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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.
\n\n\\( \\Delta\\)u=0 in a unit sphere
\n\n\u2202u(r, \\(\\zeta\\))/\u2202r|r=1 + h u(1, \\(\\zeta\\))= f(\\(\\zeta\\)) on a unit sphere,
\n\n\\(\\zeta\\)\u00a0= cos (\\(\\theta\\)), \\(\\theta\\)\u00a0is the azimuthal angle, \u00a0and h\u00a0\\(\\in \\textbf{R} ^{*}_{+}\\)\u00a0
\n\nThe function f(\\(\\theta\\))\u00a0is a prescribed function and is assumed to be a square-integrable function.
\n\nMoreover the analytical expression of the integral:
\n\n\\(\\int_0^r { \\rho^{h-1}\\over \\sqrt{1-2\\zeta \\rho+\\rho^2}}d \\rho\\)
\n\nis given in terms of the Appell function F1.
\n\nIn 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.
", "publication_date": "2016-01-11", "publisher": "Zenodo", "references": [ { "reference": "Butkovski\u012d AG. Green s functions and transfer functions handbook. Chichester: Ellis Horwood; 1982" }, { "reference": "Li YL, Liu CH, Franke SJ. Three-dimensional Green s function for wave propagation in a linearly inhomogeneous medium\u2014the exact analytic solution. J Acoust Soc Am. 1990;87:2285\u20132291 ; DOI 10.1121/1.399072" }, { "reference": "Churchill RV. The operational calculus of Legendre transforms. J Math Phys Camb. 1954;33:165\u2013178 ; DOI: 10.1002/sapm1954331165" }, { "reference": "Kellogg OD. Foundations of potential theory. New York: Dover Publications; 1953." }, { "reference": "Lanzani L, Shen Z. On the Robin boundary condition for Laplace s equation in Lipschitz domains. Commun Part Diff Eq. 2005;29:91\u2013109. Doi: 10.1081/PDE-120028845" }, { "reference": "Dassios G, Fokas A. Methods for solving elliptic PDEs in spherical coordinates. SIAM J Appl Math. 2008;68:1080\u20131096" }, { "reference": "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" }, { "reference": "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\u20131560" }, { "reference": "Carslaw HS, Jaeger JC. Conduction of heat in solids. Oxford: Clarendon Press; 1959." }, { "reference": "\u00d6zisik MN. Heat transfer: a basic approach. New York: McGraw-Hill; 1985." }, { "reference": "Zauderer E. Partial differential equations of applied mathematics. New York: Wiley; 1983." }, { "reference": "Carslaw HS. Introduction to the theory of Fourier s series and integrals. New York: Dover Publications; 1950." }, { "reference": "Suda R, Takami M. A fast spherical harmonics transform algorithm. Math Comput. 2002;71:703\u2013716" }, { "reference": "Arfken GB. Mathematical methods for physicists. New York: Academic Press; 1970." }, { "reference": "Bailey WN. On the reducibility of Appell s function F4. Q J Math. 1934; 5:291\u2013292" }, { "reference": "Gradshtein IS, Ryzhik IM. Table of integrals, series, and products. New York: Academic Press; 1965." }, { "reference": "Cuesta FJ, Lam\u00faa 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\u20134062" }, { "reference": "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\u2013875" }, { "reference": "H\u00e0o DN, Thanh PX, Lesnic D. Determination of the ambient temperature in transient heat conduction. IMA J Appl Math. 2015;80:24\u201346" }, { "reference": "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\u20131792" }, { "reference": "Fasino D, Inglese G. An inverse Robin problem for Laplace s equation: theoretical results and numerical methods. Inverse Probl. 1999;15:41\u201348" }, { "reference": "Grebenitcharsky RS, Sideris MG. The compatibility conditions in altimetry\u2013gravimetry boundary value problems. J Geodesy. 2005;78:626\u2013636 ; DOI: 10.1007/s00190-004-0429-7" }, { "reference": "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\u2013570 ; DOI: 10.1007/s001900050124" } ], "related_identifiers": [ { "identifier": "https://cv.archives-ouvertes.fr/stephane-mottin", "relation_type": { "id": "iscitedby", "title": { "de": "Wird zitiert von", "en": "Is cited by" } }, "scheme": "url" }, { "identifier": "10.1080/10652469.2015.1121255", "relation_type": { "id": "iscitedby", "title": { "de": "Wird zitiert von", "en": "Is cited by" } }, "scheme": "doi" } ], "resource_type": { "id": "publication-article", "title": { "de": "Zeitschriftenartikel", "en": "Journal article" } }, "rights": [ { "description": { "en": "" }, "icon": "cc-by-nc-nd-icon", "id": "cc-by-nc-nd-4.0", "props": { "scheme": "spdx", "url": "https://creativecommons.org/licenses/by-nc-nd/4.0/legalcode" }, "title": { "en": "Creative Commons Attribution Non Commercial No Derivatives 4.0 International" } } ], "subjects": [ { "subject": "Robin boundary conditions" }, { "subject": "integral transform" }, { "subject": "Legendre transform" }, { "subject": "inverse problems" }, { "subject": "Appell function" }, { "subject": "evaluation of definite integrals" } ], "title": "An analytical solution of the Laplace equation with Robin conditions by applying Legendre transform" }, "parent": { "access": { "owned_by": { "user": 27171 } }, "communities": { "default": "68b0db73-9e9c-4d7b-9109-f67d93dbd5e1", "entries": [ { "access": { "member_policy": "open", "members_visibility": "public", "record_policy": "open", "review_policy": "open", "visibility": "public" }, "children": { "allow": false }, "created": "2017-03-16T19:10:59.832633+00:00", "custom_fields": {}, "deletion_status": { "is_deleted": false, "status": "P" }, "id": "68b0db73-9e9c-4d7b-9109-f67d93dbd5e1", "links": {}, "metadata": { "curation_policy": "Only my articles/bookSections uploaded to Zenodo.
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