Vamos a compactificar el operador Laplaciano en coordenadas esféricas y en coordenadas cartesianas:

(\bar{r}, \theta, \varphi) \longrightarrow (r,\theta,\varphi) \longrightarrow (x,y,z)

\phi(\bar{r},\theta,\varphi) = (\frac{\bar{r}}{\bar{r}+a} \sin \theta \cos \varphi, \frac{\bar{r}}{\bar{r}+a} \sin \theta \sin \varphi, \frac{\bar{r}}{\bar{r}+a} cos \theta)

de manera que, con la formula ya vista en este post, nos queda:

(\bar{r}+a)^2 \bigg [ \frac{(\bar{r}+a)^2}{a^2} \frac{\partial^2}{\partial \bar{r}^2} + \frac{2}{\bar{r}} \frac{(\bar{r}+a)^2}{a^2} \frac{\partial}{\partial \bar{r}} + \frac{1}{\bar{r}^2} \frac{\partial^2}{\partial \theta^2} + \frac{\cot \theta}{\bar{r}^2} \frac{\partial}{\partial \theta} + \frac{\csc^2 \theta}{\bar{r}^2} \frac{\partial^2}{\partial \varphi^2} \bigg ] u =

= f(x(\bar{r},\theta,\varphi),y(\bar{r},\theta,\varphi),z(\bar{r},\theta,\varphi)).

De la misma manera:

(\bar{x}, \bar{y}, \bar{z}) \longrightarrow (x,y,z)

\phi(\bar{x},\bar{y},\bar{z}) = (\tanh \frac{\bar{x}}{a}, \tanh \frac{\bar{y}}{b}, \tanh \frac{\bar{z}}{c})

de manera que:

\bigg [ a^2 \cosh^4 \frac{\bar{x}}{a} \frac{\partial^2}{\partial \bar{x}^2} + 2 a \cosh^3 \frac{\bar{x}}{a} \sinh \frac{\bar{x}}{a} \frac{\partial}{\partial \bar{x}} +

+ b^2 \cosh^4 \frac{\bar{y}}{a} \frac{\partial^2}{\partial \bar{y}^2} + 2 b \cosh^3 \frac{\bar{y}}{b} \sinh \frac{\bar{y}}{b} \frac{\partial}{\partial \bar{y}} +

+ c^2 \cosh^4 \frac{\bar{z}}{c} \frac{\partial^2}{\partial \bar{z}^2} + 2 c \cosh^3 \frac{\bar{z}}{c} \sinh \frac{\bar{z}}{c} \frac{\partial}{\partial \bar{z}} \bigg ] u(\bar{x},\bar{y},\bar{z}) = f(x(\bar{x}),y(\bar{y}),z(\bar{z}).

O, con el otro cambio:

(\bar{x}, \bar{y}, \bar{z}) \longrightarrow (x,y,z)

\phi(\bar{x},\bar{y},\bar{z}) = \frac{2}{\pi}(\arctan \frac{\bar{x}}{a}, \arctan \frac{\bar{y}}{b}, \arctan \frac{\bar{z}}{c})

de manera que:

\frac{\pi^2}{2} \bigg [\frac{(\bar{x}^2+a^2)^2}{2a^2} \frac{\partial^2}{\partial \bar{x}^2} + \frac{\bar{x}(\bar{x}^2+a^2)}{a^2} \frac{\partial}{\partial \bar{x}} +

+ \frac{(\bar{y}^2+b^2)^2}{2b^2} \frac{\partial^2}{\partial \bar{y}^2} + \frac{\bar{y}(\bar{y}^2+b^2)}{b^2} \frac{\partial}{\partial \bar{y}} +

+ \frac{(\bar{z}^2+c^2)^2}{2c^2} \frac{\partial^2}{\partial \bar{z}} + \frac{\bar{z}(\bar{z}^2+c^2)}{c^2} \frac{\partial}{\partial \bar{z}} \bigg ] u(\bar{x},\bar{y},\bar{z}) = f(x(\bar{x}),y(\bar{y}),z(\bar{z})

Anuncios