Numerical Methods for Solving Partial Differential Equations by George F. Pinder

Author:George F. Pinder
Language: eng
Format: epub, mobi
ISBN: 9781119316381
Publisher: Wiley
Published: 2017-01-29T00:00:00+00:00

Chapter 6

Initial Boundary-Value Problems

6.1 Introduction

In this chapter we are interested in solving problems that have both a spatial and temporal component. As a point of departure, let us consider the equation

(6.1)

A typical initial condition for this equation would be to specify the state of the system at some time designated as zero such as

(6.2)

and a typical boundary condition that could be a function of time as

(6.3)

6.2 Two Dimensional Polynomial Approximations

To achieve this goal we need to extend our early Lagrange polynomial approximation strategy to consider two dimensions, namely space and time. To this end we define the following quantities:

(6.4)

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