Next: Initial-boundary-value Problems Up: Method of Characteristics Previous: Method of Characteristics

## Initial-value Problems

Here the problem is defined over an infinite range in . The general statement of the problem is
 (1.11) (1.12)

The initial value problem, IVP, defined in Equations (1.11) and (1.12), is also referred to as a Cauchy problem and the solution is determined uniquely by the single condition at . From (1.10) we have

If is continuously differentiable, then and are continuous. Thus, is a classical solution of (1.11) and (1.12).

If is only piecewise continuous (derivatives are not continuous), e.g.

then is called a weak solution.

Example 1. .7

with the initial condition

The characteristic curves are given by the solutions to

and

Hence,

The PDE becomes

At , and . Therefore,

Example 1. .8

with the initial condition

The characteristic curves are given by the solutions to

and

Thus, eliminating and re-writing in terms of and , we obtain

The PDE reduces to the ODE

Using the initial condition, , , we have

and so

The solution is shown in Figure 1.5.

From Figure 1.5 it is clear that the wave moves to the left. This allows us to summarise the results of the first order wave equation.

Next: Initial-boundary-value Problems Up: Method of Characteristics Previous: Method of Characteristics
Prof. Alan Hood
2000-10-30