Mathematical Methods for Physics and Engineering : A Comprehensive Guide

PDES: GENERAL AND PARTICULAR SOLUTIONS

0

ct

x+ct=constant

x−ct=constant

L x

Figure 20.5 The characteristics for theone-dimensional wave equation. The

shaded region indicates the region over which the solution is determined by

specifying Cauchy boundary conditions att= 0 on the line segmentx=0to

x=L.

Find the characteristics of the one-dimensional wave equation

∂^2 u

∂x^2

−

1

c^2

∂^2 u

∂t^2

=0.

This is a hyperbolic equation withA=1,B=0andC=− 1 /c^2. Therefore from (20.44)
the characteristics are given by
(
dx
dt

) 2

=c^2 ,

and so the characteristics are the straight linesx−ct=constantandx+ct=constant.

The characteristics of second-order PDEs can be considered as the curves along

whichpartialinformation about the solutionu(x, y) ‘propagates’. Consider a point

in the space that has the independent variables as its coordinates; unless both

of the two characteristics that pass through the point intersect the curve along

which the boundary conditions are specified, the solution will not be determined

at that point. In particular, if the equation is hyperbolic, so that we obtain two

families of real characteristics in thexy-plane, then Cauchy boundary conditions

propagate partial information concerning the solution along the characteristics,

belonging to each family, that intersect the boundary curveC.Thesolutionu

is then specified in the region common to these two families of characteristics.

For instance, the characteristics of the hyperbolic one-dimensional wave equation

in the last example are shown in figure 20.5. By specifying Cauchy boundary