ABSTRACT
This chapter presents some basic elements from various areas of applied mathematics: linear algebra, space-time functions, functional analysis, function spaces, theory of functions, space-time differential operators, and calculus of variations. These elements provide the necessary concepts and basic principles for developing a mathematical framework and associated computational infrastructure for finite element processes associated with initial value problems. A space is a collection of objects that share a certain common feature or property and follow a set of rules. Integration by parts in space and time is an important calculus tool that allows one to transfer differentiation from one function to another with respect to space as well as time in an integral representation. Space-time functionals are naturally functions of spatial coordinates as well as time. Such functionals play an important role in obtaining approximations of the theoretical solutions of initial value problems encountered in mathematical physics, science, and engineering.
