This paper studies the initial-boundary value problem of a class of nonlinear time-fractional parabolic equations, where the fractional derivative used is in the sense of \(\psi \) -Caputo derivative of order \(\alpha \in (0,1)\) . By using the modified \(\psi \) -Laplace transform and Fourier sine transform, the mild solution of the equation is derived. When the initial value is in an appropriate space and small enough, the global existence and uniqueness of this mild solution are proved. Furthermore, under some appropriate assumptions on the initial conditions, it is proved that when \(\alpha \rightarrow 1^-\) , the mild solution of the time-fractional equation will converge to the mild solution of its classical corresponding problem. These conclusions are applicable not only to the Burgers equation but also to the Navier–Stokes equations. Finally, taking the Navier–Stokes equations as an example, the convergence is verified through numerical simulation.
