Mathematical Methods (10/24.539) General Boundary Value Problems (BVPs) Introduction

Mathematical Methods (10/24.539)

V. General Boundary Value Problems (BVPs)

Introduction

In the previous section we focused on various schemes (both analytical and numerical) for solving general IVPs. We now emphasize another important class of problems known as Boundary Value Problems (BVPs). The difference between these two problem classes is related to the specification of the n conditions needed to uniquely determine the n arbitrary coefficients in the general solution to an n^th order ODE. The n conditions are specified at a single (initial) point for IVPs. However, for a BVP, the n boundary conditions are specified at two or more points in the domain of interest. This modification results in completely different algorithms for solving these problems - especially in the numerical solution schemes used for BVPs.

Analytical schemes for linear constant coefficient non-homogeneous BVPs have already been discussed in Section II - Linear Differential Equations. Exact solution methods for general nonlinear BVPs are simply not available (although a number of ‘tricks’ can be applied to specific problems). Variable coefficient linear problems do have a general solution procedure - referred to as the Power Series Solution Method - but it is often quite tedious and not overly useful for solving general engineering design problems. The Power Series technique is quite powerful, however, and it is very useful for deriving analytical solutions for several important problems. We will discuss this method in some detail later in this course (see Section VII - Power Series Solution Method and Section VIII - Special Functions). Note also that Eigenvalue Problems, a special class of homogeneous BVPs, are discussed later in these notes (see Section IX - The Sturm-Liouville Problem). For now, we are interested in the solution of non-homogeneous BVPs that occur in real engineering applications, and the only general solution method for these problems rely on numerical schemes.

In particular, in this section of notes we focus on two numerical techniques for treating general non-homogeneous BVPs with a single independent variable - the Shooting Method and the Finite Difference Method.

The Shooting Method is essentially an iterative application of the numerical integration techniques used for IVPs. It is easy to apply in most cases, but it is not appropriate for solving BVPs with more that one independent variable (PDEs are discussed later in this course).

The Finite Difference Method takes a completely different approach to the problem. It essentially converts the ODE into a coupled set of algebraic equations, with one balance equation for each finite volume or node in the system. This technique is very powerful, and it is easily extended to multidimensional and space-time situations. Thus, this method will be used again when discussing the numerical solution techniques appropriate for general PDEs (later in the course).

The general notation associated with BVPs and an overview of the Shooting and Finite Difference methods are discussed within the following subsections: