Mathematical Methods (10/24.539) General Initial Value Problems (IVPs) Introduction

Mathematical Methods (10/24.539)

IV. General Initial Value Problems (IVPs)

Introduction

Engineering systems are described in mathematical terms via balance equations (energy balances, mass balances, force balances, etc.) and these conservation relationships invariably lead to differential equations. For systems with only one independent variable, we only have to work with ordinary differential equations (as done in Section I and Section II). However, we often have several dependent variables of interest, and these usually give rise to a set of coupled ODEs. The matrix notation and fundamental matrix operations reviewed in Section III give us the tools necessary to deal with a coupled system of differential equations in an efficient manner (the matrix equations can consist of either algebraic or differential equations).

In this section we focus on a particular class of problems known as Initial Value Problems (IVPs) which occur frequently in practical applications. This subject was included indirectly as part of our discussions in Sections I and II, but here we emphasize how to work with coupled systems of first order ordinary differential equations. We can also solve IVPs written in the form of high order ODEs by simply converting the n^th order system into n coupled first order equations. This approach gives us a common basis for solving all IVPs with only one independent variable (PDEs which deal with two or more independent variables are discussed in later sections).

IVPs that are linear and have constant coefficients can be solved analytically. This is an important class of problems and applications from this set generally have relatively straightforward solution schemes that are based on the matrix eigenvalue and eigenvector concepts reviewed in Section III. However, many real engineering systems of interest have variable coefficients or nonlinear interactions and, in general, these are not readily amenable to analytical techniques. Thus, this section of notes also emphasizes general numerical schemes that can be applied in most situations that arise. The full treatment of both analytical and numerical solution strategies in tandem should give the student all the necessary tools to handle just about any situation that would arise in practice. Detailed illustrative examples of both the analytical and numerical solution methods are also given to help develop confidence in this area. A good understanding of this subject is essential to engineers and scientists involved in the modeling and simulation of physical systems.

This section of notes overviews the basic concepts, analytical and numerical solution techniques, and the accompanying illustrative IVP examples within several subsections, as follows: