Rigorous $C_1$ integration of dissipative PDEs

We introduce a new $C^1$ algorithm for the rigorous integration of dissipative partial differential equations. The algorithm is designed for computer-assisted proofs that require rigorous control of both solutions and their derivatives with respect to initial data. As applications, we establish the existence of locally attracting periodic orbits for initial and boundary value problems for two non-autonomous dissipative PDEs: the Chafee-Infante equation and the Burgers equation with a fractional Laplacian.