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## 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