Differential Equations

Abstract

Real systems change with time, so engineers must understand the analysis and modeling of dynamic systems. Real systems typically have values of interest that depend on the exact point where the value is evaluated. Engineering modeling often leads to equations that are called Ordinary Differential Equations (ODEs). ODEs have solutions that depend on one variable. The dependent variable could be time, position in reactor, distance from the particle center, distance from the center of a pipe, or any continuous value. For many ODEs, given the initial value for the variable and the ODEs that define the variable, numerical methods can be used to calculate the variable value as it changes with time. In some cases, the initial and final values may be known, but the initial velocity for the variable is unknown; shooting methods can be used to solve these types of equations. Partial Differential Equations (PDEs) result from more advanced modeling of real-world systems. When a model results in variables having more than one independent variable, the problem is described as a PDE. The variable of interest could depend on multiple spatial dimensions. In dynamic systems, the variable could additionally depend on time. The simplest approach to numerically solve PDEs is often to approximate derivative terms with finite difference approximations at a number of fixed locations. More advanced methods such as finite element methods will generalize the solution to non-regular geometry. Flowing systems can be modeled using finite volume approximations. Rather than developing these methods in great detail, software such as COMSOL can be used to readily simulate complex PDE systems.