Recent content by OliviaB

Apply the method of images to derive the solution \displaystyle \phi(x,y,z) = \frac{z}{2 \pi} \int_{- \infty}^{\infty} \int_{- \infty}^{\infty} \frac{f(x_0, y_0)}{((x - x_0)^2 + (y - y_0)^2 + z^2)^{\frac{3}{2}}} dx_0 dy_0 from \displaystyle \bigtriangledown^2 \phi (x,y,z) = 0 \phi(x,y,0)...

It's taken me ages, and I really am struggling to understand this, so I am not sure if this is correct but here goes... \bigtriangledown^2 G = \delta(\underline{x} - \underline{x}_0) \frac{\partial G(x,0)}{\partial y} = 0 for y \geq 0 \ - \infty < x < \infty Let r = |\underline{x} -...

Use the method of images to find a Green's function for the problem in the attached image. Demonstrate the functions satisfies the homogenous boundary condition.

I think this is Poisson's Equation (and inhomogenous). I think I need to use Green's Identity. Let \mathcal{R} be a bounded region in \mathbb{R}^3, and suppose p(x) > 0 on \mathcal{R}. (i) If u is a solution of \bigtriangledown^2 u = p(x) u \ \ x \in \mathcal{R} \ \...