Methods of Mathematical Physics Green functions
Methods of Mathematical Physics Green Functions
Method of green's functions a General concepts of Green's function ■ Fundamental solution Green's function of the evolution problems a Fundamental solutions of the evolution problems a Green's function of the evolution problems Conclusion of the charpter
Method of Green’s functions ◼ General concepts of Green’s function ◼ Fundamental solution ◼ Green’s function of the evolution problems ◼ Fundamental solutions of the evolution problems ◼ Green’s function of the evolution problems ◼ Conclusion of the charpter
General Concepts of Green's function ■ Concept Definition: o The field comes from a point resource a Example: △G=8(r-r),G|r=0 (ot-a2A)G=8(r-r)8(t-t), Glr=Glt=0=0 General Form LG(x)=δ(x-×1) ● gounda= Initial=0
General Concepts of Green’s Function ◼ Concept ◼ Definition： • The field comes from a point resource ◼ Example ： • △ G = (r-r’)，G|=0 • (t – a 2△) G = (r-r’)(t-t’)， G|= G|t=0=0 ◼ General Form • L G(xi) = (xi-xi ’) • G|boundary= G|initial=0
General Concepts of Green's function Classification a According to the universal equation Green' s function of the steady problem L=A Green's function of the heat problem L =(0 -a2A Green's function of the wave problem L =(Ot-a2A) a According to the boundary condition: The green's function for a unboundary space, namely, the fundamental solution The Green's function for a homogeneous bounding condition
General Concepts of Green’s Function ◼ Classification： ◼ According to the universal equation： • Green’s function of the steady problem L = △ • Green’s function of the heat problem L = (t – a2△) • Green’s function of the wave problem L = (tt – a2△) ◼ According to the boundary condition: • The Green’s function for a unboundary space, namely, the fundamental solution. • The Green’s function for a homogeneous bounding condition
General Concepts of Green's function Steady Heat problems Wave problems Green's problems(a-a2△)G(at-a2△)G function△G =8(r-r)6(tt)|=6(r-r)6(tt 6(r-r) G|t=0=0 G|t=0=0 Gt|t=0=0 Unboundin g space Boundary condition G|r=0
General Concepts of Green’s Function Green’s function Steady problems △ G = (r-r’) Heat problems (t – a2△) G = (r-r’)(t-t’) G|t=0=0 Wave problems (tt – a2△) G = (r-r’)(t-t’) G|t=0=0 Gt|t=0=0 Unboundin g space Boundary condition G|= 0