Transcription of GREEN’S FUNCTION FOR LAPLACIAN - University of Michigan
{{id}} {{{paragraph}}}
Example: quiz answers
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.
and for x on the boundary of D, we have u(x) = 0 because G(x,x0) = 0 by the definition of G in (0.5). Verification of (0.3) for u and G satisfying Neumann or Robin conditions can be done similarly. Now let’s see how to find the Green’s function for some particular domains. 1
Tags:
Information
Domain:
Source:
Link to this page:
Please notify us if you found a problem with this document: