Modern methods for solving partial differential equations of parabolic type and their mathematical theoretical foundations

In this paper, the existence problem of weak solutions in partial differential equations is first discussed. Secondly, the parabolic-type partial differential equation (PDE) model with non-chiral terms is used to portray the coupling principle of microwave heating dielectric medium, and the approximate models of TEM and TE10-wave microwave heating are constructed and verified by comparing with the COMSOL model in simulation. An open-loop type iterative learning control algorithm is proposed for the distributed parameter system, and the effectiveness of the control algorithm proposed in this paper is theoretically demonstrated. It is concluded that the TEM and TE10-wave microwave heating approximation models are more successful in describing the microwave heating process than the COMSOL model, and the consistency of the simulation results fully proves the correctness and effectiveness of the proposed model. The paradigm of the system output error decreases with the increase in the number of system runs, and the paradigm of the system output error tends to be close to zero.