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I would like to numerically solve an equation of a type as shown below. Does anyone of you have an idea how to approach such a problem? Any links to literature or for further reading would be greatly appreciated.

Thank you!


I search a function $f(\vec{x})$ with $f \in \mathbb{R}$ and $\vec{x} \in \mathbb{R}^n$, where $n \in\{2,3\}$.

$f$ is positive:

$f > 0~ \forall ~\vec{x}$

and radial-symetric with $r=\left|\vec{x}\right|$:

$f(\vec{x}) = f(\left| \vec{x}\right|) = f(r)$

It is bound in infinity:

$\lim_{r\rightarrow \infty} f(r) = 1$

The differential equation looks like:

$\Delta f(\vec{x}) = \nabla \cdot \left[ a \vec{G}(\vec{x})f(\vec{x}) + b f(\vec{x}) \left( \left( \vec{G}(\vec{x})f(\vec{x}) \right) * f(\vec{x}) \right) \right]$,

where $\Delta$ is the Laplace Operator, $\nabla$ the Nabla operator, both acting on $\vec{x}$. $\cdot$ denotes the dot-product, and $*$ a convolution, with $a,b \in \mathbb{R}$, and $\vec{G}(\vec{x}) \in \mathbb{R}^n$

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  • $\begingroup$ If the function is radially symmetric then you should be able to write everything as just a function of $r$. Can you state how the equation would look like for $f(r)$? $\endgroup$ – Wolfgang Bangerth Mar 29 '14 at 1:13

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