Computational Materials Engineering

9

Finite Elements for Microstructure Evolution

Dierk Raabe

9.1 Fundamentals of Differential Equations

9.1.1 Introduction to Differential Equations

Many of the laws encountered in materials science are most conveniently formulated in terms of differential equations. Deriving and solving differential equations are thus among the most common tasks in modeling material systems.

Differential equations are equations involving one or more scalar or tensorial dependent variables, independent variables, unknown functions of these variables, and their corresponding derivatives. Equations which involve unknown functions that depend on only one independent variable are referred to as ordinary differential equations. If the equations involve unknown ...