Introduction to Numerical Methods for Time Dependent Differential Equations

CHAPTER 8

1-PERIODIC SOLUTIONS OF TIME DEPENDENT PARTIAL DIFFERENTIAL EQUATIONS WITH CONSTANT COEFFICIENTS

We start this chapter by studying three simple examples of partial differential equations and their initial value problems in the 1-periodic case. These examples are fundamental as model equations and to introduce the concepts of well posedness and stability under lower-order perturbations. At the end we generalize these concepts to more general equations and systems of equations.

8.1 Examples of equations with simple wave solutions

8.1.1 One-way wave equation

Perhaps the simplest example is the one-way wave equation

(8.1)

with initial data:

(8.2)

We solve the problem by separation of variables; that is, we assume that

(8.3)

By (8.2),

Introducing (8.3) into (8.1) gives us

(8.4)

Therefore,

and by (8.3),

(8.5)

We can now use the results of Section 7.2 and the principle of superposition ...