A method entitled the Elzaki Substitution Method (ESM) will be generalized in this article, an overview of the ESM for nth-order non-linear partial differential equations (NPDEs) involving the following three types of mixed derivatives: (where and are positive integers such that ) will be included. ESM technique often includes Adomian polynomial for non-linear terms to find exact solutions of non-linear homogeneous and non-homogeneous higher-order initial value problems containing mixed partial derivatives. Illustrative examples are presented in order to demonstrate the utility and applicability of the proposed procedure.