Mathematical Methods for Physics and Engineering - Matematica.NET

20.6 CHARACTERISTICS AND THE EXISTENCE OF SOLUTIONSwe may differentiate the two first derivatives ∂u/∂x and ∂u/∂y along the boundaryto obtain the pair of equationsddsdds( ) ∂u∂x( ) ∂u∂y= dx ∂ 2 uds= dxds∂x 2 + dy ∂ 2 uds ∂x∂y ,∂ 2 u∂x∂y + dy ∂ 2 uds ∂y 2 .We may now solve these two equations, together with the original PDE (20.43),for the second partial derivatives of u, except where the determinant of theircoefficients equals zero,A B Cdx dy0∣ ds ds∣ =0.∣Expanding out the determinant,A( ) 2 dy− Bds0( dxdsdxdsMultiplying through by (ds/dx) 2 we obtainAdyds∣)( ) dy+ Cds( ) 2 dx=0.ds( ) dy 2− B dy + C =0, (20.44)dx dxwhich is the ODE for the curves in the xy-plane along which the second partialderivatives of u cannot be found.As for the first-order case, the curves satisfying (20.44) are called characteristicsof the original PDE. These characteristics have tangents at each point given by(when A ≠0)dydx = B ± √ B 2 − 4AC. (20.45)2AClearly, when the original PDE is hyperbolic (B 2 > 4AC), equation (20.45)defines two families of real curves in the xy-plane; when the equation is parabolic(B 2 =4AC) it defines one family of real curves; and when the equation is elliptic(B 2 < 4AC) it defines two families of complex curves. Furthermore, when A,B and C are constants, rather than functions of x and y, the equations of thecharacteristics will be of the form x + λy = constant, which is reminiscent of theform of solution discussed in subsection 20.3.3.703