Mathematical Methods for Physics and Engineering - Matematica.NET

PDES: GENERAL AND PARTICULAR SOLUTIONSwere discussed. Comparing (20.41) with (20.12) we see that the characteristics aremerely those curves along which p is constant.Since the partial derivatives ∂u/∂x and ∂u/∂y may be evaluated provided theboundary curve C does not lie along a characteristic, defining u(x, y) =φ(s)along C is sufficient to specify the solution to the original problem (equationplus boundary conditions) near the curve C, in terms of a Taylor expansionabout C. Therefore the characteristics can be considered as the curves alongwhich information about the solution u(x, y) ‘propagates’. This is best understoodby using an example.◮Find the general solution ofx ∂u ∂u− 2y∂x ∂y = 0 (20.42)that takes the value 2y +1 on the line x =1betweeny =0andy =1.We solved this problem in subsection 20.3.1 for the case where u(x, y) takes the value2y +1 along the entire line x = 1. We found then that the general solution to the equation(ignoring boundary conditions) is of the formu(x, y) =f(p) =f(x 2 y),for some arbitrary function f. Hence the characteristics of (20.42) are given by x 2 y = cwhere c is a constant; some of these curves are plotted in figure 20.2 for various values ofc. Furthermore, we found that the particular solution for which u(1,y)=2y +1for all ywas given byu(x, y) =2x 2 y +1.In the present case the value of x 2 y is fixed by the boundary conditions only betweeny =0andy = 1. However, since the characteristics are curves along which x 2 y, and hencef(x 2 y), remains constant, the solution is determined everywhere along any characteristicthat intersects the line segment denoting the boundary conditions. Thus u(x, y) =2x 2 y +1is the particular solution that holds in the shaded region in figure 20.2 (corresponding to0 ≤ c ≤ 1).Outside this region, however, the solution is not precisely specified, and any function ofthe formu(x, y) =2x 2 y +1+g(x 2 y)will satisfy both the equation and the boundary condition, provided g(p) = 0 for0 ≤ p ≤ 1. ◭In the above example the boundary curve was not itself a characteristic andfurthermore it crossed each characteristic once only. For a general boundary curveC this may not be the case. Firstly, if C is itself a characteristic (or is just a singlepoint) then information about the solution cannot ‘propagate’ away from C, andso the solution remains unspecified everywhere except on C.The second possibility is that C (although not a characteristic itself) crossessome characteristics more than once, as in figure 20.3. In this case specifying thevalue of u(x, y) along the curve PQdetermines the solution along all the characteristicsthat intersect it. Therefore, also specifying u(x, y) along QR can overdeterminethe problem solution and generally results in there being no solution.700