1 Introduction 1 Introduction Partial differential equations are at the foundation of much of computational science.
Most physical phenomena depend in complex ways on space and time.
Examples include fluid flow, heat transfer, nuclear and chemical and reactions and population dynamics. Computational scientists often seek to gain understanding of such phenomena by casting fundamental principles, such as conservation of mass, momentum and energy in the form of mathematical models of the underlying physical phenomena. Usually a mathematical model requires more than one independent variable to characterize the state of the physical system. For example, to describe a general fluid flow usually requires that the physical variables of interest, say pressure, density and velocity, be dependent on time and three space variables. This chapter is not intended to be a complete discussion of partial differential equations.