Mathematical Methods for Physics and Engineering - Matematica.NET

PDES: GENERAL AND PARTICULAR SOLUTIONSctx − ct =constant0Lxx + ct =constantFigure 20.5 The characteristics for the one-dimensional wave equation. Theshaded region indicates the region over which the solution is determined byspecifying Cauchy boundary conditions at t = 0 on the line segment x =0tox = L.◮Find the characteristics of the one-dimensional wave equation∂ 2 u∂x − 1 ∂ 2 u2 c 2 ∂t =0. 2This is a hyperbolic equation with A =1,B =0andC = −1/c 2 . Therefore from (20.44)the characteristics are given by( ) 2 dx= c 2 ,dtand so the characteristics are the straight lines x − ct =constantandx + ct =constant.◭The characteristics of second-order PDEs can be considered as the curves alongwhich partial information about the solution u(x, y) ‘propagates’. Consider a pointin the space that has the independent variables as its coordinates; unless bothof the two characteristics that pass through the point intersect the curve alongwhich the boundary conditions are specified, the solution will not be determinedat that point. In particular, if the equation is hyperbolic, so that we obtain twofamilies of real characteristics in the xy-plane, then Cauchy boundary conditionspropagate partial information concerning the solution along the characteristics,belonging to each family, that intersect the boundary curve C. Thesolutionuis then specified in the region common to these two families of characteristics.For instance, the characteristics of the hyperbolic one-dimensional wave equationin the last example are shown in figure 20.5. By specifying Cauchy boundary704