48.
In the paper they also prove $C^0$ continuity of $\beta$ (where the topology on the domain and the range of $\beta$ is that of uniform convergence) and that, moreover, higher derivatives of $\beta(f)$ on the open disk depend continuously on $f$.
Remark. The extension operator $\beta$ is defined on a larger class of homeomorphisms of the circle than diffeomorphisms, namely, quasisymmetric maps.
The fact that for each $C^1$-smooth diffeomorphism $f\in \operatorname{Diff}^+(\mathbb{S}^1)$ the map $\beta(f)$ is $C^1$-smooth on the closed disk is proven in PhD thesis of S.Pal:
Boundary and Holder regularities of Douady-Earle extensions and eigenvalues of Laplace operators acting on Riemann surfaces.
Once you have this global section, it follows that the mapping $\pi$ is a trivial fiber bundle: The projection $$ \operatorname{Diff}^+(\mathbb{D}^2)\to \operatorname{Diff}^+(\mathbb{D}^2_\partial) $$ is given by $$ F\mapsto F\circ (\beta(\pi(F)))^{-1}. $$
If you want to get higher regularity, I suggest going through the proofs of the cited papers and checking the estimates.
Edit. As you can see from the references, proving local triviality of infinite-dimensional fibrations is a nontrivial task. But one is frequently interested in only proving that a certain map is a fibration in the sense of Hurewicz or Serre. This is much easier. For instance, the following holds (I work in $C^\infty$ category):
Let $M$ be a smooth compact manifold with boundary. Then the sequence $$ Diff(M,\partial M)\to Diff(M)\to Diff(\partial M) $$ is a Hurewicz fibration.