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
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
Green's Functions for the steady problems a Basic procedure △a=f(r) problem ul==0 Green's fn △G= IGI2=0 Relation f()=f()o(")dr )=fG7)G(77)d
Green’s Functions for the steady problems problem = = | 0 ( ) u u f r Green’s fn. = = − | 0 ( ') G G r r Relation = = − ( ) ( ') ( , ') ' ( ) ( ') ( ') ' u r f r G r r d f r f r r r d ◼ Basic procedure
Green's Functions for the steady problems ■求解方法 稳定问题的格林函数也可以利用静电场类比法得到 点源问题可以看成接地的导体边界内在r处有 个电量为-:0的点电荷 边界内部的电场由点电荷与导体中的感应电荷共同 生 在一些情况下,导体中所有感应电荷的作用可以用 个设想的等效电荷来代替,该等效电荷称为点电 荷的电像 这种方法称为电像法
Green’s Functions for the steady problems ◼ 求解方法 • 稳定问题的格林函数也可以利用静电场类比法得到。 • 点源问题可以看成接地的导体边界内在 r’ 处有一 个电量为 - 0 的点电荷。 • 边界内部的电场由点电荷与导体中的感应电荷共同 产生。 • 在一些情况下，导体中所有感应电荷的作用可以用 一个设想的等效电荷来代替，该等效电荷称为点电 荷的电像。 • 这种方法称为电像法
Green's Functions for the steady problems ■ Example 在半空间内求解稳定问题的格林函数 解:根据题目,定解问题为 △G=d(x-x)(y-y3)6(二-21)2z>0 0 这相当于在接地导体平面上方点M(x2,y2z)处放 置一个电量为-so的点电荷,求电势。 设想在M的对称点N(xy2,-z)处放置一个电量为 +Eo的点电荷,容易看出在平面z=0上电势为零, 这表明在N点的点电荷就是电像
Green’s Functions for the steady problems ◼ Example 在半空间内求解稳定问题的格林函数 = = − − − | = 0 ( ') ( ') ( '), 0 G z 0 G x x y y z z z 解：根据题目，定解问题为 这相当于在接地导体平面上方点 M(x’,y’,z’) 处放 置一个电量为 - 0 的点电荷，求电势。 设想在M的对称点 N (x’,y’ ,-z’)处放置一个电量为 + 0 的点电荷，容易看出在平面 z=0上电势为零， 这表明在N点的点电荷就是电像