Summary: An operator-based approach is used here to prove the existence and uniqueness of a strong solution \(u\) to the time-varying nonlinear Fokker-Planck equation \[ \begin{aligned} u_t(t,x)-\Delta (a(t,x,u(t,x))u(t,x))+\operatorname{div}(b(t,x,u(t,x))u(t,x))=0 \text{ in } (0,{\infty})\times \mathbb{R}^d, u(0,x)=u_0(x), x\in \mathbb{R}^d \end{aligned} \] in the Sobolev space \(H^{-1}(\mathbb{R}^d)\), under appropriate conditions on \(a:[0,T]\times \mathbb{R}^d\times \mathbb{R}\to \mathbb{R}\) and \(b:[0,T]\times \mathbb{R}^d\times \mathbb{R}\to \mathbb{R}^d.\) It is proved also that if \(u_0\) is a density of a probability measure, so is \(u(t,\cdot)\) for all \(t\geq 0\). Moreover, we construct a weak solution to the McKean-Vlasov SDE associated with the Fokker-Planck equation such that \(u(t)\) is the density of its time marginal law.