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

Mottin, Stéphane

doi:10.5281/zenodo.439037

January 11, 2016 Journal article Open Access

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

Mottin, Stéphane

Contact person(s)

Mottin, Stéphane

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.

\( \Delta\)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 \(\in \textbf{R} ^{*}_{+}\)

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

Moreover the analytical expression of the integral:

\(\int_0^r { \rho^{h-1}\over \sqrt{1-2\zeta \rho+\rho^2}}d \rho\)

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.

License CC-BY-NC-ND. --------- French law about open access and open science: https://www.legifrance.gouv.fr/affichTexte.do?cidTexte=JORFTEXT000033202746&categorieLien=id ----------------------- LOI n° 2016-1321 du 7 octobre 2016 pour une République numérique - Article 30.

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References

Butkovskiĭ AG. Green s functions and transfer functions handbook. Chichester: Ellis Horwood; 1982
Li YL, Liu CH, Franke SJ. Three-dimensional Green s function for wave propagation in a linearly inhomogeneous medium—the exact analytic solution. J Acoust Soc Am. 1990;87:2285–2291 ; DOI 10.1121/1.399072
Churchill RV. The operational calculus of Legendre transforms. J Math Phys Camb. 1954;33:165–178 ; DOI: 10.1002/sapm1954331165
Kellogg OD. Foundations of potential theory. New York: Dover Publications; 1953.
Lanzani L, Shen Z. On the Robin boundary condition for Laplace s equation in Lipschitz domains. Commun Part Diff Eq. 2005;29:91–109. Doi: 10.1081/PDE-120028845
Dassios G, Fokas A. Methods for solving elliptic PDEs in spherical coordinates. SIAM J Appl Math. 2008;68:1080–1096
Mottin S, Panasenko G, Ganesh SS. Multiscale modeling of light absorption in tissues: limitations of classical homogenization approach. Plos One. 2010;5(12):e14350. doi:10.1371/journal.pone.0014350
Arridge SR, Cope M, Delpy DT. The theoretical basis for the determination of optical pathlengths in tissue: temporal and frequency analysis. Phys Med Biol. 1992;37:1531–1560
Carslaw HS, Jaeger JC. Conduction of heat in solids. Oxford: Clarendon Press; 1959.
Özisik MN. Heat transfer: a basic approach. New York: McGraw-Hill; 1985.
Zauderer E. Partial differential equations of applied mathematics. New York: Wiley; 1983.
Carslaw HS. Introduction to the theory of Fourier s series and integrals. New York: Dover Publications; 1950.
Suda R, Takami M. A fast spherical harmonics transform algorithm. Math Comput. 2002;71:703–716
Arfken GB. Mathematical methods for physicists. New York: Academic Press; 1970.
Bailey WN. On the reducibility of Appell s function F4. Q J Math. 1934; 5:291–292
Gradshtein IS, Ryzhik IM. Table of integrals, series, and products. New York: Academic Press; 1965.
Cuesta FJ, Lamúa M, Alique R. A new exact numerical series for the determination of the Biot number: application for the inverse estimation of the surface heat transfer coefficient in food processing. Int J Heat Mass Transfer. 2012;55:4053–4062
Kondjoyan A. A review on surface heat and mass transfer coefficients during air chilling and storage of food products. Int J Refrig. 2006;29:863–875
Hào DN, Thanh PX, Lesnic D. Determination of the ambient temperature in transient heat conduction. IMA J Appl Math. 2015;80:24–46
Schweiger M, Arridge SR, Hiraoka M, Delpy DT. The finite element method for the propagation of light in scattering media: boundary and source conditions. Med Phys. 1995;22:1779–1792
Fasino D, Inglese G. An inverse Robin problem for Laplace s equation: theoretical results and numerical methods. Inverse Probl. 1999;15:41–48
Grebenitcharsky RS, Sideris MG. The compatibility conditions in altimetry–gravimetry boundary value problems. J Geodesy. 2005;78:626–636 ; DOI: 10.1007/s00190-004-0429-7
Martinec Z, Grafarend EW. Construction of Green s function to the external Dirichlet boundary-value problem for the Laplace equation on an ellipsoid of revolution. J Geodesy. 1997;71:562–570 ; DOI: 10.1007/s001900050124