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Green FunctionalSolving of Inhomogeneous Differential EquationsLet L be a differential operator. Say we want to solve the equation Then what we'd like to say is However, if So if L is surjective, then at least there is a way to do: So we make the problem worse, so to speak, in order to solve for u. One of the possible ways to construct G (which you need in order to make u worse) and (G u) is the Convolution. (G is called the Green Function) ConvolutionProperties of Convolutionf*g = g*f δ*f = f*δ = f Let's return to the Green Function. Let the solution to the following be known: Then this G(x) is a possible Green Function. ExampleLet's say we want to solve a linear differential equation with constant coefficients (i.e. a translation invariant equation) Let's say we have a inhomogenity f(x) acting on it: If we know a functional G so that the following holds, we can go much further: So maybe Author: Danny (remove the ".nospam" to send) Last modification on: Thu, 09 May 2013 . |