Extended two-dimensional DTM and its application on nonlinear PDEs with proportional delay

In this work, we successfully extended two-dimensional differential transform method and their reduced form, by presenting and proving some theorems, to obtain the solution of partial differential equations (PDEs) with proportional delay in t and shrinking in x. Theorems are presented in the most general form to cover a wide range of PDEs, being linear or nonlinear and constant or variable coefficient. In order to show the power and robustness of the present methods and to illustrate the pertinent features of related theorems, some examples are presented.