- #1
fog37
- 1,568
- 108
- TL;DR Summary
- Poisson's equation solution and Green's function method
Hello,
Poisson equation and Laplace equation (which is the homogeneous version of Poisson PDE) are important equations in electrostatics where both the electric field ##E## and scalar potential ##\phi## don't depend on time. Poisson's equation is $$\nabla^2 \phi(x,y,z) = - \frac{\rho(x,y,z)} {\epsilon_0}$$ The inhomogeneous term, also called the source, is ##- \frac{\rho(x,y,z)} {\epsilon_0}##. One of the possible methods (not the only one, correct?) to find a solution to Poisson's equation is the Green function method which involves finding the solution when the source is just a Diract delta function ##\delta(x,y,z)##. By linearity, the solution when the source is ##- \frac{\rho(x,y,z)} {\epsilon_0}## is just a superposition integral adding all the responses from many delta sources.
Poisson equation and Laplace equation (which is the homogeneous version of Poisson PDE) are important equations in electrostatics where both the electric field ##E## and scalar potential ##\phi## don't depend on time. Poisson's equation is $$\nabla^2 \phi(x,y,z) = - \frac{\rho(x,y,z)} {\epsilon_0}$$ The inhomogeneous term, also called the source, is ##- \frac{\rho(x,y,z)} {\epsilon_0}##. One of the possible methods (not the only one, correct?) to find a solution to Poisson's equation is the Green function method which involves finding the solution when the source is just a Diract delta function ##\delta(x,y,z)##. By linearity, the solution when the source is ##- \frac{\rho(x,y,z)} {\epsilon_0}## is just a superposition integral adding all the responses from many delta sources.
- My dilemma: since Poisson's equation is a PDE and like every PDE it is a pointwise relation between partial derivatives at specific spatial points, i.e. the equation is valid all the spatial points where the source term ##- \frac{\rho(x,y,z)} {\epsilon_0} \neq 0## , correct? However, the Green's function method provides a solution ##\phi(x,y,z)=\int \frac{\rho(x,y,z)} {\epsilon_0 r}dxdydx## that is valid at spatial points where ##- \frac{\rho(x,y,z)} {\epsilon_0} =0##, i.e. the solution ##\phi(x,y,z)## exists and has values at spatial points where the source is zero. How can the PDE solution ##\phi(x,y,z)## exist at spatial points where the PDE is not satisfied?
- Does the Green function method require the initial conditions (ICs) to be zero while he boundary conditions (BCs) can be arbitrary? If the PDE has nonzero ICs, then the Green function method provides only portion of the solution, correct?