The Painlevé equation is a typical nonlinear differential equation and has a wide range of applications. The complex method combines complex analysis and differential equation theory, and is an effective method for solving meromorphic solutions of nonlinear differential equations. The complex method is used to prove that meromorphic solutions of generalized algebraic differential equations related to the third Painlevé equation belong to class $W$, and meromorphic solutions of the mentioned equation were obtained. Through appropriate transformations and application of the obtained results, class $W$ meromorphic exact solutions for the modified double sine Gordon equation and modified Kortweg de Vries (mKdV) equation were derived, and non class $W$ meromorphic exact solutions for the relevant nonlinear partial differential equations were obtained. It is well known that based on the suitable transformations and these results, it is convenient to seek meromorphic exact solutions of other related mathematical and physical equations. In future work, further research can be conducted on the meromorphic solutions and applications of generalized algebraic differential equations related to other categories of the Painlevé equation.