With the development of the big data era, the need for computation power is dramatically growing, especially for solving partial differential equations (PDEs), because PDEs are often used to describe complex systems and phenomena in both science and engineering. However, it is still a great challenge for on-chip photonic solving of time-evolving PDEs because of the difficulties in big coefficient matrix photonic computing, high accuracy, and error accumulation. We overcome these challenges by realizing a microcomb-driven photonic chip and introducing time-division multiplexing and matrix partition techniques into PDE photonic solving, which can solve PDEs with a large coefficient matrix on a photonic chip with a limited size. Time-evolving PDEs, including the heat equation with the first order of time derivative, the wave equation with the second order of time derivative, and the nonlinear Burgers equation, are solved with an accuracy of up to 97%. Furthermore, the parallel solving of the Poisson equation and Laplace’s equation is demonstrated experimentally on a single chip, with an accuracy of 95.9% and 95.8%, respectively. We offer a powerful photonic platform for solving PDEs, which takes a step forward in the application of photonic chips in mathematical problems and will promote the development of on-chip photonic computing.