GREEN’S FUNCTION FOR LAPLACIAN - University of Michigan

GREEN S FUNCTION FOR LAPLACIANThe Green s FUNCTION is a tool to solve non-homogeneous linear equations. We will illus-trate this idea for the LAPLACIAN .Suppose we want to find the solutionuof the Poisson equation in a domainD Rn: u(x) =f(x),x Dsubject to some homogeneous boundary condition. Imaginefis the heat source anduis thetemperature. The idea of Green s FUNCTION is that if we know the temperature respondingto an impulsive heat source at any pointx0 D, then we can just sum up the result withthe weight functionf(x0) (it specifies the strength of the heat at pointx0) to obtain thetemperature responding to the heat sourcef(x) inD. Mathematically, one may expressthis idea by defining the Green s FUNCTION as the following:Letu=u(x),x= (x1, x2.)

homogeneous boundary condition that nullifies the effect of Γ on the boundary of D. Sim-ilarly we can construct the Green’s function with Neumann BC by setting G(x,x0) = 0)+v(x,x0) where v is a solution of the Laplace equation with a Neumann bound-ary condition that nullifies the heat flow coming from Γ.

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  Bound, Functions, Boundary, Bound ary

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