In this chapter we consider space-time coupled classical methods of approximation for initial value problems (IVPs) in which the space-time domain Ω ¯ x t $ \bar{\iOmega }_{xt} $ https://s3-euw1-ap-pe-df-pch-content-public-u.s3.eu-west-1.amazonaws.com/9781351270007/9d4f0e6c-f4ce-4b85-935c-7f3bf23290eb/content/inline-math3_1.tif"/> of the IVP is not discretized. By space-time coupled methods we mean concurrent dependence of all quantities of interest on spatial coordinates as well as time, which is in agreement with the physics of the evolution described by the governing differential equations (GDEs) constituting the IVPs. In order to present development of a general mathematical framework for space-time classical methods of approximation for all IVPs regardless of their origin or field of application, we must consider mathematical classification of all space-time differential operators into distinct categories (Chapter 2), non-self-adjoint and non-linear, and then undertake development of the methods of approximation for these using nondiscretized space-time domain Ω ¯ x t $ \bar{\iOmega }_{xt} $ https://s3-euw1-ap-pe-df-pch-content-public-u.s3.eu-west-1.amazonaws.com/9781351270007/9d4f0e6c-f4ce-4b85-935c-7f3bf23290eb/content/inline-math3_2.tif"/> .