General Concepts of Green's function ■ Properties: a Let the equation is L u(x=f(x) a The eq. of Green's fn. is L G(x)=8(x-X) a Under the same conditions Since:f()=∫f(x)δ(x×)dx Therefore: u(x= f(X,)G(x-X)dx Applications range: nonhomogeneous universal eqs Homogeneous condition procedure find the correspondding green's fn then integra
General Concepts of Green’s Function ◼ Properties： ◼ Let the equation is L u(x) = f (x) ◼ The eq. of Green’s fn. is L G(x) = (x-x’) ◼ Under the same conditions ◼ Since： f (x)=∫ f (x’) (x-x’) dx’ ◼ Therefore： u (x) =∫ f (x’) G(x-x’) dx’ ◼ Applications ◼ range： • nonhomogeneous universal eqs. • Homogeneous condition ◼ procedure： • find the correspondding Green’s fn, then integral
Fundamental solutions for the steady problems Fundamental Solutions for the steady problems can be obtained from the electric field Problem Green's field △V Eq Au=f() AG=S(-F') q6(7-P)/E0 f(r)dr Sol G 4x|r- 4 r-r 40 r-r
Fundamental Solutions for the steady problems Problem Green’s field Eq. Sol. u f (r) = G (r r') = − 0 q (r r')/ V − − = 4 | '| 0 r r q V − = 4 | '| 1 r r G − − = − − = 4 | '| ( ') ' r r f r d u Fundamental Solutions for the steady problems can be obtained from the electric field