An analytical solution of the Laplace equation with Robin conditions by applying Legendre transform

We derived the analytical solution of the Laplace equation with Robin conditions on a sphere with azimuthal symmetry by applying Legendre transform, which was expressed in terms of the Appell hypergeometric function.

$\mathrm{\Delta }$u=0 in a unit sphere

∂u(r, $\zeta$)/∂r|r=1 + h u(1, $\zeta$)= f($\zeta$) on a unit sphere,

$\zeta$ = cos ($\theta$), $\theta$ is the azimuthal angle,  and h $\in {\mathbf{\text{R}}}_{+}^{\ast }$ 

The function f($\theta$) is a prescribed function and is assumed to be a square-integrable function.

Moreover the analytical expression of the integral:

${\int }_0^r\frac{{\rho }^{h-1}}{\sqrt{1-2\zeta \rho +{\rho }^2}}d\rho$

is given in terms of the Appell function F1.

In many experimental approaches, the Robin coefficient h is the main unknown parameter for example in transport phenomena where the Robin coefficient is the dimensionless Biot number. The usefulness of this formula is illustrated by some examples of inverse problems.