User:Amsanville/sandbox

Article Evaluations: Method of Images
Initially only about physics and charged particles, this has since been updated. It also lacks the entire mathematical basis for why this is a valid method for solving the boundary conditions of certain ODEs. Note, this method can be applied generally to equations that are:

1.) Linear

2.) Have some geometric center to the solution

3.) Have an evenness or oddness with respect to that geometric center

For example, consider two solutions: $$y_{1}$$ and $$y_{2}$$ that satisfy some linear PDE:

$$ L[y] = 0 $$

Then, because the PDE is a linear differential equation, linear combinations of these solutions also satisfy the PDE:

$$ L[ay_{1} + by_{2}] = aL[y_{1}] + bL[y_{2}] = a(0) + b(0) = 0 $$

where $$a$$ and $$b$$ are, in general, complex numbers. We then see that if at some boundary in the problem that either $$y_{1} = y_{2}$$, but some derivative $$y'_{1} = -y'_{2}$$ then it is easy to satisfy either conditions so that $$y = 0$$ or $$y' = 0$$. Typically, the derivative conditions are no flux or no stress conditions and the derivatives of interest are the derivatives normal to the boundary. If we have, solutions to our PDE of the form:

$$ y(\mathbf{x}, \mathbf{x}_{0}) $$ where $$\mathbf{x}_{0}$$ is an initial point for our function and $$\mathbf{x}$$ is a general point in the domain it might be possible to construct $$y_{1}$$ and $$y_{2}$$ above so that we have reflections across the boundary. This works particularly well with Gaussian-like distributions which typically governor things like inverse $$r^{2}$$ laws like charge distribution and diffusion. It depends a lot on your PDE and your solution. I don't know when in general the method of images can be applied. It seems to require at a minimum though:

1.) A finite domain of interest, so that we can place artificial images outside the domain as to not interfere with initial conditions.

2.) A linear differential equation, ordinary or partial, so that solutions can be added.

3.) A geometric center to the solution that has a bell curve-like shape so that the mirror image will have the opposite slope. This allows you to put one image on one side of the boundary and the other image on the other and have a positive image cancel out the slopes and a negative image cancel out the primary function. This allows for you to satisfy reflective and absorbing boundary conditions.

I'm not an expert in the exact mathematics of this by any means, but it certainly is a cute and useful trick to building these idealized models in both mass transport and E&M (I always think those point charge in space problems are kind of funny, you will either never have a point charge that's standing still or you will never have a standing still charge that is well modeled as a point charge). Anyway, I will likely add some of this to the method of images page.

Final evaluation:

-Was only about physics, has since been updated

-Lacks mathematical basis for the method, which will require good sourcing on how to generalize it

-Also lacks more general boundary conditions, currently the mass transport section only mentions the reflective boundaries, but we also mentioned absorbing boundaries. I'll write out some details on that.

Mathematics of the Method of Images
This method is a specific application of Green's functions. The method of images works well when the boundary is a flat surface and the distribution has a geometric center. This allows for simple mirror-like reflection of the distribution to satisfy a variety of boundary conditions. Consider the simple 1D case illustrated in the graphic where there is a distribution of $$\langle c \rangle$$ as a function of $$x$$ and a single boundary located at $$x_{b}$$ with the real domain such that $$x \ge x_{b}$$ and the image domain $$x < x_{b} $$. Consider the solution $$f(\pm x + x_{0}, t)$$ to satisfy the linear differential equation for any $$x_{0}$$, but not necessarily the boundary condition.

Note these distributions are typical in models that assume a normal distribution. This is particularly common in environmental engineering, especially in atmospheric flows that use Outline of air pollution dispersion.

Perfectly Reflecting Boundary Conditions
The mathematical statement of a perfectly reflecting boundary condition is as follows:

$$ \nabla y(\mathbf{x}) \cdot \mathbf{n} = 0 $$

This states that the derivative of our scalar function $$y$$ will have no derivative in the normal direction to a wall. In the 1D case, this implifies to:

$$ \frac{d\langle c \rangle}{dx} = 0 $$

This condition is enforced with positive images so that :

$$ \langle c \rangle = f(x - x_{0}, t) + f(-x + (x_{b} - (x_{0} - x_{b})), t) $$

where the $$-x + (x_{b} - (x_{0} - x_{b}))$$ translates and reflects the image into place. Taking the derivative with respect to $$x$$:

$$ \frac{d\langle c \rangle}{dx}\bigg|_{x_{b}} = \frac{df(x - x_{0}, t)}{dx}\bigg|_{x_{b}} + \frac{df(-x + (x_{b} - (x_{0} - x_{b})), t)}{dx}\bigg|_{x_{b}} = \frac{df(x,t)}{dx}\bigg|_{x_{b} - x_{0}} - \frac{df(x, t)}{dx}\bigg|_{x_{b} - x_{0}} = 0 $$

Thus, the perfectly reflecting boundary condition is satisfied.

Perfectly Absorbing Boundary Conditions
The statement of a perfectly absorbing boundary condition is as follows :

$$ y(x_{b}) = 0 $$

This condition is enforced using a negative mirror image:

$$ \langle c \rangle = f(x - x_{0}, t) - f(-x + (x_{b} - (x_{0} - x_{b})), t) $$

And:

$$ \langle c \rangle\bigg|_{x_{b}} = f(x_{b} - x_{0}, t) - f(-x_{b} + (x_{b} - (x_{0} - x_{b})), t) = f(x_{b} - x_{0}, t) - f(x_{b} - x_{0}, t) = 0 $$

Thus this boundary condition is also satisfied.

UPDATE: Turns out the method of images is based on Green's functions. I'll start there...