Green's Functions

Green's Functions :-

     The objective of this section to construct Green's (1793 - 1841) function and use it to solve the boundary value problem 

    L(y) + f(x) = 0, a≤x≤b .........(1)

a₁ y(a) + a₂ y'(a) = 0 , ................(2a)

b₁ y(b) + b₂ y'(b) =0, ..................(2b)

  Where L is the differential operator defined by L(y) = (py')' + qy.p , p' and q are given functions continuous on [a,b] with p(x) ≠0 on [a,b] and at least one of a₁ and a₂ and one of b₁ and b₂ are non zero . 

Definition :

       The Green's function for L(y) =0 under given homogeneous boundary conditions(2) is the function G(x,ξ) satisfying the following conditions . 

 (i) G(x,ξ) is continuous for all values of ox            but its first and second derivatives are          continuous for all 

        x≠ ξ ; a≤x≤b , a<ξ<b , 

(ii) At x= ξ the first derivative of G(x,ξ) has a jump discontinuity given by 

  d G(x,ξ)/dx | x= ξ⁺  = G'(ξ⁺,ξ)-G'(ξ⁻,ξ) = -1/p(ξ)
                          x=ξ⁻                                    ......(3)

(iii) For fixed ξ , G(x,ξ) satisfies the given boundary conditions and is the solution of the associated homogeneous equation    L(y) = 0 except at x= ξ .

  i.e a₁ G(a,ξ) + a₂ G'(a,ξ) = 0 .............(4a)

       b₁ G(b,ξ) + b₂ G'(b,ξ) = 0 ............(4b)

and L(G(x,ξ)) = 0 for all x except x= ξ ......(5)

Note :

      In equation(3) G'(ξ⁺,ξ) =

                                 lim G'(ξ+ε,ξ) and G'(ξ⁻,ξ)

                                ε-->0

                           =   lim G'(ξ-ε, ξ)

                               ε-->0 

Theorem _1 :-

          If the equation L(y) = 0 satisfying the boundary condition (2a) and (2b) has only the trivial solution , then the                 Green's function     exists and is unique .

Proof :

    Let φ₁(x) be a non- trivial solution of     L(y)=0 satisfying the condition (2a).

   Also let φ₂(x) be a nontrivial solution of L(y)=0 satisfying the condition (2b). Then φ₁ and φ₂ are linearly independent , for if they are not linearly independent , then φ₁ = C φ₂ and φ₁ would satisfy both of the conditions (2a) and (2b) implying that φ₁ is non trivial solution of L(y) = 0 with (2a) and (2b) which contradicts the hypothesis .

      Since G(x,ξ) satisfies (condition(iii) of the definition) the same conditions as φ₁(x) and φ₂(x) in the intervals [a,ξ) and (ξ,b] respectively , it follows that

                     {
                          C₁ φ₁(x) for x<ξ
      G(x,ξ) = {
                           C₂ φ₂(x) for x>ξ        ...........(5)
                     {

for some choice of C₁ and C₂ .

      Since G(x,ξ) is continuous at x= ξ (condition (i) of the definition), we have from equation(5) ,

      C₂ φ₂(ξ) - C₁ φ₁(ξ) = 0                  ...............(6)

  The discontinuity in the derivative of G at x= ξ , (condition(ii) of the definition) yields

       C₂ φ₂'(ξ) - C₁ φ₁'(ξ) = -1/p(ξ) ..............(7)

Solving equations (6) and (7) for C₁ and C₂ we obtain ,

   C₁ = -φ₂(ξ)/[p(ξ) W(φ₁,φ₂;ξ)]

   C₂ = -φ₁(ξ)/[p(ξ) W(φ₁,φ₂;ξ)] ..........(8)

 where W(φ₁,φ₂;ξ) is the Wronskian of        φ₁ and φ₂  at ξ

i.e. W(φ₁,φ₂;ξ) = φ₁(ξ) φ₂'(ξ) - φ₂(ξ)φ₁'(ξ) .

   Since φ₁ and φ₂ are linearly independent, the Wronskian W (φ₁,φ₂;ξ) ≠0 which implies in turn that p(ξ) W(φ₁,φ₂;ξ)≠0 so that C₁ and C₂ are well defined in equation(8) .

    Since φ₁ and φ₂ are solutions of L(y)=0 we have

   d/dx (p φ₁) + q φ₁ = 0 ,

and d/dx (p φ₂') + q φ₂= 0 ,

Multiplying the first equation by φ₂ and the second by φ₁ and subtracting , we obtain

      φ₁ d/dx (p φ₂') - φ₂ d/dx (p φ₁') = 0,

which can be written in the form

   d/dx [p(φ₁ φ₂' - φ₂ φ₁')] = 0

 Integration of this equation yields

     p(φ₁ φ₂' - φ₂ φ₁') = constant = A  .......(9)

 Hence from(5) , (8) and (9) the Green's function is uniquely determined as

                 
                     { - φ₁(x) φ₂(ξ)/A for x≤ ξ
      G(x,ξ) = {
                     {-φ₂(x) φ₁(ξ)/A for x≥ξ    ......(10)


Using the definition of the Green's function we can obtain a solution of the boundary value problem (1) and (2) .