EDIT: Thanks to the first answer, I may see that there is differences between a giant solenoid and a completely uniform magnetic field. Therefore it would be great if one can explain to me both the case.
Let's say there is an uniform magnetic field inside a very long and large solenoid. The magnetic field is slowly decreased. How can we find the electric field?
From the Maxwell-Faraday Equation we have $\nabla \times E=-\frac{\partial B}{\partial t}$ and it's can be solved by a curly electric field with the value of electric field is constant (proportional to $|\frac{\partial B}{\partial t}|$) and directed to circles around a centre. But the centre is arbitrarily (one may say it's the central of the solenoid but I find it uncompelling since any centre will satisfy Maxwell equation). So how can one find the electric field in this situation?