Mathquery

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 |