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## Green Functional## Solving of Inhomogeneous Differential EquationsLet L be a differential operator. Say we want to solve the equation for u Then what we'd like to say is , that is . However, if has nontrivial solutions (i.e. more than one solution: the trivial solution and another one), then L apparently cannot be injective. Hence this doesn't work. So (called right inverse)
(the right-inverse, note that 1 is an operator)
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## Properties 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: Sat, 04 May 2024 . |